Multiple Hilbert transform associated with polynomials
MMULTIPLE-HILBERT TRANSFORMS ASSOCIATEDWITH POLYNOMIALS
JOONIL KIM
Abstract.
Let Λ = (Λ , · · · , Λ d ) with Λ ν ⊂ Z n + , and set P Λ the family of all vectorpolynomials, P Λ = P Λ : P Λ ( t ) = (cid:88) m ∈ Λ c m t m , · · · , (cid:88) m ∈ Λ d c d m t m with t ∈ R n . Given P Λ ∈ P Λ , we consider a class of multi-parameter oscillatory singular integrals, I ( P Λ , ξ, r ) = p.v. (cid:90) (cid:81) [ − r j ,r j ] e i (cid:104) ξ,P Λ ( t ) (cid:105) dt t · · · dt n t n where ξ ∈ R d , r ∈ R n + . When n = 1, the integral I ( P Λ , ξ, r ) for any P Λ ∈ P Λ is bounded uniformly in ξ and r .However, when n ≥
2, the uniform boundedness depends on each indivisual polynomial P Λ . In this paper, we fix Λ and find a necessary and sufficient condition on Λ such thatfor all P Λ ∈ P Λ , sup ξ, r |I ( P Λ , ξ, r ) | ≤ C P Λ < ∞ . (0.1)The condition is described by faces and their cones of polyhedrons associated with Λ ν ’s. Contents
1. Introduction 22. Polyhedra, Their Faces and Cones 63. Main Theorem and Background 144. Representation of faces and their cones 225. Preliminaries Estimates 356. Cone Type Decompositions 407. Descending Faces v.s. Ascending Cones 49
Mathematics Subject Classification.
Primary 42B20, 42B25.
Key words and phrases.
Multiple Hilbert transform, Newton polyhedron, Face, Cone, Oscillatory Sin-gular Integral. a r X i v : . [ m a t h . C A ] F e b JOONIL KIM
8. Proof of Sufficiency 599. Necessity Theorem 6610. Proof of Necessity 7711. Proofs of Corollary 3.1 and Main Theorem 3 88References 931.
Introduction
Let Z + denote the set of all nonnegative integers and let Λ ν ⊂ Z n + be the finite set ofmulti-indices for each ν = 1 , · · · , d . Given Λ = (Λ , · · · , Λ d ), we set P Λ the family of allvector polynomials P Λ of the following form: P Λ = P Λ : P Λ ( t ) = (cid:88) m ∈ Λ c m t m , · · · , (cid:88) m ∈ Λ d c d m t m with t ∈ R n (1.1)where c ν m ’s are nonzero real numbers. Given P Λ ∈ P Λ , ξ = ( ξ , · · · , ξ d ) ∈ R d and r =( r , · · · , r n ) ∈ R n + , we define a multi-parameter oscillatory singular integral: I ( P Λ , ξ, r ) = p.v. (cid:90) (cid:81) [ − r j ,r j ] e i (cid:104) ξ,P Λ ( t ) (cid:105) dt t · · · dt n t n where the principal value integral is defined bylim (cid:15) → (cid:90) (cid:81) { (cid:15) j < | t j |
0. The existence of this limit follows by the Taylorexpansion of t → e i (cid:104) ξ,P Λ ( t ) (cid:105) and the cancelation property (cid:82) dt ν /t ν = 0 with ν = 1 , · · · , n .We see that whether sup ξ |I ( P Λ , ξ, r ) | is finite or not depends on(1) Sets Λ ν of exponents of monomials in P Λ ( t ).(2) Coefficients of polynomial P Λ ( t ).(3) Domain of integral (cid:81) [ − r j , r j ].(1) The dependence on set Λ ν of exponents is observed in the following simple cases:sup ξ ∈ R |I ( P Λ , ξ, (1 , | = sup ξ ∈ R (cid:12)(cid:12)(cid:12)(cid:82) − (cid:82) − sin( ξt t ) dt t dt t (cid:12)(cid:12)(cid:12) = ∞ if Λ = { (1 , } sup ξ ∈ R (cid:12)(cid:12)(cid:12)(cid:82) − (cid:82) − sin( ξt t ) dt t dt t (cid:12)(cid:12)(cid:12) = 0 if Λ = { (2 , } ULTIPLE HILBERT TRANSFORMS 3 (2) The dependence on coefficients of polynomials P Λ first appeared in [12], later in [1] and[13]. There exist two different polynomials P Λ and Q Λ in P Λ having the same exponentset Λ, with sup ξ I ( P Λ , ξ, r ) finite but sup ξ I ( Q Λ , ξ, r ) infinite. We can check this for P Λ ( t ) = t t − t t and Q Λ ( t ) = t t + t t . However, in this paper, we do not concernwith this coefficient dependence. We rather search for a condition of Λ valid for universal P Λ ∈ P Λ that for all P Λ ∈ P Λ , sup ξ ∈ R d |I ( P Λ , ξ, r ) | ≤ C P Λ < ∞ . (1.2)(3) The dependence on the domain (cid:81) [ − r j , r j ] is observed for the case Λ = { (2 , , (3 , } ,sup ξ ∈ R , 1) or an infinite interval (0 , ∞ ),we set up our problem by first fixing the range of r according to S ⊂ N n = { , · · · , n } : r ∈ I ( S ) = n (cid:89) j =1 I j where I j = (0 , 1) for j ∈ S and I j = (0 , ∞ ) for j ∈ N n \ S .(1.3)Instead of (1.2), we shall find the necessary and sufficient condition on Λ and S thatfor all P Λ ∈ P Λ , sup ξ ∈ R d , r ∈ I ( S ) |I ( P Λ , ξ, r ) | ≤ C P Λ < ∞ . (1.4)For each Schwartz function f on R d and a vector polynomial P Λ ∈ P Λ , the multipleHilbert transform of f associated to P Λ is defined to be (cid:0) H P Λ r f (cid:1) ( x ) = p.v. (cid:90) (cid:81) nj =1 [ − r j ,r j ] f (cid:0) x − P Λ ( t ) (cid:1) dt t · · · dt n t n . Here r j = 1 with j ∈ S corresponds to a local Hilbert transform, and r j = ∞ with j ∈ N n \ S corresponds to a global Hilbert transform. Since I ( P Λ , ξ, r ) is the Fourier JOONIL KIM multiplier of the Hilbert transform H P Λ r , the boundedness (1.4) is equivalent to thatfor all P Λ ∈ P Λ , sup r ∈ I ( S ) (cid:13)(cid:13) H P Λ r (cid:13)(cid:13) L p ( R d ) → L p ( R d ) ≤ C P Λ where p = 2 . (1.5)In this paper, we show (1.4) and (1.5) with 1 < p < ∞ for all n and d when S ⊂ N n . Toseek and manifest the condition to determine (1.4) and (1.5), we study the concept of facesand their cones of the Newton Polyhedron associated with Λ and S ⊂ N n . It is noteworthyin advance that the necessary and sufficient condition of (1.4) is not determined by onlyfaces but also by cones of the Newton polyhedron, which has not appeared explicitly inthe graph case Λ = ( e , · · · , e n , Λ n +1 ) or low dimensional case n ≤ Scheme and Organization . As a motive for this problem, we remark the result ofA. Carbery, S. Wainger and J. Wright in [3]: Given a polynomial P Λ ∈ P Λ with Λ =( { e } , { e } , Λ ) with n = 2 , d = 3 and S = { , } , a necessary and sufficient condition for (cid:13)(cid:13) H P Λ r (cid:13)(cid:13) L p ( R ) → L p ( R ) ≤ C where r = (1 , m in a Newton polyhedron N (Λ ) = Ch(Λ + R ) has at least oneeven component. The idea of the proof in [3] is to split the sum of dyadic pieces H P Λ r = (cid:80) J ∈ Z H P Λ J into finite sums of cones { J ∈ m ∗ } associated with vertices m of N (Λ ):(1.6) (cid:88) m is a vertex of N (Λ ) (cid:32) (cid:88) J ∈ m ∗ H P Λ J (cid:33) with m ∗ = { α q + α q : α , α ≥ } where q j is a normal vector of the supporting line π q j of an edge F j of N (Λ ) such that m = (cid:84) j =1 F j . They proved that for Λ (cid:48) = ( { e } , { e } , { m } ),(1.7) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ m ∗ (cid:16) H P Λ J − H P Λ (cid:48) J (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R ) → L p ( R ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ m ∗ H P Λ (cid:48) J (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R ) → L p ( R ) ≤ C by using the vertex dominating property (1) and the vanishing property (2):(1) vertex dominating property: J ∈ m ∗ ⇒ − J · m ≥ − J · n for n ∈ N (Λ ) \ { m } ,(2) vanishing property: at least one component of m is even, that implies H P ( ∅ , ∅ , m ) J ≡ n ≥ 3, we shall establish the corresponding cone type decomposition (1.6)and the reduction estimate (1.7) together with (1) and (2). As an analogue of (1.6), we ULTIPLE HILBERT TRANSFORMS 5 split H P Λ r = (cid:80) J ∈ Z n + H P Λ J with r = (1 , · · · , 1) into(1.8) (cid:88) ( F ν ); F ν is a face of N (Λ ν ) (cid:88) J ∈ (cid:84) dν =1 F ∗ ν H P Λ J with F ∗ ν = N ν (cid:88) j =1 α j q j : α j ≥ . Here q j is normal vector of the supporting plane π q j of a face F ν in the Newton polyhedron N (Λ ν ), where F ν = (cid:84) N ν j =1 π q j . For this purpose, we introduce in Section 2 the concept of aface F and its cone F ∗ in a Polyhedron. In Section 3, we state our main results and somebackground for this problem. In Sections 4, we provide properties of faces and their conesrelated with their representations. In Sections 5, we give a few basic L p estimation tools.In Section 6, we make (1.8). As an analogue of (1.7), we prove in Section 8 that(1.9) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ (cid:84) dν =1 F ∗ ν (cid:16) H P Λ J − H P Λ (cid:48) J (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) → L p ( R d ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ (cid:84) dν =1 F ∗ ν H P Λ (cid:48) J (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) → L p ( R d ) ≤ C where Λ (cid:48) = ( F ν ∩ Λ ν ) dν =1 . To show (1.9), we use the dominating and vanishing properties:(1) If J ∈ (cid:84) dν =1 F ∗ ν , then 2 − J · m ≥ − J · n where m ∈ F ν and n ∈ N (Λ ν ) \ F ν ,(2) If sum of elements in (cid:83) dν =1 F ν ∩ Λ ν has at least one even component, H P Λ (cid:48) J ≡ n ≥ d (cid:92) ν =1 ( F ∗ ν ) ◦ (cid:54) = ∅ and rank (cid:32) d (cid:91) ν =1 F ν (cid:33) ≤ n − . Note that the cones F ∗ ν as well as faces F ν of the Newton polyhedra associated with Λ ν areinvolved in determining (1.4). Thus, a difficulty in showing (1.9) is to keep the above coneoverlapping condition until the low ranked faces occurs. For this purpose, we construct inSection 7 a sequence of faces and cones such that N (Λ ν ) = F ν (0) ⊃ · · · ⊃ F ν ( s ) ⊃ · · · ⊃ F ν ( N ) = F ν , N ∗ (Λ ν ) = F ∗ ν (0) ⊂ · · · ⊂ F ∗ ν ( s ) ⊂ · · · ⊂ F ∗ ν ( N ) = F ∗ ν . (1.10)This sequence plays crucial roles to keep (cid:84) dν =1 ( F ∗ ν ( s )) ◦ (cid:54) = ∅ with s = 1 , · · · , N and give anefficient size control of J · m with J ∈ (cid:84) dν =1 F ∗ ν and m ∈ F ν ( s ). In Sections 9-10, we provenecessity parts of main theorems. In Section 11, we finish the proof for general situations. JOONIL KIM Notations. For the sake of distinction, we shall use the notations ı · = ı + · · · + ı n n , (cid:104) x, y (cid:105) = x y + · · · + x d y d for the inner products on Z n , R d , respectively. Note that a constant C may be differenton each line. As usual, the notation A (cid:46) B for two scalar expressions A, B will mean A ≤ CB for some positive constant C independent of A, B and A ≈ B will mean A (cid:46) B and B (cid:46) A . Polyhedra, Their Faces and Cones Throughout this paper, we show detailed proof for basic properties about faces andcones of polyhedra by using an easy tool such as elementary linear algebra. For furtherstudy, we refer readers to [7].2.1. Polyhedron.Definition 2.1. Let U ⊂ R n be a subspace endowed with an inner product (cid:104) , (cid:105) in R n .Then V is called an affine subspace in R n if V = p + U for some p ∈ R n . Definition 2.2. Let V be an affine subspace in R n . A hyperplane in V is a set π q ,r = { y ∈ V : (cid:104) q , y (cid:105) = r } where q ∈ R n and r ∈ R .The corresponding closed upper half-space and lower half-space are π + q ,r = { y ∈ V : (cid:104) q , y (cid:105) ≥ r } and π − q ,r = { y ∈ V : (cid:104) q , y (cid:105) ≤ r } . The open upper half-space and lower half space are( π + q ,r ) ◦ = { y ∈ V : (cid:104) q , y (cid:105) > r } and ( π − q ,r ) ◦ = { y ∈ V : (cid:104) q , y (cid:105) < r } . Definition 2.3 (Polyhedron in V ) . Let V be an affine subspace in R n and let Π = { π q j ,r j } Nj =1 be a collection of hyperplanes in V . A polyhedron P in V is defined to be anintersection of closed upper half-spaces π + q j ,r j : P = N (cid:92) j =1 π + q j ,r j = N (cid:92) j =1 { y ∈ V : (cid:104) q j , y (cid:105) ≥ r j for 1 ≤ j ≤ N } . ULTIPLE HILBERT TRANSFORMS 7 We call the above collection Π = Π( P ) the generator of P . We denote the polyhedron P by P (Π) indicating its generator Π. Sometimes, we mean also the generator Π of P to bethe collection of normal vectors { q j } Nj =1 instead of hyperplanes { π q j ,r j } Nj =1 . Definition 2.4. Let B = { q , · · · , q M } ⊂ R n be a finite number of vectors. Then thespan of B is the set Sp( B ) = M (cid:88) j =1 c j q j : c j ∈ R . The convex span of B and its interior are defined byCoSp( B ) = M (cid:88) j =1 c j q j : c j ≥ and CoSp ◦ ( B ) = M (cid:88) j =1 c j q j : c j > respectively. Finally the convex hull of B is the setCh( B ) = M (cid:88) j =1 c j q j : c j ≥ M (cid:88) j =1 c j = 1 . If B ⊂ R n is not a finite set, then the span of B is defined by the collection of all finitelinear combinations of vectors in B . Definition 2.5 (Ambient Space of Polyhedron) . Let P ⊂ R n and p , q ∈ P . ThenSp( P − p ) = Sp( P − q ) for all p , q ∈ P . We denote the vector space Sp( P − p ) by V ( P ). The dimension of P is defined bydim( P ) = dim( V ( P )) . From the fact p − q ∈ V ( P ), V ( P ) + p = V ( P ) + q . We call V ( P ) + p the ambient affine space of P in R n and denote it by V am ( P ) : V am ( P ) = V ( P ) + p , (2.1)which is the smallest affine space containing P . JOONIL KIM Definition 2.6. Let B ⊂ R n . Then the rank of a set B is the number of linearly inde-pendent vectors in B : rank( B ) = dim(Sp( B )) . Faces of Polyhedron.Definition 2.7 (Faces) . Let V be an affine subspace in R n . Given a class Π of hyperplanein V , let P = P (Π) be a polyhedron in V . A subset F ⊂ P is a face if there exists ahyperplane π q ,r in V (which does not have to be in Π) such that F = π q ,r ∩ P and P \ F ⊂ π + q ,r . (2.2)We may replace P \ F by P , or π + q ,r by ( π + q ,r ) ◦ in (2.2). Thus F is a face of P if and only ifthere exists a vector q ∈ R n and r ∈ R satisfying (cid:104) q , u (cid:105) = r < (cid:104) q , y (cid:105) for all u ∈ F and y ∈ P \ F . (2.3)When F is a face of P , it is denoted by F (cid:22) P . The above hyperplane π q ,r is called thesupporting hyperplane of the face F . The dimension of a face F of P is the dimension ofan ambient affine space V am ( F ) of F where V am ( F ) is defined in (2.1). We denote the setof all k -dimensional faces of P by F k ( P ), and (cid:83) F k ( P ) by F ( P ). By convention, an emptyset is − P ) = m . Then we call, a face F whose dimension isless than m , a proper face of P and denote it by F (cid:22) P . Lemma 2.1. Let P = P (Π) be a polyhedron in an affine space V . Then (1) If F , G (cid:22) P and G ⊂ F , then G (cid:22) F . (2) Let π q ,r ∈ Π . Then F = π q ,r ∩ P is a face of P . (3) Let ∆ ⊂ Π . Then F = (cid:84) π q ,r ∈ ∆ F q ,r with F q ,r = π q ,r ∩ P , is a face of P .Proof. Since G (cid:22) P , there exist q , r satisfying (2.3) where F replaced by G . Next we canalso replace P by F . This proves (1). By Definitions 2.2 and 2.3 for π q ,r and P , (cid:104) q , u (cid:105) = r < (cid:104) q , y (cid:105) for all u ∈ F = π q ,r ∩ P and y ∈ P \ F ULTIPLE HILBERT TRANSFORMS 9 which shows (2.3). Thus (2) is proved. Let ∆ = { π q j ,r j : 1 ≤ j ≤ M } ⊂ Π. Then by (2),for every j = 1 , · · · , M (cid:104) q j , u (cid:105) = r j ≤ (cid:104) q j , y (cid:105) for all u ∈ F = M (cid:92) j =1 F q j ,r j and y ∈ P \ F .(2.4)For y ∈ P \ F = (cid:83) Mj =1 ( P \ F q j ,r j ) above, there exists j = (cid:96) such that y ∈ P \ F q (cid:96) ,r (cid:96) . Thus ≤ in (2.4) is replaced by < for j = (cid:96) . Hence we sum (2.4) in j to obtain that (cid:42) M (cid:88) j =1 c j q j , u (cid:43) = M (cid:88) j =1 c j r j < (cid:42) M (cid:88) j =1 c j q j , y (cid:43) for all u ∈ F and y ∈ P \ F (2.5)where q = (cid:80) Mj =1 c j q j and r = (cid:80) Mj =1 c j r j . Hence this with (2.3) yields (3). (cid:3) Definition 2.8. Let F be a face of a convex polyhedron P . Then the boundary ∂ F of F is defined to be (cid:83) G , where the union is over all face G (cid:22) F . When dim( F ) = k , ∂ F = (cid:91) dim G = k − , G (cid:22) F G , since faces whose dimensions < k − k − F . Notethat ∂ F is the boundary of F with respect to the usual topology of V am ( F ) in (2.1). Lemma 2.2. Let P = P (Π) be a polyhedron. Then ∂ P ⊂ (cid:83) π ∈ Π π .Proof. Let x ∈ ∂ P . Assume x ∈ (cid:84) π ∈ Π ( π + ) ◦ . Then a ball B ( x , (cid:15) ) with some (cid:15) > (cid:84) π ∈ Π ( π + ) ◦ ⊂ (cid:84) π ∈ Π ( π + ) = P in V am ( P ). Thus, x / ∈ ∂ P because ∂ P is aboundary of P with respect to the usual topology of V am ( P ). Hence x / ∈ (cid:84) π ∈ Π ( π + ) ◦ .Combined with x ∈ ∂ P ⊂ P = (cid:84) π ∈ Π π + , we have x ∈ (cid:83) π ∈ Π π . (cid:3) Definition 2.9. Let F be a face of a convex polyhedron P . Then the interior F ◦ of P isdefined to be F ◦ = F \ ∂ F . Note also that F ◦ is the interior of F with respect to the usualtopology defined on V am ( F ) in (2.1). Example 2.1. Observe that CoSp ( p , · · · , p N ) ◦ = (cid:110)(cid:80) Nj =1 α j p j : α j > (cid:111) . Lemma 2.3. Let P be a polyhedron and F (cid:22) P with dim ( F ) = k . Suppose that B ⊂ ∂ F isa convex set. Then there is a k − dimensional face G such that B ⊂ G (cid:22) F .Proof. Assume contrary. Then B is not contained in one proper face of F , that is, thereexists p , p ∈ B such that Ch( p , p ) (cid:42) G for any G (cid:22) F . We shall find a contradiction.Given a plane π and a line segment Ch( p , p ) with p + p ∈ π , we have only two cases:(2.6) (1) Ch( p , p ) ⊂ π , or (2) p ∈ ( π + ) ◦ and p ∈ ( π − ) ◦ where p , p may be switched. By Definition 2.8, (cid:91) G(cid:22)F G = ∂ F where dim( G ) = k − p , p ⊂ B implies that Ch( p , p ) ⊂ G for some face G in (2.7).By Ch( p , p ) ⊂ B ⊂ ∂ F and (2.7), we have ( p + p ) / ∈ G for some G in (2.7). Let π bea supporting plane of G such that G = P ∩ π and P ⊂ π + . Then p , p ∈ B ⊂ F ⊂ P ⊂ π + .This implies that (2) in (2.6) is impossible. So we have (1) in (2.6), that is, Ch( p , p ) ⊂ π .Thus Ch( p , p ) ⊂ π ∩ P = G . (cid:3) A Cone of Face.Definition 2.10 (Cones, Dual Face) . Let F be a face of a polyhedron P in R n . Then thecone F ∗ of F is defined by F ∗ | P = { q ∈ R n : ∃ r ∈ R such that F ⊂ π q ,r ∩ P and P \ F ⊂ π + q ,r } = { q ∈ R n : ∃ r ∈ R such that (cid:104) q , u (cid:105) = r ≤ (cid:104) q , y (cid:105) for all u ∈ F , y ∈ P \ F } . (2.8)The interior of a cone F ∗ is the set of all nonzero normal vectors q satisfying (2.2):( F ∗ ) ◦ | P = { q ∈ R n : ∃ r ∈ R such that F = π q ,r ∩ P and P \ F ⊂ π + q ,r } = { q ∈ R n : ∃ r ∈ R such that F = π q ,r ∩ P and P \ F ⊂ ( π + q ,r ) ◦ } (2.9) = { q ∈ R n : ∃ r ∈ R such that (cid:104) q , u (cid:105) = r < (cid:104) q , y (cid:105) for all u ∈ F , y ∈ P \ F } . ULTIPLE HILBERT TRANSFORMS 11 We use the notation F ∗ | ( P , V ) when we restrict q in a given vector space V . Thus F ∗ | P = F ∗ | ( P , R n ) in (2.8). If not confused, we write just F ∗ instead of F ∗ | P or F ∗ | ( P , R n ). Wenote that F ∗ itself is a polyhedron in R n and ( F ∗ ) ◦ is an interior of F ∗ . Remark 2.1. To understand a cone F ∗ as a dual face of F , it is likely that a cone of F isto be defined by the collection of all normal vectors q satisfying (2.2) as in (2.9). If so, thecollection (2.9) is an open set, not a polyhedron anymore. To make F ∗ itself a polyhedron,we define a cone of F by (2.8) instead of its interior (2.9). Lemma 2.4. Let P be a polyhedron and F , G ∈ F ( P ) . Then F (cid:22) G if and only if G ∗ (cid:22) F ∗ .Proof. We first show F (cid:22) G implies that G ∗ (cid:22) F ∗ . If F = G , we are done. Let F (cid:22) G . Itsuffices to show that there exists q ∈ R n and r ∈ R such that (cid:104) q , u (cid:105) = r < (cid:104) q , v (cid:105) for all u ∈ G ∗ and v ∈ F ∗ \ G ∗ , (2.10)which means that G ∗ (cid:22) F ∗ by (2.3) in Definition 2.7. Choose q = n − m with n ∈ G \ F and m ∈ F . Then (cid:104) q , u (cid:105) = 0 because m , n ∈ G and u ∈ G ∗ . By v ∈ F ∗ \ G ∗ with m ∈ F and n ∈ G \ F , (cid:104) q , v (cid:105) > G ∗ (cid:22) F ∗ implies that F (cid:22) G . Observe that if q ∈ G ∗ , then there exists unique ρ = inf {(cid:104) x , q (cid:105) : x ∈ P } such that π q ,ρ is a supporting plane of a face containing G . Since G is a face, there exists q ∈ ( G ∗ ) ◦ ⊂ G ∗ . By Definition 2.10, π q ,ρ ∩ P = G . From q ∈ G ∗ ⊂ F ∗ ,it follows that F ⊂ π q ,ρ ∩ P = G , which yields F (cid:22) G by (1) of Lemma 2.1. (cid:3) Generalized Newton Polyhedron. For each S ⊂ N n = { , · · · , n } , we define R S + = { ( u , · · · , u n ) : u j ≥ j ∈ S and u j = 0 for j ∈ N n \ S } . Definition 2.11. Let Ω be a finite subset of Z n + and S ⊂ N n = { , · · · , n } . We definea Newton polyhedron N (Ω , S ) associated with Ω and S by the convex hull containing(Ω + R S + ) in R n : N (Ω , S ) = Ch (cid:0) Ω + R S + (cid:1) . By R ∅ + = { } and R N n + = R n + , we see that N (Ω , ∅ ) = Ch(Ω), and N (Ω , N n ) = Ch (cid:0) Ω + R n + (cid:1) that is the usual Newton Polyhedron denoted by N (Ω). Note that N (Ω , S ) is a polyhedronin the sense of Definition 2.3. See Figure 1. N N N , N Figure 1. Newton Polyhedra N (Ω , S ) for n = 2. Definition 2.12. Let Λ = (Λ ν ) with Λ ν ⊂ Z n + and S ⊂ { , · · · , n } . Then, the ordered d -tuple of Newton polyhedra N (Λ ν , S )’s is defined by (cid:126) N (Λ , S ) = ( N (Λ ν , S )) dν =1 . To indicate a given polynomial P = ( P ν ) ∈ P Λ , we also denote (cid:126) N (Λ , S ) by (cid:126) N ( P, S ). Definition 2.13. Let Λ = (Λ ν ) with Λ ν ⊂ Z n + and S ⊂ { , · · · , n } . We define thecollection of d -tuples of faces F ν ∈ F ( N (Λ ν , S )) by F ( (cid:126) N (Λ , S )) = { F = ( F , · · · , F d ) : F ν ∈ F ( N (Λ ν , S )) } . For each F ∈ F ( (cid:126) N (Λ , S )), we denote d -tuple of cones by F ∗ = ( F ∗ ν ).2.5. Basic Decompositions According to Faces and Cones. Choose ψ ∈ C ∞ c ([ − , ≤ ψ ≤ ψ ( u ) = 1 for | u | ≤ / . Put η ( u ) = ψ ( u ) − ψ (2 u ) and h ( u ) = η ( u ) /u for u (cid:54) = 0 . Let Λ = (Λ , · · · , Λ d ) and P Λ ∈ P Λ . For each F = ( F ν ) ∈ F ( (cid:126) N (Λ , S )) ULTIPLE HILBERT TRANSFORMS 13 and J ∈ Z n , define I J ( P F , ξ ) = (cid:90) R n exp (cid:32) i d (cid:88) ν =1 (cid:32) (cid:88) m ∈ F ν ∩ Λ ν c ν m − J · m t m (cid:33) ξ ν (cid:33) n (cid:89) (cid:96) =1 h ( t (cid:96) ) dt (2.11) = (cid:90) R n exp i (cid:88) m ∈ (cid:83) F ν ∩ Λ ν − J · m (cid:104) ξ, c m (cid:105) t m n (cid:89) (cid:96) =1 h ( t (cid:96) ) dt where c m = ( c ν m ) defined in (1.1). We shall write I J ( P Λ , ξ ) instead of I J ( P N (Λ ,S )) , ξ ). Definition 2.14. Given S ⊂ N n = { , · · · , n } , we define1 S = ( r j ) where r i = 1 for i ∈ S and r i = ∞ for i ∈ N n \ S and Z ( S ) = n (cid:89) i =1 Z i where Z i = R + if i ∈ S and Z i = R if i ∈ N n \ S. Then by using Z i above, we write (cid:89) i ∈ S {− < t i < } (cid:89) i ∈ N n \ S {−∞ < t i < ∞} = n (cid:89) i =1 (cid:91) k i ∈ Z i ∩ Z {| t i | ≈ − k i } , and make the following dyadic decomposition: I ( P Λ , ξ, S ) = (cid:88) J ∈ Z ( S ) ∩ Z n I J ( P Λ , ξ ) . (2.12)As the name (dual face) tells, each J ∈ F ∗ ν ∩ Z n can be understood as a linear functionalmapping n ∈ R n to J · n ∈ R satisfying the following dominating property:(2.13) 2 − J · m = 2 − r ( J ) ≥ − J · n for all m ∈ F ν and n ∈ P ν \ F ν . Thus, for J ∈ (cid:84) dν =1 F ∗ ν in (2.12) with the property (2.13) in (2.11),(2.14) ∃ α ∈ Z n + , (cid:18) ∂∂t (cid:19) α (cid:32) d (cid:88) ν =1 (cid:32) (cid:88) m ∈ Λ ν c ν m − J · m t m (cid:33) ξ ν (cid:33) ≈ − J · m ν ξ ν for all m ν ∈ F ν ∩ Λ ν . This combined with Z ( S ) = (cid:83) F =( F ν ) ∈F ( (cid:126) N (Λ ,S )) (cid:16)(cid:84) dν =1 F ∗ ν (cid:17) suggests us to decompose inSection 6 (cid:88) J ∈ Z ( S ) ∩ Z n I J ( P Λ , ξ ) = (cid:88) F ∈F ( (cid:126) N (Λ ,S )) (cid:88) J ∈ (cid:84) dν =1 F ∗ ν ∩ Z n I J ( P Λ , ξ ) , and next prove in Sections 7 and 8 that for each F = ( F ν ) ∈ F ( (cid:126) N (Λ , S )), N (cid:88) s =1 (cid:88) J ∈ (cid:84) dν =1 F ∗ ν ∩ Z n (cid:12)(cid:12) I J ( P F ( s − , ξ ) − I J ( P F ( s ) , ξ ) (cid:12)(cid:12) + (cid:88) J ∈ (cid:84) dν =1 F ∗ ν ∩ Z n |I J ( P F , ξ ) | ≤ C. (2.15)Here F ( s ) = ( F ν ( s )) will be chosen in an suitable way so that F ν ( s − (cid:23) F ν ( s ) with ν = 1 , · · · , d where (cid:126) N (Λ , S ) = F (0) and F ( N ) = F as in (1.10).3. Main Theorem and Background In order to state main results, we first try to find an appropriate condition on anexponent set (cid:83) dν =1 F ν ∩ Λ ν which guarantees I J ( P F , ξ ) ≡ Even Sets. Let (cid:83) dν =1 F ν ∩ Λ ν = { m , · · · , m N } . Suppose every vector m of the form α m + · · · + α N m N with α j = 0 or 1has at least one even component. Then, the Taylor expansion of the exponential functionin (2.11) yields that I J ( F , ξ ) = ∞ (cid:88) k =0 (cid:90) R n (cid:16) i (cid:80) m ∈ (cid:83) F ν ∩ Λ ν − J · m (cid:104) ξ, c m (cid:105) t m (cid:17) k k ! n (cid:89) (cid:96) =1 h ( t (cid:96) ) dt = ∞ (cid:88) k =0 (cid:90) R n (cid:88) α + ··· + α N = k C ( J, m , α, ξ ) t α m + ··· + α m N k ! n (cid:89) (cid:96) =1 h ( t (cid:96) ) dt = 0(3.1)since h ( t (cid:96) ) is an odd function for each (cid:96) = 1 , · · · , n . This observation leads to the followingnotions of even and odd sets in Z n + . Let Ω = { m , · · · , m N } ⊂ Z n + and let the class ofsum of vectors in Ω beΣ(Ω) = { α m + · · · + α N m N : α j = 0 or 1 } . Definition 3.1. A finite subset Ω = { m , · · · , m N } of Z n + is said to be odd iff there existsat least one vector m ∈ Σ(Ω) all of whose components are odd numbers such that m = (odd , · · · , odd) . Definition 3.2. A finite subset Ω of Z n + is said to be even iff Ω is not odd, that is, every m = ( m , · · · , m n ) ∈ Σ(Ω) has at least one even numbered component m j . ULTIPLE HILBERT TRANSFORMS 15 Example 3.1. In Z , let A = { (1 , , , (3 , , } , and B = { (1 , , , (0 , , } . Then A isan even set and B an odd set. Notice that A is an even set, though there is no k ∈ { , , } such that k th component of every vector in A is even. In (3.1), we have proved the following proposition: Proposition 3.1. Suppose that (cid:83) dν =1 ( F ν ∩ Λ ν ) is an even set. Then I J ( F , ξ ) ≡ (cid:83) F ν ( s ) (formu-lated in Proposition 5.1) or vanishing property in Propositions 3.1. Thus, the evennesscondition in Propositions 3.1 shall be imposed on the only faces contained in the subclass A of F ( (cid:126) N (Λ , S )) satisfying the following two conditions: Low Rank Condition : rank (cid:32) d (cid:91) ν =1 F ν (cid:33) ≤ n − F ∈ A ,(3.2) Overlapping Cone Condition : d (cid:92) ν =1 ( F ∗ ν ) ◦ (cid:54) = ∅ for F ∈ A (3.3)where the overlapping cone condition comes from the decompositions in J and the domi-nating condition (2.13).3.2. Statement of Main Results. We start with the simplest case d = 1. Observe forthis case that (cid:84) dν =1 ( F ∗ ν ) ◦ (cid:54) = ∅ always holds whenever rank (cid:16)(cid:83) dν =1 F ν (cid:17) ≤ n − . Main Theorem 1. Let Λ ⊂ Z n + and S ⊂ N n . Suppose that d = 1 in (1.1). Let < p < ∞ .Then for all P ∈ P Λ , ∃ C P > such that sup r ∈ I ( S ) (cid:13)(cid:13) H Pr (cid:13)(cid:13) L p ( R ) → L p ( R ) ≤ C P if and only if F ∩ Λ is an even set for F ∈ F ( N (Λ , S )) whenever rank ( F ) ≤ n − . For d > 1, the overlapping condition (3.3) is crucial as well as the rank condition (3.2). Definition 3.3. Given (cid:126) N (Λ , S ) = ( N (Λ ν , S )) dν =1 , we set the collection of all d -tuples offaces satisfying both low rank condition (3.2) and overlapping (3.3) by F lo ( (cid:126) N (Λ , S )) = (cid:40) ( F ν ) ∈ F (cid:16) (cid:126) N (Λ , S ) (cid:17) : rank (cid:32) d (cid:91) ν =1 F ν (cid:33) ≤ n − d (cid:92) ν =1 ( F ∗ ν ) ◦ (cid:54) = ∅ (cid:41) . We assume first that Λ ν ’s are mutually disjoint such that Λ µ ∩ Λ ν = ∅ for any µ (cid:54) = ν . Main Theorem 2. Let Λ = (Λ , · · · , Λ d ) with Λ ν ⊂ Z n + and S ⊂ N n . Suppose that Λ ν ’sare mutually disjoint. Let < p < ∞ . Thenfor all P ∈ P Λ , ∃ C P > such that sup r ∈ I ( S ) (cid:13)(cid:13) H Pr (cid:13)(cid:13) L p ( R d ) → L p ( R d ) ≤ C P if and only if (cid:83) dν =1 ( F ν ∩ Λ ν ) is an even set for F = ( F ν ) ∈ F lo ( (cid:126) N (Λ , S )) . Remark 3.1. Main Theorems 1 and 2 do not give a criteria for the boundedness with agiven individual polynomial P Λ , but enables us to determine the boundedness for uni-versal polynomials P Λ with a set Λ of exponents fixed. Also, Main Theorems 1 and2 do not give a condition for the boundedness of (cid:13)(cid:13) H Pr (cid:13)(cid:13) L p ( R d ) → L p ( R d ) with fixed r , butfor the uniform boundedness sup r ∈ I ( S ) (cid:13)(cid:13) H Pr (cid:13)(cid:13) L p ( R d ) → L p ( R d ) . It is interesting to know if sup r ∈ I ( S ) (cid:13)(cid:13) H Pr (cid:13)(cid:13) L p ( R d ) → L p ( R d ) can be replaced by (cid:13)(cid:13) H P S (cid:13)(cid:13) L p ( R d ) → L p ( R d ) in the above theoremswhere S is defined in Definition 2.14. Let P Λ be a form of graph ( t , · · · , t n , P n +1 ( t )) so that Λ = ( { e } , · · · , { e n } , Λ n +1 ).For this case, we are able to show that the L p boundedness of H P Λ S and the uniform L p boundedness of H P Λ r in r ∈ I ( S ) are equivalent. Moreover, we do not need the overlappingcondition (3.3), since we can make the condition (cid:84) dν =1 ( F ∗ ν ) ◦ (cid:54) = ∅ always hold. Corollary 3.1. Let < p < ∞ and let Λ = ( { e } , · · · , { e n } , Λ n +1 ) and S ⊂ N n . Thenfor all P ∈ P Λ , ∃ C P > such that (cid:13)(cid:13) H P S (cid:13)(cid:13) L p ( R d ) → L p ( R d ) ≤ C P if and only if ( F n +1 ∩ Λ n +1 ) ∪ A , for F n +1 ∈ F ( N (Λ n +1 , S )) and A ⊂ { e , · · · , e n } , is aneven set whenever rank ( F n +1 ∪ A ) ≤ n − . Remark 3.2. The above evenness condition in Corollary 3.1 is equivalent to d (cid:91) ν =1 ( F ν ∩ Λ ν ) is an even set whenever rank (cid:32) d (cid:91) ν =1 F ν (cid:33) ≤ n − for F ∈ F ( (cid:126) N (Λ , S )) . We exclude the assumption of mutually disjointness of Λ ν ’s in the hypotheses of MainTheorem 2. Let P = ( P ν ) dν =1 be a vector polynomial. For each ν = 1 , · · · , d , we define a ULTIPLE HILBERT TRANSFORMS 17 set Λ( P ν ) to be a set of all exponents of the monomials in P ν :Λ( P ν ) = (cid:110) m ∈ Z n + : c ν m (cid:54) = 0 in P ν ( t ) = (cid:88) c ν m t m (cid:111) . Moreover, we denote a d -tuple (Λ( P ν )) dν =1 by Λ( P ). Denote the set of d × d invertiblematrices by GL ( d ). For A ∈ GL ( d ) and P ∈ P Λ with P ( t ) = ( P ( t ) , · · · , P d ( t )), we let AP be a vector polynomial given by the matrix multiplication AP ( t ) = (cid:88) m ∈ Λ(( AP ) ν ) a ν m t m dν =1 for some a ν m (cid:54) = 0where we regard P ( t ) and AP ( t ) above as column vectors. Then AP ∈ P Λ (cid:48) where Λ (cid:48) =Λ( AP ). If A = I an identity matrix, Λ (cid:48) = (Λ(( AP ) ν )) dν =1 = (Λ( P ν )) dν =1 = (Λ ν ) dν =1 = Λ. Definition 3.4. Let P ∈ P Λ where Λ = (Λ ν ) with Λ ν ⊂ Z n + and S ⊂ { , · · · , n } . Let A ∈ GL ( d ). Given a vector polynomial AP , we consider the d -tuple of Newton polyhedrons (cid:126) N ( AP, S ) = ( N (( AP ) ν , S )) dν =1 and d -tuple of their faces F ( (cid:126) N ( AP, S )) = { F A = (( F A ) , · · · , ( F A ) d ) : ( F A ) ν ∈ F ( N (( AP ) ν , S )) } . Main Theorem 3. Let Λ = (Λ , · · · , Λ d ) with Λ ν ⊂ Z n + and S ⊂ N n . Let < p < ∞ .For all P ∈ P Λ ∃ C P > such that sup r ∈ I ( S ) (cid:13)(cid:13) H Pr (cid:13)(cid:13) L p ( R d ) → L p ( R d ) ≤ C P if and only if for all A ∈ GL d and P ∈ P Λ , d (cid:91) ν =1 ( F A ) ν ∩ Λ(( AP ) ν ) is an even set whenever F A = (( F A ) ν ) ∈ F lo ( (cid:126) N ( AP, S ))(3.4) where the class F lo ( (cid:126) N ( AP, S )) is defined as in Definition 3.3. Remark 3.3. For the case n = 2 in Main Theorems 2 and 3, the overlapping conditionof cones in F lo ( (cid:126) N (Λ , S )) does not have to appear explicitly. By omitting the overlappingcone condition in F lo ( (cid:126) N (Λ , S )) , we let F l ( (cid:126) N (Λ , S )) = (cid:40) ( F ν ) ∈ F (cid:16) (cid:126) N (Λ , S ) (cid:17) : rank (cid:32) d (cid:91) ν =1 F ν (cid:33) ≤ n − (cid:41) ⊃ F lo ( (cid:126) N (Λ , S )) . Then, for the case n = 2 , the evenness condition for F lo ( (cid:126) N (Λ , S )) is equivalent to thecondition for F l ( (cid:126) N (Λ , S )) . It suffices to show ⇒ . Suppose that (cid:83) dν =1 F ν ∩ Λ ν is an odd setwith rank (cid:16)(cid:83) dν =1 F ν (cid:17) ≤ . Then there exists µ such that F µ ∩ Λ µ has a point ( odd, odd ) , be-cause both of two points ( even, odd ) , ( odd, even ) can not lie in the one line passing throughthe origin. Therefore, G = ( G ν ) defined by G µ = F µ and G ν = ∅ for ν (cid:54) = µ satisfies that G ∈ F lo ( (cid:126) N (Λ , S )) and (cid:83) dν =1 G ν ∩ Λ ν is an odd set. Remark 3.4. For the case n ≥ , the overlapping condition is crucial in Main Theorems2 and 3. Moreover, we note that it is not just cones (cid:84) dν =1 F ∗ ν , but their interiors (cid:84) dν =1 ( F ∗ ν ) ◦ that satisfy the overlapping condition (3.3). The example 4.1 in Section 4 shows that thereexists F ∈ F l ( (cid:126) N (Λ , S )) such that (cid:84) dν =1 F ∗ ν (cid:54) = ∅ and (cid:83) dν =1 ( F ν ∩ Λ ν ) is an odd set, butfor all P ∈ P Λ sup r ∈ I ( S ) (cid:13)(cid:13) H Pr (cid:13)(cid:13) L p ( R d ) → L p ( R d ) ≤ C P . Background. In the one parameter case ( n = 1), the operator H P Λ r with r =(1 , · · · , 1) can be regarded as a particular instance of singular integrals along curves sat-isfying finite type condition in E. M. Stein and S. Wainger [21]. The L p theory of thosesingular integrals has been developed quite well. For example, see M. Christ, A. Nagel, E.M. Stein and S. Wainger [6] for singular Radon transforms with the curvature conditionsin a very general setting. See also M. Folch-Gabayet and J. Wright [8] for the case thatphase functions P Λ are given by rational functions.In the multi-parameter case ( n ≥ L boundedness. In [17], F. Ricci and E. M. Stein establishedan L p theorem for multi-parameter singular integrals whose kernels satisfy more generaldilation structure. A special case of their results implies that if Λ = ( { e } , · · · , { e n } , { m } )where at least n − m are even, then (cid:107)H P Λ S (cid:107) L p ( R n +1 ) → L p ( R n +1 ) are boundedfor 1 < p < ∞ . In [3], A. Carbery, S. Wainger and J. Wright obtained a necessary andsufficient condition for L p ( R ) boundedness of H Λ1 S with S = { , } , Λ = ( { e } , { e } , Λ )where d = 3 and n = 2. Their theorem states that ULTIPLE HILBERT TRANSFORMS 19 Theorem 3.1 (Double Hilbert transform [3]) . Let Λ = ( { e } , { e } , Λ ) and S = { , } with n = 2 and d = 3 . For < p < ∞ , the local double Hilbert transform H P Λ S is boundedin L p ( R ) if and only if every vertex m in N (Λ , S ) has at least one even component. S. Patel [14] extends this result to S = ∅ corresponding to the global Hilbert transform. Theorem 3.2 (Double Hilbert transform [14]) . Let Λ = ( { e } , { e } , Λ ) and S = {∅} with n = 2 and d = 3 . For < p < ∞ , the global double Hilbert transform H P Λ S is boundedin L p ( R ) if and only if every vertex m in Ch (Λ ) and every edge E in Ch (Λ ) passingthrough the origin has at least one even component. S. Patel [13] also studies the case n = 2 and d = 1. He has shown that the necessaryand sufficient condition for the L p boundedness of H P Λ S cannot be determined by only thegeometry of N (Λ) but by coefficients of the given polynomial P Λ ( t ). More precisely, thecondition is described in terms of not a single vertex m and its coefficient c m in P Λ , butthe sum of quantities associated with many vertices and their corresponding coefficients: Theorem 3.3 (Double Hilbert transform [13]) . Let Λ = (Λ ) and S = { , } with n = 2 and d = 1 . Then, the local double Hilbert transform H P Λ S is bounded in L p ( R ) for
Λ = ( { e } , { e } , Λ , · · · , Λ k ) and S = { , } with n = 2 and d = k + 2 . Suppose that P ( t , t ) = ( P Λ ( t ) , · · · , P Λ k ( t )) and P Λ = ( t , t , P ( t , t )) . For < p < ∞ , the local double Hilbert transform H P Λ S isbounded in L p ( R d ) if and only if for every A ∈ GL ( k ) , every vertex m ∈ N (( AP ) ν , S ) with ν = 1 , · · · , k has at least one even numbered component. Remark 3.5. The results of Main Theorems 2 and 3 for n = 2 and S = { , } followsfrom Theorem 3.4 by a slight modification of its necessary proof. The triple Hilbert transforms H P Λ S with S = { , , } and Λ = ( { e } , { e } , { e } , Λ )were studied in the two papers [3] [4] published in 2009. In [3], A. Carbery, S. Waingerand J. Wright have discovered a remarkable differences between the triple and the doubleHilbert transforms. The L boundedness of the triple Hilbert transform H P Λ S depends onthe coefficients of P Λ as well as the Newton polyhedron N (Λ ), whereas that of the doubleHilbert transform depends only on the Newton polygon N (Λ ). They establish two typesof theorems. First one gives the necessary and sufficient condition that the operators H P Λ S are bounded in L for all class of polynomials P Λ ∈ P (Λ) when Λ is given. This theorem iscalled the universal theorem. The second theorem is to inform the necessary and sufficientcondition that the one individual operator H P Λ S is bounded in L when a polynomial P Λ isgiven. This theorem is called the individual theorem. The condition of the first theorem isexpressed solely in terms of N (Λ ) but that of the second in terms of individual coefficientsof given polynomial P Λ in question. Here we only state their universal theorem. Theorem 3.5 (Triple Hilbert transform [3]) . Let < p < ∞ . Given S = { , , } and Λ = ( { e } , { e } , { e } , Λ ) , suppose that (H1) Every entry of a vertex in N (Λ , S ) is positive. (H2) (a) Each edge N (Λ , S ) is not contained on any hyperplane parallel to a coordinateplane. (b) The projection of the line containing an edge N (Λ , S ) onto a coordinate planedoes not pass through the origin. (H3) The plane determined by any three vertices in N (Λ , S ) does not contain the origin. ULTIPLE HILBERT TRANSFORMS 21 Then the triple Hilbert transform H P Λ S is bounded in L ( R ) for all P Λ ∈ P Λ if and onlyif every vertex in N (Λ , S ) has at least two even entries, and every edges E of N (Λ , S ) satisfies that there exists a one component such that the entry of that component of everyvector in E ∩ Λ is even. Remark 3.6. They found a vector polynomial P Λ ( t ) = ( t , t , t , P Λ ( t )) such that thecorresponding triple Hilbert transform H P Λ S is bounded on L ( R ) although N (Λ , S ) breaksthe above evenness condition. In [4], Y.K. Cho, H. Hong, C.W. Yang and the author proved the theorem withoutassuming the three hypotheses H1-H3 so that Theorem 3.6 (Triple Hilbert transform [4]) . Let < p < ∞ . Given S = { , , } and Λ = ( { e } , { e } , { e } , Λ ) , the triple Hilbert transform H P Λ S is bounded in L p ( R ) for all P Λ ∈ P Λ if and only if every F ∈ F ( N (Λ )) for rank (( F ∩ Λ ) ∪ A ) ≤ with A ⊂ { e , e , e } satisfies that there exists a one component such that the entry of thatcomponent of every vector in F ∩ Λ is even. Remark 3.7. The hypotheses in Theorems 3.1, 3.2, 3.5 and 3.6 are same as those ofCorollary 3.1 for n = 2 , . It is interesting to find for n ≥ an asymptotic behavior of I ( P Λ , ξ, S ) with a coefficient of logarithm in ξ having a similar form to (3.5) in Theorem3.3. As a variable coefficient version, we define H P ( f )( x ) by (cid:90) (cid:81) nj =1 [ − r j ,r j ] f (cid:0) x − t , · · · , x n − t n , x n +1 − P ( x , · · · , x n , t , · · · , t n ) (cid:1) dt t · · · dt n t n whose corresponding oscillatory singular integral operator is given by T Pλ ( f )( x ) = p.v. (cid:90) { y : | x j − y j | We study representations of faces F and their cones F ∗ of a polyhedron P = P (Π) in R n .It is well known that every face F has an expression (cid:84) Nj =1 π q j ,r j ∩ P with some generators π q j ,r j ∈ Π and its cone F ∗ expressed as CoSp( { q j } Nj =1 ). We shall prove this representationformula and give some detailed description of generators for the case dim( P ) < n .4.1. Low Dimensional Polyhedron in R n . A polyhedron P = P (Π) in R n with dim( P ) = m ≤ n is regarded as a m dimensional polyhedron in the affine space V am ( P ) of dimension m defined in (2.1). Since V am ( P ) itself is a polyhedron in R n , we choose the generator Πof P and split it into two parts Π = Π a ∪ Π b : Π b a generator of V am ( P ) and Π a a generatorof P in V am ( P ). See the left picture in Figure 2. Lemma 4.1. Let P ⊂ R n be a polyhedron with dim ( P ) = dim ( V am ( P )) = m ≤ n . Then P = P (Π a ∪ Π b ) such that V am ( P ) = P (Π b ) in R n with Π b = { π ± n i , ± s i } n − mi =1 with n i ⊥ n j for i (cid:54) = j , (4.1) P = P (Π (cid:48) a ) in V am ( P ) with Π a = { π q j ,r j } Mj =1 and Π (cid:48) a = { π q j ,r j ∩ V am ( P ) } Mj =1 , (4.2) where q j ∈ V ( P ) = Sp ⊥ ( { n i } n − mi =1 ) . ULTIPLE HILBERT TRANSFORMS 23 Face Cone PF nnq − = πππ n n − q q q { } , nnb − =Π ππ { } ,, qqqa πππ=Π ),,(][ nnqCoSpF −= )( Π= PP PF nnq − = πππ q n − q n q Polyhedron )( PV am Figure 2. Low Dimensional Polyhedron, Face and Cone. Proof. If n = m , then let Π a = Π and Π b = ∅ so that P (Π b ) = R n . Let m < n . Thereexist n − m orthonormal vectors n j ’s and some constants s j ’s such that V ( P ) = n − m (cid:92) i =1 π n i , so that V am ( P ) = n − m (cid:92) i =1 π n i ,s i . where V ( P ) = Sp ⊥ ( { n i } n − mi =1 ). By (2.1), V am ( P ) = V ( P ) + r with r ∈ P and s i = r · n i . Thiscombined with π + n i ,s i ∩ π + − n i , − s i = π n i ,s i implies V am ( P ) = P (Π b ) with Π b = { π ± n i , ± s i : i = 1 , · · · , n − m } , which yields (4.1). By Definition 2.3, there are p j ∈ R n such that P = (cid:92) j { x ∈ V am ( P ) : (cid:104) p j , x (cid:105) ≥ ρ j } . (4.3) Let x ∈ P and P V ( P ) be a projection map to the vector space V ( P ). Then from x − r ∈ V ( P ), (cid:104) p j , x (cid:105) = (cid:104) P V ( P ) ( p j ) , x (cid:105) + (cid:104) P V ⊥ ( P ) ( p j ) , x (cid:105) = (cid:104) P V ( P ) ( p j ) , x (cid:105) + (cid:104) P roj V ⊥ ( P ) ( p j ) , r (cid:105) . We put P roj V ( P ) ( p j ) = q j and r j = ρ j − (cid:104) P roj V ⊥ ( P ) ( p j ) , r (cid:105) and rewrite (4.3) as P = M (cid:92) j =1 { x ∈ V am ( P ) : (cid:104) q j , x (cid:105) ≥ r j } where q j = P roj V ( P ) ( p j ) ∈ V ( P ) = Sp ⊥ ( { n i } n − mi =1 ) . This proves (4.2). Finally, P = P (Π a ∪ Π b ) follows from (4.1) and (4.2). (cid:3) Representation of Face.Lemma 4.2. Let P = P (Π) with dim ( V am ( P )) = m and F be a proper face of P . Then ∃ π ∈ Π such that F ⊂ π. Proof. By Definition 2.8 and by Lemma 2.2, F ⊂ ∂ P ⊂ (cid:91) π ∈ Π π ∩ P . (4.4)By (4.2) of Lemma 4.1, we may take Π = Π (cid:48) a in (4.4). Thus from (2) of Lemma 2.1, each π ∩ P with π ∈ Π is a face of P with dimension m − 1. Therefore the second ⊂ in (4.4) isreplaced by =. We next use the same argument as in the proof of Lemma 2.3 to obtainthat F in (4.4) is contained in one face π ∩ P in (4.4). (cid:3) Definition 4.1 (Facet) . Let P = P (Π) be a polyhedron in an affine space V such thatdim( P ) = dim( V am ( P )) = m . Then m − F of P is called a facet of P . Lemma 4.3. Let P = P (Π) be a polyhedron in an affine space V such that dim ( P ) = dim ( V am ( P )) = m where Π = Π a ∪ Π b as in Lemma 4.1. Then every facet F of P isexpressed as F = π ∩ P and P \ F ⊂ ( π + ) ◦ for some π ∈ Π a ⊂ Π . Proof. By Lemma 4.1, we regard P = P (Π (cid:48) a ) as a polyhedron in the m dimensional affinespace V am ( P ). Here π (cid:48) = π ∩ V am ( P ) ∈ Π (cid:48) a is a m − V am ( P ).By Lemma 4.2, ∃ π (cid:48) ∈ Π (cid:48) a such that F ⊂ π (cid:48) = π ∩ V am ( P ) where π ∈ Π a . (4.5) ULTIPLE HILBERT TRANSFORMS 25 On the other hand, by Definition 2.7, there exists an m − π q ,r in V am ( P ) such that F = π q ,r ∩ P and P \ F ⊂ ( π + q ,r ) ◦ .(4.6)In view of (4.5) and (4.6), both m − π (cid:48) and π q ,r in V am ( P )contain the m − F . Thus π (cid:48) = π q ,r . By this and (4.6), F = π q ,r ∩ P = π (cid:48) ∩ P = ( π ∩ V am ( P )) ∩ P = π ∩ P where π ∈ Π a and P \ F ⊂ ( π + q ,r ) ◦ = (( π (cid:48) ) + ) ◦ ⊂ ( π + ) ◦ . (cid:3) Proposition 4.1 (Face Representation) . Let P = P (Π) be a polyhedron in R n where Π = Π a ∪ Π b as in Lemma 4.1. Let dim ( P ) = m ≤ n. Then every face F of P withdim ( F ) ≤ n − has the expression F = (cid:92) π ∈ Π( F ) F π where F π = π ∩ P where Π( F ) = { π p j } Nj =1 ⊂ Π . (4.7) Remark 4.1. We split the generator Π( F ) = { π p j } Nj =1 = Π a ( F ) ∪ Π b ( F ) in (4.7) so that Π a ( F ) = Π( F ) ∩ Π a = { π q j } (cid:96)j =1 and Π b ( F ) = Π( F ) ∩ Π b = Π b = { π ± n i } n − mi =1 . (4.8) See the left side of Figure 2. Denote only normal vectors { p j } Nj =1 in Π( F ) by Π( F ) also.Proof of Proposition 4.1. Let dim( F ) = m ≤ n − 1. An improper face F = P has anexpression P = (cid:92) π ∈ Π b π ∩ P where π ∈ Π b . It suffices to show that each proper face F of P (Π) is expressed as F = M (cid:92) j =1 F j where F j = π j ∩ P with π j ∈ Π a ⊂ Π are facets of P . (4.9)To show (4.9), we first let F be a face of codimension 1 of the m -dimensional ambientaffine space V am ( P ). Then F itself is a facet of P such that F = π ∩ P with π ∈ Π a ⊂ Πfrom Lemma 4.3. Let F be a face of codimension 2 of the m -dimensional ambient affinespace V am ( P ). By Lemma 4.2, π q ,r ∈ Π a such that F ⊂ π q ,r . By (2) of Lemma 2.1, P (cid:48) = π q ,r ∩ P is a facet of P such that dim( P (cid:48) ) = m − P (cid:48) itself is an m − a ( P (cid:48) ) ⊂ { π q ,r ∩ π : π ∈ Π a ( P ) } . By F ⊂ P (cid:48) and (1) of Lemma 2.1, m − F of P is a facet of an m − P (cid:48) . Hence by Lemma 4.3, there exists π (cid:48) ∈ Π a ( P (cid:48) ) in (4.11) suchthat F = π (cid:48) ∩ P (cid:48) . Thus, by (4.11) there exists π ∈ Π a ( P ) such that π (cid:48) = π q ,r ∩ π and F = π (cid:48) ∩ P (cid:48) = ( π q ,r ∩ π ) ∩ P (cid:48) = ( π q ,r ∩ P ) ∩ ( π ∩ P ) = F π q ,r ∩ F π where F π and F π q ,r are facets of P . We finish the proof of (4.9) inductively. (cid:3) Representation of Cone.Proposition 4.2 (Cone representation) . Every proper face F (cid:22) P = P (Π) having agenerator Π( F ) = { p j } Nj =1 with expression (4.7) has its cone of the form: ( F ∗ ) ◦ | P = ( F ∗ ) ◦ | ( P , R n ) = CoSp ◦ ( { p i : i = 1 , · · · , N } ) . Here F ∗ | P = F ∗ | ( P , R n ) = CoSp( { p i : i = 1 , · · · , N } ) similarly. Remark 4.2. See the right side of Figure 2, which elucidate the relation between facesand their cones. In the above, F ∗ | ( P , R n ) = F ∗ ( P , V ( P )) ⊕ V ⊥ ( P ) where F ∗ | ( P , V ( P )) =CoSp( { p i : p i ∈ Π a ( F ) } ) with Π a ( F ) in (4.8). From this, we also obtain that dim ( F ) + dim ( F ∗ | ( P , R n )) = n whereas dim ( F ) + dim ( F ∗ | ( P , V ( P )) = dim ( V ( P )) . Lemma 4.4. Let P = P (Π) be a polyhedron in an inner product space V with dim ( P ) = dim ( V ) = n . Let F ∈ F ( P ) be a facet expressed as (4.12) F = π q ,r ∩ P where π q ,r ∈ Π and P \ F ⊂ ( π + q ,r ) ◦ . Then ( F ∗ ) ◦ | ( P , V ) = CoSp ◦ ( q ) . ULTIPLE HILBERT TRANSFORMS 27 Proof. Let q (cid:48) = c q ⊂ CoSp ◦ ( q ) with q in (4.12) and c > 0. Then q (cid:48) satisfies (2.9) inDefinition 2.10. So q (cid:48) ∈ ( F ∗ ) ◦ . Thus CoSp ◦ ( q ) ⊂ ( F ∗ ) ◦ . Let p ∈ ( F ∗ ) ◦ . Then by (2.9), π p ,r ∩ P = F . This combined with (4.12) implies that F = ( π p ,r ∩ P ) ∩ F = π p ,r ∩ ( π q ,r ∩ P ). So,dim( F ) < n − p / ∈ CoSp ◦ ( q ). Thus p ∈ CoSp ◦ ( q ), which proves ( F ∗ ) ◦ ⊂ CoSp ◦ ( q ). (cid:3) Lemma 4.5. Let P = P (Π) be a polyhedron in an inner product space V with dim ( P ) = dim ( V ) = n . Let F ∈ F ( P ) be a k dimensional face of P with a generator Π( F ) = { q j } Mj =1 ,that is, F = M (cid:92) j =1 F j where F j = π q j ,r j ∩ P is a facet of P so that ( F ∗ j ) ◦ = CoSp ◦ ( { q j } ) for j = 1 , · · · , M . Then, ( F ∗ ) ◦ | ( P , V ) = CoSp ◦ ( { q j } Mj=1 ) . (4.13) Proof. Let q ∈ CoSp ◦ ( { q j } Mj=1 ). Then (2.5) with q = (cid:80) Mj =1 c j q j yields (2.9). Thus q ∈ ( F ∗ ) ◦ , which proves ⊃ of (4.13). We next show ⊂ of (4.13). Let U = Sp { q j : j = 1 , · · · , M } .Subtract a vector r ∈ F = (cid:84) Mj =1 π q j ,r j ∩ P , V M (cid:92) j =1 π q j , = V M (cid:92) j =1 π q j , ∩ ( P − r ) = V ( F − r ) = V ( F ) and dim( V ( F )) = k. Thus dim (cid:0) U ⊥ (cid:1) = k and dim ( U ) = n − k . Let q ∈ ( F ∗ ) ◦ | ( P , V ) ⊂ V = U ⊕ U ⊥ . Then, q = N (cid:88) j =1 s j q j + u for some s j ∈ R n and u ∈ U ⊥ . (4.14)Since F is a k dimensional face, we can choose k linearly independent vectors { u − u , · · · , u k +1 − u : u (cid:96) ∈ F } . Since q ∈ ( F ∗ ) ◦ and q j ∈ ( F ∗ j ) ◦ where ( F ∗ ) ◦ = ( F ∗ ) ◦ | ( P , V ), by (2.9) and u (cid:96) , u ∈ F = (cid:84) F j , (cid:104) q , u (cid:96) − u (cid:105) = (cid:104) q j , u (cid:96) − u (cid:105) = 0 for all (cid:96) = 2 , · · · , k + 1.(4.15)This implies that { u − u , · · · , u k +1 − u } ⊂ U ⊥ and forms a basis of U ⊥ becausedim( U ⊥ ) = k . Hence u ∈ U ⊥ is expressed as u = k +1 (cid:88) (cid:96) =2 c (cid:96) ( u (cid:96) − u ) . Thus by (4.15), we have (cid:104) q , u (cid:105) = 0 and (cid:104) q j , u (cid:105) = 0 for j = 1 , · · · , M. Therefore, in (4.14)0 = (cid:104) q , u (cid:105) = N (cid:88) j =1 s j (cid:104) q j · u (cid:105) + (cid:104) u , u (cid:105) = | u | , which implies that q = M (cid:88) j =1 s j q j for q ∈ ( F ∗ ) ◦ .(4.16)We now fix (cid:96) and show s (cid:96) > 0. Since F j ’s are facets of one polyhedron, we can choose y ∈ (cid:92) ≤ j ≤ M,j (cid:54) = l F j \ F l ⊂ P \ F l ⊂ P \ F and u ∈ F = (cid:92) F j . Thus for q ∈ ( F ∗ ) ◦ and q l ∈ ( F ∗ l ) ◦ , (cid:104) q , y − u (cid:105) > (cid:104) q l , y − u (cid:105) > . (4.17)Since y , u ∈ F j and q j ∈ ( F ∗ j ) ◦ for all j ∈ { , · · · , M } \ { l } , (cid:104) q j , y − u (cid:105) = 0 . (4.18)By (4.17)-(4.18) in (4.16), we obtain that s l > 0. Similarly s j > ≤ j ≤ M .Therefore q ∈ CoSp ◦ ( { q j } Mj=1 ). Thus ( F ∗ ) ◦ ⊂ CoSp ◦ ( { q j } Mj=1 ) . (cid:3) We note that F ∗ is translation-invariant in the following sense. Lemma 4.6. Let m ∈ V . Then [( F + m ) ∗ ] ◦ | ( P + m , V ) = ( F ∗ ) ◦ | ( P , V ) .Proof. Note that q ∈ [( F + m ) ∗ ] ◦ | ( P + m , V ) if and only if there exists ρ such that (cid:104) q , u + m (cid:105) = ρ < (cid:104) q , y + m (cid:105) for u + m ∈ F + m and y + m ∈ ( P + m ) \ ( F + m ) , that is equivalent to the following: ∃ ρ (cid:48) = ρ − (cid:104) q , m (cid:105) such that (cid:104) q , u (cid:105) = ρ (cid:48) < (cid:104) q , y (cid:105) for u ∈ F and y ∈ P \ F which means that q ∈ ( F ∗ ) ◦ | ( P , V ). (cid:3) ULTIPLE HILBERT TRANSFORMS 29 Proof of Proposition 4.2. We rewrite (4.7) and (4.8) as F = (cid:92) π ∈ Π a ( F ) F π (cid:92) (cid:92) π ∈ Π b F π = (cid:92) π ∈ Π a ( F ) F π (4.19)where(1) Π a ( F ) = { π q j } (cid:96)j =1 ⊂ Π a , F π = π ∩ P = π (cid:48) ∩ P a facet of P with π (cid:48) = π ∩ V am ( P ) ∈ Π (cid:48) a (2) Π b = { π ± n i , ± s i } n − mi =1 , F π = π ∩ P = P where V ( P ) = Sp ⊥ ( { n i } n − mi =1 ) . We claim that F has a cone of the following form:( F ∗ ) ◦ | P = CoSp ◦ ( { q j : j = 1 , · · · , (cid:96) } ) ⊕ V( P ) ⊥ = CoSp ◦ ( { q j : j = 1 , · · · , (cid:96) } ) ⊕ CoSp ◦ ( { n i , − n i } n − mi =1 ) . By (2.1), ∃ m ∈ V such that V am ( P ) = m + V ( P ) . We first work with m = 0. By of (4.2) of Lemma 4.1, we regard P as a polyhedron P (Π (cid:48) a )defined in V am ( P ). Thus by (1) of (4.19) and Lemma 4.5,( F ∗ ) ◦ | P , V ( P )) = CoSp ◦ ( { q j : j = 1 , · · · , (cid:96) } ) . This means that q ∈ CoSp ◦ ( { q j : j = 1 , · · · , (cid:96) } ) if and only if q ∈ ( F ∗ ) ◦ | ( P , V ( P )), that is, q ∈ V ( P ) and r such that (cid:104) q , u (cid:105) = r < (cid:104) q , y (cid:105) for all u ∈ F , y ∈ P \ F . By this combined with (cid:104) n , u (cid:105) = (cid:104) n , y (cid:105) = 0 for all n ∈ V ( P ) ⊥ and u , y ∈ V ( P ) , we see that q ∈ CoSp ◦ ( { q j : j = 1 , · · · , (cid:96) } ) ⊕ V ( P ) ⊥ if and only if ∃ q ∈ V ( P ) ⊕ V ( P ) ⊥ = R n and r such that (cid:104) q , u (cid:105) = r < (cid:104) q , y (cid:105) for all u ∈ F , y ∈ P \ F . Hence we have for a proper face F ,( F ∗ ) ◦ | ( P , R n ) = CoSp ◦ ( { q j : j = 1 , · · · , (cid:96) } ) ⊕ V ⊥ ( P ) . (4.20)The case m (cid:54) = 0 follows from the case m = 0 in (4.20) and Lemma 4.6. Similarly, F ∗ | ( P , R n ) = CoSp( { q j : j = 1 , · · · , (cid:96) } ) ⊕ V ⊥ ( P ) . (4.21) We finished the proof of Proposition 4.2. (cid:3) Remark 4.3. By (2) of (4.19), an improper face P has the expression that P = (cid:92) π ∈ Π b F π = (cid:92) π ∈ Π b π ∩ P where Π b = { π ± n i } n − mi =1 .Then we see that P ∗ | P = V ⊥ ( P ) = CoSp ( {± n i } n − mi =1 ) and ( P ∗ ) ◦ | P = V ⊥ ( P ) \ { } . (4.22) This accords with Definition 2.10 together with (4.20) and (4.21). Finally, when P = P (Π) with Π = { p j } Nj =1 , we take F ∗ = CoSp ( { p j } Nj =1 ) if F = ∅ . In Example 4.1, we construct the faces and cones for the Newton Polyhedrons N (Λ )and N (Λ ) associated with a polynomial P Λ ( t ) = ( P Λ ( t , t , t ) , P Λ ( t , t , t )) and checkthe hypotheses of Main Theorem 2 for n = 3 and d = 2 with S = { , , } . Example 4.1. Consider the polynomial P Λ ( t ) = ( c m t m + c n t n , c m t m + c n t n ) where Λ = { m = (0 , , , n = (3 , , } , Λ = { m = (0 , , , n = (3 , , } . Normal vectors { q νj } j =1 of facets of N (Λ ν ) for ν = 1 , are q νj = e j for j = 1 , , , q ν = (2 , , √ , q = (0 , , √ , and q = (0 , , √ .See Figure 3, where normal vectors q νj are written without the superscript ν = 1 forsimplicity. All the faces of N (Λ ν ) for ν = 1 , are written as F ( N (Λ ν )) = (cid:110) F ( q νj ) = π q νj ∩ N (Λ ν ) : j = 1 , · · · , (cid:111) , F ( N (Λ ν )) = (cid:8) F ( q νi ) ∩ F ( q νj ) : ( i, j ) = (1 , , (1 , , (2 , , (3 , , (3 , , (4 , (cid:9) , F ( N (Λ ν )) = { F ( q ν ) ∩ F ( q ν ) ∩ F ( q ν ) ∩ F ( q ν ) , F ( q ν ) ∩ F ( q ν ) ∩ F ( q ν ) } = { m ν , n ν } . Cones of 0-faces (vertices) are (cid:0) F ( N (Λ ν )) (cid:1) ∗ = { m ∗ ν = CoSp ( q ν , q ν , q ν , q ν ) and n ∗ ν = CoSp ( q ν , q ν , q ν ) } . Cones of 1-faces (edges) are (cid:0) F ( N (Λ ν )) (cid:1) ∗ = (cid:8) [ F ( q νi ) ∩ F ( q νj )] ∗ = CoSp (cid:0) q νi , q νj (cid:1)(cid:9) . ULTIPLE HILBERT TRANSFORMS 31 Faces of Cones of and ( ) Λ N *1 )]([ qF *2 )]([ qF *4 )]([ qF *5 )]([ qF *3 )]([ qF *25 )]([ qF *345 )]()()([ qFqFqF )( qF )( qF )( qF )( qF )( qF )()()( qFqFqF ( ) Λ N ( ) Λ N Figure 3. Faces and their cones of N (Λ ) and N (Λ ). Cones of 2-faces are (cid:0) F ( N (Λ ν )) (cid:1) ∗ = (cid:8) ( F ( q νj )) ∗ = CoSp (cid:0) q νj (cid:1)(cid:9) . All possible combinations (cid:83) ν F ν ∩ Λ ν with rank ( (cid:83) ν F ν ) ≤ are even sets except the fol-lowing two odd set: (1) odd set { n , m } , (2) odd set { m , n , m } .We can check the following in view of Figure 3, where we add the cone ( F ( q )) ∗ = CoSp ( q ) for the face F ( q ) of N (Λ ) .From { n , m } where n ∗ = CoSp (cid:0) q , q , q (cid:1) and m ∗ = CoSp (cid:0) q , q , q , q (cid:1) , ( n ∗ ) ◦ ∩ ( m ∗ ) ◦ = ∅ and n ∗ ∩ m ∗ = CoSp (cid:18) (2 , , √ (cid:19) . From { m n , m } where m n ∗ = CoSp (cid:0) q , q (cid:1) and m ∗ = CoSp (cid:0) q , q , q , q (cid:1) , ( m n ∗ ) ◦ ∩ ( m ∗ ) ◦ = ∅ and m n ∗ ∩ m ∗ = CoSp (cid:18) (2 , , √ (cid:19) . As we point out in Remark 3.4, it is not just cones (cid:84) F ∗ ν , but their interiors (cid:84) ( F ∗ ν ) ◦ thatsatisfy the overlapping condition (3.3). Thus even if { n , m } and { m , n , m } are oddsets, it does not prevent the uniform boundedness of the integrals: sup r j ∈ (0 , ,ξ ∈ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:81) ( − r j ,r j ) e iξ ( c m t m + c n t n )+ ξ ( c m t m + c n t n ) dt t dt t dt t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C. Representations of Unbounded Faces.Lemma 4.7. Let Λ ⊂ Z n + and S ⊂ S ⊂ N n . Suppose that F ∈ F ( N (Λ , S )) such that q = ( q j ) ∈ ( F ∗ ) ◦ | N (Λ , S ) where q j = 0 if j ∈ S and q j > if j ∈ S \ S . Then, F = F + R S + , (4.23) and F = N (Λ ∩ F , S ) . (4.24) Here S can be an empty set.Proof of (4.23). Since 0 ∈ R S + , F ⊂ F + R S + . Let m + (cid:80) j ∈ S a j e j ∈ F + R S + where m ∈ F .Assume that m + (cid:80) j ∈ S a j e j ∈ N (Λ , S ) \ F . By Definition 2.10, (cid:104) m , q (cid:105) < (cid:42) m + (cid:88) j ∈ S a j e j , q (cid:43) for q ∈ ( F ∗ ) ◦ | N (Λ , S ) , which is impossible because q j = 0 for j ∈ S in the hypothesis. Thus m + (cid:88) j ∈ S a j e j ∈ F . This implies that F + R S + ⊂ F . (cid:3) ULTIPLE HILBERT TRANSFORMS 33 Proof of (4.24). By definition, N (Λ ∩ F , S ) is the smallest convex set containing (Λ ∩ F ) + R S + . In view of (4.23), F contains the set set (Λ ∩ F ) + R S + . Thus, N (Λ ∩ F , S ) ⊂ F . We next show that F ⊂ N (Λ ∩ F , S ) . Let x ∈ F ⊂ N (Λ , S ) = Ch(Λ + R S + ). Then, x = (cid:88) m ∈ Ω c m m with Ω is a finite subset of Λ + R S + where (cid:80) m ∈ Ω c m = 1 and c m > 0. Assume that m ∈ Ω ∩ F c (cid:54) = ∅ . Then by Definition 2.10,for q = ( q j ) ∈ ( F ∗ ) ◦ | N (Λ , S ), (cid:104) m , q (cid:105) > (cid:104) x , q (cid:105) . Thus (cid:104) x , q (cid:105) = (cid:88) m ∈ F ∩ Ω c m (cid:104) m , q (cid:105) + (cid:88) m ∈ F c ∩ Ω c m (cid:104) m , q (cid:105) > (cid:32) (cid:88) m ∈ Ω c m (cid:33) (cid:104) x , q (cid:105) = (cid:104) x , q (cid:105) which is a contradiction. So Ω ∩ F c = ∅ . Hence x = (cid:88) m ∈ Ω c m m where Ω ⊂ F ∩ (cid:0) Λ + R S + (cid:1) . (4.25)Here each m ∈ Ω ⊂ F ∩ (cid:0) Λ + R S + (cid:1) above is expressed as m = z + (cid:88) j ∈ S a j e j ∈ F where z ∈ Λ and a j ≥ . (4.26)By Definition 2.10, for q = ( q j ) ∈ ( F ∗ ) ◦ | N (Λ , S ), m ∈ F and z ∈ N (Λ , S ), (cid:42) z + (cid:88) j ∈ S a j e j , q (cid:43) ≤ (cid:104) z , q (cid:105) . (4.27)If z ∈ N (Λ , S ) \ F , then the inequality in (4.27) is strict. This is impossible because q j with j ∈ S in q is nonnegative in the above hypothesis. Thus z ∈ F in (4.26). Moreover a j = 0 for j ∈ S \ S in (4.27) because q j > j ∈ S \ S . Therefore z ∈ F ∩ Λ and j ∈ S in (4.26). Hence, in (4.25), x = (cid:88) m ∈ Ω c m m where Ω ⊂ ( F ∩ Λ) + R S + and (cid:88) m ∈ Ω c m = 1 , that is x ∈ Ch(( F ∩ Λ) + R S + ) = N ( F ∩ Λ , S ), which implies F ⊂ N (Λ ∩ F , S ) . (cid:3) PBF )( qqqChBF )( ppChB ),,,( qqqqCh q pq q q q p )( qqqChp p ),,,( qqqqCh p p q q q )( ppChB q q q q p p p ),,,( qqqqCoSpF ),()1( qqCoSpF ),()2( qqCoSpF ),,,()3( qqqqCoSpF Figure 4. Essential Faces.4.5. Essential Faces. In constructing a sequence { F ∗ ν ( s ) } Ns =0 in (1.10), we need the fol-lowing concept of faces. Definition 4.2. Let P be a polyhedron such that B ⊂ P . Then a set F ( B | P ) is defined tobe the smallest face of P containing B in the sense that B ⊂ F ( B | P ) (cid:22) P and B (cid:42) G for any G (cid:22) F ( B | P ) . We call F ( B | P ) the essential face of P containing B . See the first and second pictures inFigure 4. Lemma 4.8. Let P be a polyhedron such that B ∩ P ◦ (cid:54) = ∅ . Then F ( B | P ) = P . ULTIPLE HILBERT TRANSFORMS 35 Proof. We see that F ( B | P ) (cid:22) P . Assume that F ( B | P ) (cid:22) P . Then B ⊂ F ( B | P ) ⊂ ∂ P . Thisis a contradiction to the hypothesis B ∩ P ◦ (cid:54) = ∅ . (cid:3) Lemma 4.9. Let P be a polyhedron such that B ⊂ P . Then ( F ( B | P )) ◦ ∩ Ch ( B ) (cid:54) = ∅ . Proof. If not, Ch( B ) ⊂ ∂F ( B | P ). By Lemma 2.3, Ch( B ) ⊂ G (cid:22) F ( B | P ), which is impossi-ble by Definition 4.2. (cid:3) Lemma 4.10. Let P be a polyhedron in R n and let B ⊂ P be a convex set. Then B ◦ ⊂ F ( B | P ) ◦ . Proof. We need the following observation: If two affine spaces V and V meet at z ∈ V ∩ V with V (cid:42) V (transversally), then B V ( z, (cid:15) ) ∩ ( V +2 ) ◦ (cid:54) = ∅ and B V ( z, (cid:15) ) ∩ ( V − ) ◦ (cid:54) = ∅ for any (cid:15) > B V ( z, (cid:15) ) = { v ∈ V : | v − z | < (cid:15) } is an (cid:15) -neighborhood of z in V . Let z ∈ B ◦ .Then we show that z ∈ ( F ( B | P )) ◦ . Since z ∈ B ◦ ⊂ B ⊂ F ( B | P ), it suffices to prove that z ∈ ∂F ( B | P ) leads to a contradiction that z / ∈ B ◦ . If z ∈ ∂F ( B | P ), then by Definition 2.8, z ∈ G (cid:22) F ( B | P ) where G ⊂ ∂F ( B | P ). Let V am ( G ) be the plane containing G withdim( V am ( G )) = k − ≤ k = dim( F ( B | P )) and F ( B | P ) ⊂ V + am ( G ) . (4.29)By Definition 4.2 and Lemma 4.9, B ∩ F ( B | P ) ◦ (cid:54) = ∅ , that is, V am ( B ) (cid:42) V am ( G ). From z ∈ B ◦ and z ∈ G , it follows that z ∈ V am ( B ) ∩ V am ( G ). By (4.28) and (4.29) with V = V am ( B ) and V = V am ( G ), B V am ( B ) ( z, (cid:15) ) (cid:42) F ( B | P ) , which implies that B V am ( B ) ( z, (cid:15) ) (cid:42) B for any (cid:15) > z / ∈ B ◦ . (cid:3) Preliminaries Estimates In this section, we prove Proposition 5.1, which is an elementary tool for the L p esti-mation driven by the finite type conditions in the same spirit of [6] and [21]. Proposition5.1 and Proposition 3.1 are two basic L p estimation tools used for the proof of sufficiencyparts of Main Theorems 1-3. Preliminary Inequalities. Under the same setting as in the definition of multipleHilbert transforms (1.1), we consider the multi-parameter maximal function(5.1) M Λ f ( x ) = sup r , ··· ,r n > r · · · r n (cid:90) r − r · · · (cid:90) r n − r n | f ( x − P Λ ( t )) | dt defined for each locally integrable function f on R d . Theorem 5.1. For < p ≤ ∞ , M Λ is a bounded operator from L p ( R d ) into itself andthere exists a bound C p depending only on p, n, d and the maximal degree of the polynomials P ν such that (cid:107) M Λ f (cid:107) L p ( R d ) ≤ C p (cid:107) f (cid:107) L p ( R d ) . Remark 5.1. This result can be proved by combining a theorem of Ricci and Stein ( [17] ,Theorem 7.1) and the so-called lifting argument (see Chapter 11 of [18] ). Remark 5.2. B. Street in [20] showed the L p boundedness for a variable coefficient versionof M Λ associated with analytic functions. Furthermore, A. Nagel and M. Pramanik in [11] obtain the L p boundedness for a different kind of multi-parameter maximal operators,that were motivated by the study of several complex variables. This maximal average istaken over family of sets (balls) that are defined by finite number of monomial inequalities.In particular, to establish the L p theory in [11] , the geometric properties of the associatedpolyhedra are also systematically studied. Take a function ψ ∈ C ∞ c ([ − , ≤ ψ ≤ ψ ( u ) = 1 for | u | ≤ / . Put η ( u ) = ψ ( u ) − ψ (2 u ) . Given an integer k ∈ Z and α, β, γ ∈ { , · · · , n } , we considerthe measures A α,βk and P γk defined in terms of Fourier transforms(5.2) (cid:16) A α,βk (cid:17)(cid:98) ( ξ ) = ψ (cid:18) ξ α k ξ β (cid:19) , (cid:0) P γk (cid:1)(cid:98) ( ξ ) = η (cid:16) k ξ γ (cid:17) . Lemma 5.1. Suppose that { m k } Mk =1 , { q j } Nj =1 ⊂ Z n where rank (cid:104) { q j } Nj =1 (cid:105) = n . Given α k , β k , γ j ∈ R , define (5.3) A J = A α ,β J · m ∗ · · · ∗ A α M ,β M J · m M and P J = P γ J · q ∗ · · · ∗ P γ N J · q N ULTIPLE HILBERT TRANSFORMS 37 for each J ∈ Z n . Then for < p < ∞ , (5.4) (cid:13)(cid:13)(cid:13)(cid:13) (cid:32) (cid:88) J ∈ Z n | P J ∗ f | (cid:33) / (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) , and (5.5) (cid:13)(cid:13)(cid:13)(cid:13) (cid:32) (cid:88) J ∈ Z n | A J ∗ P J ∗ f | (cid:33) / (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) . Proof. It suffices to deal with the sum over Z n + . We show (5.5). With ( r J ( t )) denoting theRademacher functions of product form, (cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Z n + | A J ∗ P J ∗ f | / (cid:13)(cid:13)(cid:13)(cid:13) pL p ( R d ) ≈ (cid:90) U (cid:13)(cid:13) (cid:88) J ∈ Z n + r J ( t ) A J ∗ P J ∗ f (cid:13)(cid:13) pL p ( R d ) dt where U = [0 , n . Consider the symbol m ( ξ ) = (cid:18) (cid:88) J ∈ Z n + r J ( t ) A J ∗ P J (cid:19) (cid:98) ( ξ ) . Using the full rank condition for the q j and the support conditions, it can be shown that m satisfies (cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:96) ∂ ν · · · ∂ ν (cid:96) m ( ξ , · · · , ξ d ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:96) | ξ ν | · · · | ξ ν (cid:96) | for every (cid:96) = 1 , · · · , d , where 1 ≤ ν < · · · < ν (cid:96) ≤ d . Thus the desired conclusion followsfrom the multi-parameter Marcinkiewicz multiplier theorem. (5.4) follows similarly. (cid:3) Lemma 5.2. Let ( σ J ) J ∈ Z n be a sequence of positive measures on R d with the followingproperties : (i) (cid:13)(cid:13) σ J ∗ f (cid:13)(cid:13) L ( R d ) (cid:46) (cid:107) f (cid:107) L ( R d ) ( J ∈ Z n )(ii) (cid:13)(cid:13)(cid:13)(cid:13) sup J ∈ Z n | σ J ∗ f | (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) for some < p ≤ . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:32) (cid:88) J ∈ Z n | σ J ∗ f J | (cid:33) / (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:13)(cid:13)(cid:13)(cid:13)(cid:32) (cid:88) J ∈ Z n | f J | (cid:33) / (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) for p determined by /p ≤ / /p ) . Proof. For 1 ≤ p, q ≤ ∞ , consider the operator T defined by T [( f J )] = ( σ J ∗ f J ) onthe mixed-norm spaces L p ( (cid:96) q ). The condition (i) implies that T maps L ( (cid:96) ) boundedlyinto itself. The condition (ii) and the positivity of each σ J imply that T maps L p ( (cid:96) ∞ )boundedly into itself. It follows from the vector-valued Riesz-Thorin interpolation theoremthat T maps L p ( (cid:96) ) boundedly into itself. (cid:3) Basic L p estimates.Proposition 5.1. Let { H J } J ∈ Z d be a class of measures such that (cid:98) H J be the Fouriermultiplier of H J . Suppose that (cid:12)(cid:12)(cid:12) (cid:98) H J ( ξ ) (cid:12)(cid:12)(cid:12) ≤ C min (cid:110)(cid:12)(cid:12) − J · q i ξ ν i (cid:12)(cid:12) − δ , (cid:12)(cid:12) − J · q i ξ ν i (cid:12)(cid:12) δ : i = 1 , · · · , N (cid:111) (5.6) where rank { q i : i = 1 , · · · , r } = n. (5.7) Then for C = C/ (1 − − min { δ ,δ } /N ) N with C, δ , δ , N in (5.6), (cid:88) J ∈ Z (cid:12)(cid:12)(cid:12) (cid:98) H J ( ξ ) (cid:12)(cid:12)(cid:12) ≤ C where Z ⊂ Z n (5.8) which implies that for A J of the form defined as in (5.3) and for any Z ⊂ Z n , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) J ∈ Z H J ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R d ) ≤ C (cid:107) f (cid:107) L ( R d ) . Moreover, suppose that < p ≤ ∞ (cid:13)(cid:13)(cid:13)(cid:13) sup J ∈ Z | H J | ∗ f (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) ≤ C (cid:107) f (cid:107) L p ( R d ) . (5.9) Then, for < p ≤ ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) J ∈ Z H J ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) ≤ C p (cid:107) f (cid:107) L p ( R d ) . (5.10) Proof of (5.8). Let (cid:0) P − J · q i − (cid:96) i (cid:1) ∧ ( ξ ) = η (cid:16) − J · q i − (cid:96) i ξ ν i (cid:17) for i = 1 , · · · , N ULTIPLE HILBERT TRANSFORMS 39 with a view to restricting frequency variables as2 − J · q i | ξ ν i | ≈ (cid:96) i for i = 1 , · · · , N . (5.11)We use for (5.6) and (5.11) to obtain that N (cid:89) i =1 η (cid:16) − J · q i − (cid:96) i ξ ν i (cid:17) (cid:12)(cid:12)(cid:12) (cid:98) H J ( ξ ) (cid:12)(cid:12)(cid:12) ≤ C − b | L | for b = min { δ , δ } N and C in (5.6) . (5.12)where | L | = (cid:80) Ni =1 | (cid:96) i | . Then by using positivity of η and (cid:80) (cid:96) i ∈ Z η (cid:0) − J · q i − (cid:96) i ξ ν i (cid:1) = 1, (cid:88) | J |≤ R (cid:12)(cid:12)(cid:12) (cid:98) H J ( ξ ) (cid:12)(cid:12)(cid:12) = (cid:88) L ∈ Z N N (cid:89) i =1 η (cid:16) − J · q i − (cid:96) i ξ ν i (cid:17) (cid:88) | J |≤ R (cid:12)(cid:12)(cid:12) (cid:98) H J ( ξ ) (cid:12)(cid:12)(cid:12) = (cid:88) L ∈ Z N (cid:88) | J |≤ R N (cid:89) i =1 η (cid:16) − J · q i − (cid:96) i ξ ν i (cid:17) (cid:12)(cid:12)(cid:12) (cid:98) H J ( ξ ) (cid:12)(cid:12)(cid:12) ≤ (cid:88) L ∈ Z N C − b | L | ≤ C/ (1 − − min { δ ,δ } /N ) N where the first inequality follows from (5.12) and the observation that for each fixed ξ ,there exists finitely many J such that (cid:81) Ni =1 η (cid:0) − J · q i − (cid:96) i ξ ν i (cid:1) (cid:54) = 0. We proved (5.8). (cid:3) Proof of (5.10). Define P q J,L = P − J · q − (cid:96) ∗ · · · ∗ P − J · q N − (cid:96) N , where L = ( (cid:96) i ) Ni =1 ∈ Z N . We use the Littlewood-Paley decomposition for each J ∈ Z : (cid:88) L ∈ Z N P q J,L ∗ f = f. Define ˜ P q J,L by replacing η ( · ) with η ( · / 2) in (5.11). Then ˜ P q J,L ∗ P q J,L = P q J,L . Thus (cid:88) J ∈ Z H J ∗ A J ∗ P q J,L ∗ f = (cid:88) J ∈ Z ˜ P q J,L ∗ H J ∗ A J ∗ P q J,L ∗ f. By Applying the dual inequality of (5.4) in Lemma 5.1, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) J ∈ Z ˜ P q J,L ∗ H J ∗ A J ∗ P q J,L ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18)(cid:88) J ∈ Z (cid:12)(cid:12)(cid:12) H J ∗ A J ∗ P q J,L ∗ f (cid:12)(cid:12)(cid:12) (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) . Thus, it is sufficient to find a constant b > L ∈ Z N such that(5.13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:18)(cid:88) J ∈ Z (cid:12)(cid:12)(cid:12) H J ∗ A J ∗ P q J,L ∗ f (cid:12)(cid:12)(cid:12) (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) ≤ ˜ C − b | L | (cid:107) f (cid:107) L p ( R d ) , where ˜ C is a multiple of C in (5.6). By the rank condition (5.7) and (5.5) in Lemma 5.1, (cid:13)(cid:13)(cid:13)(cid:13)(cid:16)(cid:88) J ∈ Z | A σJ ∗ P q J,L ∗ f | (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) . (5.14)For p = 2, we use (5.8),(5.12) and (5.14) to obtain (5.13). Applying a standard bootstrapargument combined with (5.9), Lemmas 5.1 and 5.2, we obtain (5.13) for the other valuesof p (cid:54) = 2. The proof of (5.10) is now complete. (cid:3) Remark 5.3. The decay condition in (5.6) always holds for the case that Λ ν ’s are mutuallydisjoint. Given G = ( G ν ) ∈ F ( N (Λ , S )) , we have from the multi-dimensional Van derCorput Lemma, |I J ( P G , ξ ) | ≤ C min (cid:110) | − J · m ν ξ ν | − δ , m ν ∈ G ν ∩ Λ ν for ν = 1 , · · · , d (cid:111) . (5.15) 6. Cone Type Decompositions Cone Decompositions. Recall that Z ( S ) = (cid:81) Z i with Z i = R + for i ∈ S and Z i = R for i ∈ N n \ S as in Definition 2.14. We decompose Z ( S ) into finite number ofdifferent cones that appears in (2.12) and (2.15) as follows: Proposition 6.1. Let Λ = (Λ ν ) with Λ ν ⊂ Z n + and S ⊂ { , · · · , n } . Then, (cid:91) F ∈F ( (cid:126) N (Λ ,S ) ) Cap( F ∗ ) = Z ( S ) where Cap( F ∗ ) = d (cid:92) ν =1 F ∗ ν for F = ( F ν ) ∈ F ( N (Λ , S)) . Moreover, (cid:91) F ∈F ( (cid:126) N (Λ ,S ) ) Cap(( F ∗ ) ◦ ) = Z ( S ) \ { } where Cap(( F ∗ ) ◦ ) = d (cid:92) ν =1 ( F ∗ ν ) ◦ . Lemma 6.1. Let P = P (Π) with Π = { π q j ,r j : j = 1 , · · · , N } be a polyhedron. Then inf {(cid:104) x , e (cid:105) : x ∈ P } > −∞ if and only if e ∈ CoSp( { q j : j = 1 , · · · , N } ) . ULTIPLE HILBERT TRANSFORMS 41 Proof. Let ρ = inf {(cid:104) x , e (cid:105) : x ∈ P } > −∞ and set the plane π e ,ρ = { x ∈ V : (cid:104) x , e (cid:105) = ρ } .Since P is a closed set, F defined by π e ,ρ ∩ P is a non-empty closed set. From ρ ≤ x · e forall x ∈ P and F = π e ,ρ ∩ P ,(6.1) P \ F ⊂ (cid:0) π + e ,ρ (cid:1) ◦ . Thus F (cid:22) P and e ∈ F ∗ ⊂ CoSp( { q j : j = 1 , · · · , N } ) by using Propositions 4.1 and 4.2. Toshow the other direction, let e = (cid:80) Nj =1 c j q j ∈ CoSp( { q j : j = 1 , · · · , N } ). Then (cid:104) e , x (cid:105) = N (cid:88) j =1 c j (cid:104) q j , x (cid:105) ≥ N (cid:88) j =1 c j r j > −∞ for all x ∈ P = (cid:84) Nj =1 {(cid:104) q j , x (cid:105) ≥ r j : j = 1 , · · · , N } . (cid:3) Lemma 6.2. Let P = P (Π) with Π = { π q j ,r j : j = 1 , · · · , N } be a polyhedron. Then CoSp( { q j : j = 1 , · · · , N } ) = (cid:91) F (cid:22) P F ∗ . Proof. We first show ⊂ . Let e = (cid:80) Nj =1 c j q j ∈ CoSp( { q j : j = 1 , · · · , N } ) and let ρ = inf (cid:104) e , x (cid:105) = N (cid:88) j =1 c j (cid:104) q j , x (cid:105) : x ∈ P that exits from Lemma 6.1. Set F = π e ,ρ ∩ P . By (6.1), F is a face of P with a supportingplane π e ,ρ . Thus e ∈ F ∗ . The other direction ⊃ follows from Propositions 4.1 and 4.2. (cid:3) Lemma 6.3. Let u = ( u , · · · , u n ) ∈ F ∗ and F ∈ F ( N (Λ , S )) . Then u j ≥ for all j ∈ S .Proof. Let m ∈ F and j ∈ S . Then m + r e j ∈ N (Λ , S ) for all r ≥ 0. By Definition 2.8 and u ∈ F ∗ , u j r = (cid:104) u , m + r e j − m (cid:105) ≥ 0. Thus u j ≥ 0. See Figure 1. (cid:3) Lemma 6.4. Let S ⊂ N n and Ω ⊂ Z n + be a finite set. Suppose that P = P (Π) with Π = { π q j ,r j : j = 1 , · · · , N } is a polyhedron given by N (Ω , S ) . Then (cid:91) F ∈F ( N (Ω ,S )) F ∗ = Z ( S ) . Moreover, (cid:83) F ∈F ( N (Ω ,S )) ( F ∗ ) ◦ = Z ( S ) \ { } . Proof. It follows ⊂ from Lemma 6.3. We next show ⊃ . Put m k = min { u k : u = ( u , · · · , u n ) ∈ Ω } ,M k = max { u k : u = ( u , · · · , u n ) ∈ Ω } By N (Ω , S ) = Ch { u + R S+ : u ∈ Ω } ,(1) if k ∈ N n \ S , then m k ≤ x k ≤ M k for all x = ( x , · · · , x n ) ∈ N (Ω , S ),(2) if k ∈ S , then m k ≤ x k for all x = ( x , · · · , x n ) ∈ N (Ω , S ).Thus, if k ∈ N n \ S , then e k , − e k ∈ CoSp( { q j : j = 1 , · · · , N } ) by Lemma 6.1 and (1)above. If k ∈ S , then e k ∈ CoSp( { q j : j = 1 , · · · , N } ) by Lemma 6.1 and (2) above. Hence Z ( S ) = CoSp( {± e k : k ∈ N n \ S } ∪ { e k : k ∈ S } ) ⊂ CoSp( { q j : j = 1 , · · · , N } ) . By Lemma 6.2, Z ( S ) ⊂ (cid:83) F ∈F ( N (Ω ,S )) F ∗ . By Definitions 2.8-2.10 together with (4.22), (cid:91) F ∈F ( N (Ω ,S )) ( F ∗ ) ◦ = (cid:91) F ∈F ( N (Ω ,S )) F ∗ \ { } . (6.2)This implies the last statement. (cid:3) Proof of Proposition 6.1. By Lemma 6.4, (cid:91) F ν ∈F ( N (Λ ν ,S )) F ∗ ν = Z ( S ) for every ν = 1 , · · · , d. Hence, by taking an intersection for ν = 1 , · · · , d , (cid:91) F ∈F ( (cid:126) N (Λ ,S ) ) Cap( F ∗ ) = d (cid:92) ν =1 (cid:91) F ν ∈F ( N (Λ ν ,S )) F ∗ ν = Z ( S ) . The last statement follows from (6.2). (cid:3) Projection to Sphere; Boundary Deleted Neighborhood. We show that Proposition 6.2. Suppose that rank (cid:16)(cid:83) dν =1 N (Λ ν , S ) (cid:17) ≤ n − and the hypothesis of MainTheorems 2 holds. Then for < p < ∞ , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Z ( S ) H P Λ J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d )ULTIPLE HILBERT TRANSFORMS 43 where I J ( P Λ , ξ ) = (cid:16) H P Λ J (cid:17) ∧ ( ξ ) . To show Proposition 6.2, we consider the projective cone of F ∗ to the sphere S n − . Definition 6.1. In stead of working with the cone F ∗ of a face F , it is sometimes convenientto work with its intersection F ∗ ∩ S n − with the sphere. We denote it and its boundary by S [ F ∗ ] = F ∗ ∩ S n − and ∂ S [ F ∗ ] = ( ∂ F ∗ ) ∩ S n − . Let K ∈ S n − , then we define the (cid:15) -neighborhood of K by N (cid:15) ( K ) = { x ∈ S n − : | x − y | < (cid:15) for some y ∈ K } . Definition 6.2. [Boundary Deleted (cid:15) -neighborhood of S [ F ∗ ] Let P be a polyhedron in R n of dim( P ) = m < n and let S [ F ∗ ] ∈ S n − with F ∈ F m − k ( P ) where k = 0 , · · · , m . To givesome width to S [ F ∗ ] \ N (cid:15) ( ∂ S [ F ∗ ])), we define a boundary deleted (cid:15) -neighborhood S (cid:15) [ F ∗ ] by S (cid:15) [ F ∗ ] = N (cid:15)/M k ( S [ F ∗ ])) for k = 0 where F ∗ = CoSp(Π b ) = V ⊥ ( P ) , S (cid:15) [ F ∗ ] = N (cid:15)/M k +1 / (cid:16) S [ F ∗ ] \ N (cid:15)/M k − / ( ∂ S [ F ∗ ]) (cid:17) for 1 ≤ k ≤ m where F ∗ = CoSp(Π a ( F )) ⊕ V ⊥ ( P ) = CoSp( { q j } (cid:96)j =1 ) ⊕ V ⊥ ( P ) as in (4.21). Here M willbe chosen to be a large positive number. For the case that dim( P ) = n with F ∈ F m − k ( P )where k = 1 , · · · , n , we define a boundary deleted (cid:15) -neighborhood S (cid:15) [ F ∗ ] by S (cid:15) [ F ∗ ] = N (cid:15)/M k ( S [ F ∗ ])) for k = 1 where F ∗ = CoSp( q j ) , S (cid:15) [ F ∗ ] = N (cid:15)/M k +1 / (cid:16) S [ F ∗ ] \ N (cid:15)/M k − / ( ∂ S [ F ∗ ]) (cid:17) for 2 ≤ k ≤ n where F ∗ = CoSp( { q j } (cid:96)j =1 ). See S (cid:15) [ F ∗ ( q )] and S (cid:15) [ F ∗ ( q , q )] in the right side of Figure 3. Lemma 6.5. Let Ω ⊂ Z n + and P = N (Ω , S ) ⊂ R n be a polyhedron. Then (cid:91) F ∈F ( N (Ω ,S )) S [ F ∗ ] ⊂ (cid:91) F ∈F ( N (Ω ,S )) S (cid:15) [ F ∗ ] . Proof. We prove the case dim( P ) = m < n . By Definition 6.2, (cid:91) F ∈F m ( N (Ω ,S )) S [ F ∗ ] ⊂ (cid:91) F ∈F m ( N (Ω ,S )) S (cid:15) [ F ∗ ] = (cid:91) F ∈F m ( N (Ω ,S )) N (cid:15)/M ( S [ F ∗ ]) . Using this and Definition 6.2, (cid:91) F ∈F m − ( N (Ω ,S )) S [ F ∗ ] ⊂ (cid:91) F ∈F m − ( N (Ω ,S )) S (cid:15) [ F ∗ ] ∪ (cid:91) F ∈F m ( N (Ω ,S )) S (cid:15) [ F ∗ ] . Inductive application of this inclusion completes the proof. (cid:3) Note that by Proposition 6.1, (cid:91) F =( F ν ) ∈F ( (cid:126) N ( P,S ) ) d (cid:92) ν =1 S [ F ∗ ν ] = Z ( S ) ∩ S n − . (6.3)By Lemma 6.5 together with (6.3), we have Lemma 6.6. Let Λ = (Λ ν ) with Λ ν ⊂ Z n + and (cid:126) N (Λ , S ) = ( N (Λ ν , S )) . Then Z ( S ) ∩ S n − ⊂ (cid:91) F =( F ν ) ∈F ( (cid:126) N (Λ ,S )) d (cid:92) ν =1 S (cid:15) [ F ∗ ν ] . Using this we can decompose for sufficiently small (cid:15) > (cid:88) J ∈ Z ( S ) H P Λ J = (cid:88) F ∈F ( (cid:126) N (Λ ,S )) (cid:88) J/ | J |∈ Z ⊂ (cid:84) dν =1 S (cid:15) [ F ∗ ν ] H P Λ J . (6.4)In order to check the overlapping condition (3.3), we need the following lemma. Lemma 6.7. Let F ν ∈ F ( N (Λ ν , S )) for ν = 1 , · · · , d . Then for some sufficiently small (cid:15) > , we have the property that (cid:84) dν =1 S (cid:15) [ F ∗ ν ] (cid:54) = ∅ implies that (cid:84) dν =1 S [( F ∗ ν ) ◦ ] (cid:54) = ∅ .Proof of lemma 6.7. We prove the case dim( P ) = m < n . It suffices to find an (cid:15) > d (cid:92) ν =1 S [( F ∗ ν ) ◦ ] = ∅ implies that d (cid:92) ν =1 S (cid:15) [ F ∗ ν ] = ∅ . Suppose that d -tuple ( F ν ) of faces are given so that d (cid:92) ν =1 S [( F ∗ ν ) ◦ ] = ∅ . (6.5) ULTIPLE HILBERT TRANSFORMS 45 Note that S [ F ∗ ν ] \ N (cid:15) ( ∂ S [ F ∗ ν ]) ⊂ S [( F ∗ ν ) ◦ ] for any positive number (cid:15) > S [ F ∗ ν ] = S [( F ∗ ν ) ◦ ]for dim( F ν ) = m . From this, we splits (6.5) into two smaller parts: (cid:92) ν ; dim( F ν ) ≤ m − (cid:16) S [ F ∗ ν ] \ N (cid:15)/M k ( ν ) − / ( ∂ S [ F ∗ ν ]) (cid:17) (cid:92) (cid:92) ν ; dim( F ν )= m S [ F ∗ ν ] ⊂ d (cid:92) ν =1 S [( F ∗ ν ) ◦ ] = ∅ . (6.6)Since S [ F ∗ ν ] and S [ F ∗ ν ] \ N (cid:15)/M k ( ν ) − / ( ∂ S [ F ∗ ν ]) are closed sets in S n − in (6.6), we take a littlebit thicker intersection in ν = 1 , · · · , d with some large M and small (cid:15) to obtain that (cid:92) ν ; dim( F ν ) ≤ m − N (cid:15)/M k ( ν )+1 / (cid:16) S [ F ∗ ν ] \ N (cid:15)/M k ( ν ) − / ( ∂ S [ F ∗ ν ]) (cid:17) (cid:92) (cid:92) ν ; dim( F ν )= m S (cid:15)/M [ F ∗ ν ] = ∅ . By Definition 6.2, we have (cid:92) ν S (cid:15) [ F ∗ ν ] = ∅ . This proves Lemma 6.7. The case dim( P ) = n follows similarly. (cid:3) Lemma 6.8. Let F be a face of P = N (Ω , S ) with dim ( P ) = m < n . Suppose that ˜ m ∈ F ∩ Ω and m ∈ Ω \ F . Then for all p ∈ S (cid:15) [ F ∗ ] with dim ( F ) = m − k where k = 1 , · · · , m , p · ( m − ˜ m ) ≥ c > where c is independent of p . (6.7) Remark 6.1. We shall use Lemma 6.8 for the estimate of the difference I J ( P Ω , ξ ) −I J ( P F , ξ ) where J/ | J | = p . We do not need Lemma 6.8 if dim ( F ) = m − k with k = 0 ,since F = N (Ω , S ) for the case dim ( F ) = m so that I J ( P Ω , ξ ) − I J ( P F , ξ ) ≡ . For thecase dim ( P ) = n , (6.7) also holds for all S (cid:15) [ F ∗ ] with dim ( F ) = n − k where k = 1 , · · · , n .Proof of Lemma 6.1. By Proposition 4.2, F ∗ = F ∗ | P = CoSp( { q j } (cid:96)j =1 } ∪ {± n i } n − mi =1 )where { q j } (cid:96)j =1 and {± n i } n − mi =1 is defined as in (4.8). Here we can take q j ∈ S n − . Then S [ F ∗ ] = CoSp( { q j } (cid:96)j =1 } ∪ {± n i } n − mi =1 ) ∩ S n − = q ∈ S n − : q = (cid:88) j c j q j + r with c j > where r = (cid:80) n − mi =1 c i, ± ( ± n i ) ∈ V ( P ) ⊥ . Thus, for sufficiently large M ,(6.8) S [ F ∗ ] \ N (cid:15)/M k − / ( ∂ S [ F ∗ ]) ⊂ q ∈ S n − : q = (cid:88) j c j q j + r with c j > (cid:15)M k − / where r ∈ V ⊥ ( P ). By (4.19), F = (cid:96) (cid:92) j F j with F j = π q j ∩ P . From ˜ m ∈ F and m ∈ Ω \ F , m ∈ P \ F k for some k ∈ { , · · · , (cid:96) } and ˜ m ∈ F ⊂ F k . Thus, by Definition 2.10, q k · ( m − ˜ m ) > η k > q j · ( m − ˜ m ) ≥ j = 1 , · · · , (cid:96) (6.9)where η k depends on Ω. Let q ∈ S [ F ∗ ] \ N (cid:15)/M k − / ( ∂ S [ F ∗ ]). Then by (6.8), q = (cid:96) (cid:88) j =1 c j q j + r where c j ≥ (cid:15)/M k − / and r ∈ V ⊥ ( P ).Thus, we use (6.9) and the fact r · ( m − ˜ m ) = 0 (which follows from r ∈ V ⊥ ( P )) to have q · ( m − ˜ m ) = (cid:96) (cid:88) j =1 c j q j · ( m − ˜ m )= c k q k · ( m − ˜ m ) + (cid:96) (cid:88) j =1 ,j (cid:54) = k c j q j · ( m − ˜ m )(6.10) ≥ c k q k · ( m − ˜ m ) + 0 ≥ ( (cid:15)/M k − / ) η k ≥ (cid:15) η/M k − / > η = min { η k : k = 1 , · · · , (cid:96) } . Finally, let p ∈ S (cid:15) [ F ∗ ] = N (cid:15)/M k +1 / (cid:16) S [ F ∗ ] \ N (cid:15)/M k − / ( ∂ S [ F ∗ ]) (cid:17) . Then there exists q ∈ S [ F ∗ ] \ N (cid:15)/M k − / ( ∂ S [ F ∗ ]) satisfying (6.10) and | p − q | < (cid:15)/M k +1 / .For sufficiently large M > 0, we have p · ( m − ˜ m ) ≥ (cid:15) η/ (2 M k − / ), which proves (6.7). (cid:3) ULTIPLE HILBERT TRANSFORMS 47 In view of (6.4), (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Z ( S ) H P Λ J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) ≤ (cid:88) F ∈F ( (cid:126) N (Λ ,S )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J/ | J |∈ (cid:84) dν =1 S (cid:15) [ F ∗ ν ] H P Λ J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) . (6.11)Furthermore, Lemma 6.9. Let P ν = N (Λ ν , S ) be a polyhedron in R n with dim ( P ν ) = m ν and let F ν (cid:22) P ν for each ν = 1 , · · · , d . Suppose that there exists ν ∈ { , · · · , d } such that F ν ∈ F m ν − k ν ( P ν ) with k ν ≥ , that is, F ν (cid:22) P ν . Then, (6.12) (cid:13)(cid:13)(cid:13)(cid:16) H P Λ J − H P F J (cid:17) ∗ f (cid:13)(cid:13)(cid:13) L p ( R d ) ≤ − c | J | (cid:107) f (cid:107) L p ( R d ) for J/ | J | ∈ d (cid:92) ν =1 S (cid:15) [ F ∗ ν ] . Proof. Let B = (cid:110) ν : F ν (cid:22) N (Λ ν , S ), that is, F ν ∈ F m ν − k ν ( P ν ) where k ν ≥ (cid:111) . For each ν ∈ B , choose ˜ m ν ∈ F ν ∩ Λ ν and m ∈ Λ ν \ F ν . By Lemma 6.8, observe that forthere exists β > J/ | J | · ( m − ˜ m ν ) > β for all J/ | J | ∈ S (cid:15) [ F ∗ ν ] with ν ∈ B (6.13)where c is independent of J/ | J | . By (5.15), the Fourier multipliers of H P Λ J (= H P N (Λ ,S ) J )and H P F J are(6.14) |I J ( P Λ , ξ ) | , |I J ( P F , ξ ) | (cid:46) min (cid:26) (cid:12)(cid:12)(cid:12) − J · ˜ m ν ξ ν a ν ˜ m ν (cid:12)(cid:12)(cid:12) − δ : ˜ m ν ∈ F ν , ν = 1 , · · · , d (cid:27) . By the mean value theorem, |I J ( P Λ , ξ ) − I J ( P F , ξ ) | (cid:46) (cid:88) ν ∈ B (cid:88) m ∈ Λ ν \ F ν (cid:12)(cid:12) − J · m ξ ν a ν m (cid:12)(cid:12) δ . By (6.13)-(6.15),sup ξ |I J ( P Λ , ξ ) − I J ( P F , ξ ) | (cid:46) (cid:88) m ∈ Λ ν \ F ν (cid:12)(cid:12)(cid:12) − J · ( m − ˜ m ν ) (cid:12)(cid:12)(cid:12) δ/ (cid:46) − βδ | J | / . This implies that (6.12) holds for p = 2. Interpolation with p = 1 or p = ∞ yields therange 1 < p < ∞ . (cid:3) We sum up (6.12) of Lemma (6.9) to obtain the following lemma. Lemma 6.10. (6.15) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Z ( S ) H P Λ J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) + (cid:88) F ∈F ( N (Λ ,S )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J/ | J |∈ (cid:84) dν =1 S (cid:15) [ F ∗ ν ] H P F J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) . By using Lemma 6.10, we are now able to obtain Proposition 6.2: Under the assumptionrank (cid:32) d (cid:91) ν =1 N (Λ ν , S ) (cid:33) ≤ n − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Z ( S ) H P Λ J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) . Proof of Proposition 6.2. Since there are finitely many F = ( F ν ) ∈ F ( (cid:126) N (Λ , S )) in (6.15),it suffice to work with one fixed F on the right hand side. By (6.15) and Lemma 6.7, itsuffices to show that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J/ | J |∈ (cid:84) dν =1 S (cid:15) [ F ∗ ν ] H P F J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) only if (cid:92) S [( F ∗ ν ) ◦ ] (cid:54) = ∅ . Note that(1) rank (cid:16)(cid:83) dν =1 F ν (cid:17) ≤ rank (cid:16)(cid:83) dν =1 N (Λ ν , S ) (cid:17) ≤ n − (cid:84) dν =1 ( F ∗ ν ) ◦ ⊃ (cid:84) dν =1 S [( F ∗ ν ) ◦ ] (cid:54) = ∅ .From this and the evenness hypothesis of Main Theorem 2, it follows that (cid:83) dν =1 F ν ∩ Λ ν is an even set. Thus I J ( P F , ξ ) ≡ J . (cid:3) Sufficiency Theorem. We shall prove the sufficient part of Main Theorems 1 and2 by showing Theorem 6.1 below. Let Λ = (Λ ν ) dν =1 with Λ ν ⊂ Z n + and S ⊂ N n . To each F ∈ F ( (cid:126) N (Λ , S )) and J ∈ Z n , we recall (2.11):(6.17) I J ( P F , ξ ) = (cid:90) e i (cid:16) ξ (cid:80) m ∈ F ∩ Λ1 c m − J · m t m + ··· + ξ d (cid:80) m ∈ F d ∩ Λ d c d m − J · m t m (cid:17) (cid:89) h ( t ν ) dt · · · dt n where I J ( P (cid:126) N (Λ ,S ) , ξ ) = I J ( P Λ , ξ ). Then I J ( P F , ξ ) is the Fourier multiplier of the operator f → H F J ∗ f. ULTIPLE HILBERT TRANSFORMS 49 Theorem 6.1. Let Λ = (Λ ν ) dν =1 with Λ ν ⊂ Z n + and S ⊂ { , · · · , n } . Suppose that for G = ( G ν ) ∈ F ( (cid:126) N (Λ , S )) , |I J ( P G , ξ ) | ≤ C min (cid:110) | − J · m ν ξ ν | − δ : m ν ∈ G ν ∩ Λ ν for ν = 1 , · · · , d (cid:111) . (6.18) Suppose that d (cid:91) ν =1 ( F ν ∩ Λ ν ) is an even set for F ∈ F lo ( (cid:126) N (Λ , S )) where F lo ( (cid:126) N (Λ , S )) is defined in Definition 3.3. Then for any F ∈ F ( (cid:126) N (Λ , S )) , (cid:88) J ∈ Cap( F ∗ ) |I J ( P Λ , ξ ) | ≤ C where Cap( F ∗ ) = d (cid:92) ν =1 F ∗ ν , (6.19) and for < p < ∞ , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Z ⊂ Cap( F ∗ ) H P Λ J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) ≤ C p (cid:107) f (cid:107) L p ( R d ) . (6.20)By (5.15), the condition (6.18) is satisfied if Λ ν ’s are mutually disjoint. Thus, Theorem6.1 together with Proposition 6.1 immediately leads the sufficient part of Main Theorems1 and 2. Remark 6.2. In the above, C in (6.19) is majorized by C R (cid:89) ν (cid:89) m ∈ Λ ν ( | c ν m | + 1 / | c ν m | ) /R for some large R. (6.21) 7. Descending Faces v.s. Ascending Cones Suppose that we are given F = ( F ν ) ∈ F ( P ) with P = ( P ν ) where P ν = N (Λ ν , S ).To establish (6.19), as we have planed in (1.9), (1.10) and (2.15), we shall choose anappropriate descending chain { F ( s ) : s = 0 , · · · , N } in F ( P ) such that P = F (0) (cid:23) · · · (cid:23) F ( s ) (cid:23) · · · (cid:23) F ( N ) = F ( F ν ( s − (cid:23) F ν ( s ) for each ν ) . (7.1)We shall make the estimates: (cid:88) J ∈ Cap( F ∗ ) (cid:12)(cid:12) I J ( P F ( s − , ξ ) − I J ( P F ( s ) , ξ ) (cid:12)(cid:12) ≤ C for s = 1 , · · · , N. (7.2) To perform this estimates successfully, we need to have the full rank condition for applyingProposition 5.1: rank (cid:32) d (cid:91) ν =1 F ν ( s − (cid:33) = n. (7.3)Without the full rank condition, we need to have the overlapping condition for applyingProposition 3.1: Cap( F ∗ ( s ) ◦ ) = d (cid:92) ν =1 ( F ∗ ν ( s )) ◦ (cid:54) = ∅ . (7.4)The following technical difficulty arises for each (7.3) and (7.4). Difficulty satisfying overlapping property (7.4) . By Lemma 2.4, we see thatCap( F ∗ ( s − (cid:54) = ∅ ⇒ Cap( F ∗ ( s )) (cid:54) = ∅ whenever F ν ( s − (cid:23) F ν ( s ) for all ν .However, Cap( F ∗ ( s − ◦ ) (cid:54) = ∅ ⇒ Cap( F ∗ ( s ) ◦ ) (cid:54) = ∅ is not always true even if F ν ( s − (cid:23) F ν ( s ) for all ν . To keep (7.4), we construct (7.1) in Definition 7.2 so that Cap( F ∗ ( s ) ◦ )with every s = 1 , · · · , N contains some common portion of Cap( F ∗ ) in (7.5). For this, weuse the concept of the essential faces as defined in Definition 4.2. Difficulty satisfying the full rank condition (7.3) . Even if we have (7.3), we mighthave rank (cid:32) d (cid:91) ν =1 F ν ( s − ∩ Λ ν (cid:33) ≤ n − . For this case, in order to satisfy (5.6) in Proposition 5.1, (cid:12)(cid:12) I J ( P F ( s − , ξ ) − I J ( P F ( s ) , ξ ) (cid:12)(cid:12) in(7.2) must be dominated by | − J · m ξ ν | c not only with m ∈ F ν ( s − ∩ Λ ν exponents ofpolynomial P Λ , but also with m ∈ F ν ( s − 1) not exponents of that polynomial. To fulfillthis requirement, we shall make an efficient size control tool for (cid:8) − J · m : m ∈ F ν ( s ) (cid:9) Ns =1 with J ∈ Cap( F ∗ ) fixed , in Proposition 7.2. ULTIPLE HILBERT TRANSFORMS 51 Construction of Descending Faces and Ascending Cones. Given a a face F = ( F ν ) ∈ F ( P ), an intersection (cid:84) dν =1 F ∗ ν of cones is itself a cone type polyhedron. Thusthere exist p , · · · , p N in (cid:84) dν =1 F ∗ ν :(7.5) Cap( F ∗ ) = d (cid:92) ν =1 F ∗ ν = CoSp( p , · · · , p N ) . In order to show (6.19) and (6.20), we first split Cap( F ∗ ) asCap( F ∗ ) = (cid:91) Cap( F ∗ )( σ )where union is over all permutations σ : { , · · · , N } → { , · · · , N } andCap( F ∗ )( σ ) = { α p + · · · + α N p N ∈ Cap( F ∗ ) : α σ (1) ≥ α σ (2) ≥ · · · ≥ α σ (N) ≥ } . To prove (6.19), it suffices to show for each σ , (cid:88) J ∈ Cap( F ∗ )( σ ) |I J ( P Λ , ξ ) | ≤ C . Since the order of p , · · · , p N is random, it suffices to work with only σ = id whereCap( F ∗ )(id) = { α p + · · · + α N p N ∈ Cap( F ∗ ) : α ≥ α ≥ · · · ≥ α N ≥ } . (7.6) Definition 7.1. [Intersection of Cones] Let P = ( P ν ) so that P ν = P (Π ν ) is a polyhedronin R n and dim( P ν ) = dim( V ( P ν )) = m ν ≤ n . Suppose that Π ν = Π νa ∪ Π νb whereΠ νa = { q νj } L ν j =1 is a generator for P ν in V am ( P ), and Π νb = {± n νi } n − m ν i =1 is a generator for V am ( P ν ) in R n as in Lemma 4.1. By Propositions 4.1 and 4.2 with Remark 4.3, a face F ν having an expression: F ν = N ν (cid:92) j =1 π p νj ∩ P ν where { p νj } N ν j =1 = { q νj } (cid:96) ν j =1 ∪ {± n νi } n − m ν i =1 has its cone of the form: F ∗ ν = CoSp (cid:16) { p ν j } N ν j=1 (cid:17) where Π( F ν ) = { p ν j } N ν j=1 . (7.7)Here we remind thatCoSp (cid:0) {± n νi } n − m ν i =1 (cid:1) = V ⊥ ( P ν ) and { q νj } (cid:96) ν j =1 ⊂ V ( P ν ) . (7.8) Lemma 7.1. In proving (6.19), we may assume that Cap( F ∗ ) ∩ ( F ∗ ν ) ◦ (cid:54) = ∅ for all ν. Proof. If the cone Cap( F ∗ ) is given by (cid:84) dν =1 F ∗ ν = { } , the proof of (6.19) is done sincethere is only one term J = 0 in the summation. Thus we assume that the cone Cap( F ∗ )is not { } , that is, (cid:32) d (cid:92) ν =1 F ∗ ν (cid:33) ∩ S n − (cid:54) = ∅ . (7.9)Let (cid:16)(cid:84) dν =1 F ∗ ν (cid:17) ∩ ( F ∗ ν ) ◦ = ∅ , say ν = 1. Then from (cid:84) dν =1 F ∗ ν ⊂ ( F ∗ ) and Definitions 2.8and 2.9, we have (cid:84) dν =1 F ∗ ν ⊂ ∂ F ∗ . By Lemma 2.3 there exists F ∗ , (cid:22) F ∗ (so F (cid:22) F , )such that (cid:84) dν =1 F ∗ ν ⊂ F ∗ , . So we replace F ∗ in (cid:84) dν =1 F ∗ ν by F ∗ , with keeping (7.9). If (cid:16)(cid:84) dν =1 F ∗ ν (cid:17) ∩ ( F ∗ ) ◦ (cid:54) = ∅ where F = F , , we stop. Otherwise (cid:16)(cid:84) dν =1 F ∗ ν (cid:17) ∩ ( F ∗ ) ◦ = ∅ where F = F , , we repeat this process until we have (cid:16)(cid:84) dν =1 F ∗ ν (cid:17) ∩ ( F ∗ ) ◦ (cid:54) = ∅ satisfying (7.9)where F is taken as new F ,k such that F ∗ ,k (cid:22) · · · (cid:22) F ∗ , (cid:22) F ∗ ( F (cid:22) F , (cid:22) · · · (cid:22) F ,k ) . Assume we arrive at the final round with F ,k = P in (7.9). By (7.9) and Remark 4.3that tells ( P ∗ ) ◦ = P ∗ \ { } , (cid:32) P ∗ ∩ (cid:32) d (cid:92) ν =2 F ∗ ν (cid:33)(cid:33) ∩ ( P ∗ ) ◦ = (cid:32) P ∗ ∩ (cid:32) d (cid:92) ν =2 F ∗ ν (cid:33)(cid:33) ∩ P ∗ \ { } (cid:54) = ∅ . Hence our process ends up with (cid:16)(cid:84) dν =1 F ∗ ν (cid:17) ∩ ( F ∗ ) ◦ (cid:54) = ∅ satisfying (7.9) where F is takenpossibly as a face between the original F and the entire P . By applying the sameargument to ν = 2 , · · · , d , we finish the proof. (cid:3) Remark 7.1. Lemma 7.1 combined with Lemma 4.8 tells us Cap ( F ∗ ) is an essential partof F ∗ ν in the sense that F ( Cap ( F ∗ ) | F ∗ ν ) = F ∗ ν . This is used for proving Lemma 7.3. Definition 7.2. [Essential Cone] Suppose that polyhedrons P = ( P ν ) and faces F =( F ν ) are given as in Definition 7.1. Fix the order of { p j } Nj =1 in (7.5) and note that { p j } Nj =1 , {± n νi } n − m ν i =1 ⊂ F ∗ ν . Then for each ν = 1 , · · · , d and s = 0 , , · · · , N , define a cone: C ν ( s ) = CoSp (cid:0) {± n νi } n − m ν i =1 ∪ { p j } sj =1 (cid:1) ⊂ F ∗ ν (7.10) ULTIPLE HILBERT TRANSFORMS 53 where from (7.8) and (7.5) C ν (0) = CoSp (cid:0) {± n νi } n − m ν i =1 (cid:1) = V ⊥ ( P ν ) , (7.11) C ν ( N ) = CoSp (cid:0) {± n νi } n − m ν i =1 ∪ { p j } Nj =1 (cid:1) ⊃ CoSp (cid:0) { p j } Nj =1 (cid:1) = Cap( F ∗ ) . (7.12)In view of Definition 4.2, for each s = 0 , · · · , N and ν = 1 , · · · , d , define the smallest face F ∗ ν ( s ) of F ∗ ν containing C ν ( s ) by F ∗ ν ( s ) = F ( C ν ( s ) | F ∗ ν ) , (7.13)and call F ∗ ν ( s ) the essential cone of F ∗ ν containing C ν ( s ). See the third picture of Figure 4. Lemma 7.2. Suppose that the F ∗ ν ( s ) are defined above. Then for s = 1 , · · · , N , d (cid:92) ν =1 ( F ∗ ν ( s )) ◦ (cid:54) = ∅ . Proof. Fix s ∈ { , · · · , N } . By C ν ( s ) = CoSp (cid:16) {± n νi } n − m ν i =1 ∪ { p j } sj =1 (cid:17) in (7.10), p + · · · + p s = s (cid:88) j =1 p j + (cid:32) n − m ν (cid:88) i =1 n νi + n − m ν (cid:88) i =1 − n νi (cid:33) ∈ C ν ( s ) ◦ (7.14)for each ν = 1 , · · · , d . By Lemma 4.10, C ν ( s ) ◦ ⊂ F ( C ν ( s ) | F ∗ ν ) ◦ = ( F ∗ ν ( s )) ◦ . (7.15)By (7.14) and (7.15), p + · · · + p s ∈ (cid:84) dν =1 ( F ∗ ν ( s )) ◦ for s ≥ (cid:3) Remark 7.2. Lemma 7.2 does not hold for the case s = 0 . However, this initial caseestimate for ( F ν (0)) dν =1 = ( N (Λ ν , S )) dν =1 with the low rank condition (6.16), is alreadyfinished in Proposition 6.2. Lemma 7.3. For each ν = 1 , · · · , d , F ∗ ν ( N ) = F ∗ ν and F ∗ ν (0) = V ⊥ ( P ν ) . Proof. The first identity follows from Lemmas 4.8 and 7.1 together with (7.12) above. Thesecond from (7.11) with V ⊥ ( P ν ) = P ∗ ν in Remark 4.3 and P ∗ ν (cid:22) F ∗ ν in Lemma 2.4. (cid:3) Since F ∗ ν ( s ) is a face of a cone F ∗ ν = CoSp( { p νj } N ν j =1 ), it is also a cone expressed as: F ∗ ν ( s ) = CoSp( { p νj } j ∈ B νs ) where { p νj } j ∈ B νs ⊂ { p νj } N ν j =1 = Π( F ν ) ⊂ Π( P ν ) . (7.16)Here, by Lemma 7.3 with (7.7) and (7.8), F ∗ ν ( N ) = CoSp( { p νj } N ν j =1 ) and F ∗ ν (0) = CoSp( ± n νi } n − m ν i =1 ) . (7.17)By (7.16) combined with (3) of Lemma 2.1, we assign to each F ∗ ν ( s ), a face F ν ( s ) of P ν whose cone is F ∗ ν ( s ): F ν ( s ) = (cid:92) j ∈ B νs (cid:16) π p νj ∩ P ν (cid:17) . By (7.17), F ν (0) = n − m ν (cid:92) i =1 (cid:0) π ± n νi ∩ P ν (cid:1) = P ν and F ν ( N ) = (cid:92) j ∈ B νN (cid:16) π p νj ∩ P ν (cid:17) = F ν . (7.18) Proposition 7.1. For each fixed ν = 1 , · · · , d , we have an ascending sequence { F ∗ ν ( s ) } Ns =0 and a descending sequence { F ν ( s ) } Ns =0 . V ⊥ ( P ν ) = F ∗ ν (0) (cid:22) F ∗ ν (1) (cid:22) · · · (cid:22) F ∗ ν ( N ) = F ∗ ν , P ν = F ν (0) (cid:23) F ν (1) (cid:23) · · · (cid:23) F ν ( N ) = F ν . Proof. Since C ν ( s − ⊂ C ν ( s ) with s ≥ F ∗ ν ( s − (cid:22) F ∗ ν ( s ) . By Lemma 2.4, F ν ( s ) (cid:22) F ν ( s − s = 0 , N are in Lemma 7.3 and (7.18). (cid:3) Size Control Number. Before showing (cid:88) J ∈ Cap( F ∗ )( id ) |I J ( P F ( s − , ξ ) − I J ( P F ( s ) , ξ ) | ≤ C in Section 8, we shall investigate the size of 2 − J · m with m ∈ F ν ( s − \ F ν ( s ) and J ∈ Cap( F ∗ )( id ) = J = N (cid:88) j =1 α j p j : α ≥ · · · ≥ α s ≥ · · · ≥ α N ≥ . ULTIPLE HILBERT TRANSFORMS 55 We assert in Proposition 7.2 that α s ( s = 1 , · · · , N ) is the key number controlling sizes:2 − C α s ≤ − m · J − ˜ m · J = Effect of Mean Value PropertyEffect of Decay Property ≤ − C α s where m ∈ F ν ( s − \ F ν ( s ) and ˜ m ∈ F ν . Here C , C > J . Definition 7.3. Given a cone F ∗ = CoSp( p , · · · , p N ), its r -neighborhood is defined by D r ( F ∗ ) = N (cid:88) j =1 c j p j ∈ F ∗ : c j > r > . We shall use the following three lemmas to prove Proposition 7.2. Lemma 7.4. Suppose that CoSp ( p , · · · , p k ) ◦ ⊂ CoSp ( q , · · · , q N ) ◦ . Then there exists c > depending only on p i , q j with ≤ i ≤ k and ≤ j ≤ N such that D r ( CoSp ( p , · · · , p k )) ⊂ D cr ( CoSp ( q , · · · , q N )) . Proof. From p + · · · + p k ∈ CoSp( p , · · · , p k ) ◦ ⊂ CoSp( q , · · · , q N ) ◦ , we see that p + · · · + p k = N (cid:88) j =1 c j q j where c j > c with c depending on p i ’s . (7.19)Let p ∈ D r (CoSp( p , · · · , p k )). By using (7.19), we split p into two parts p = k (cid:88) j =1 α j p j = k (cid:88) j =1 (cid:16) α j − r (cid:17) p j + r N (cid:88) j =1 c j q j where α j > r. (7.20)Since α j − r/ ≥ r/ > 0, the first term on the right hand side of (7.20) k (cid:88) j =1 (cid:16) α j − r (cid:17) p j ∈ CoSp( p , · · · , p k ) ◦ ⊂ CoSp( q , · · · , q N ) ◦ . Also the second term on the right hand side of (7.20) is r N (cid:88) j =1 c j q j ∈ D cr (CoSp( q , · · · , q N ))because ( r/ c j ≥ rc . So p ∈ D cr (CoSp( q , · · · , q N )). (cid:3) Lemma 7.5. Let P be a polyhedron and let F be an proper face of P . Suppose that ˜ m ∈ F and m ∈ P \ F . Then for all p ∈ D r ( F ∗ ) , p · ( m − ˜ m ) ≥ c > where c depends on r, m , ˜ m . Remark 7.3. This lemma is needed only when F (cid:22) P for the same reason in Remark 6.1.The proof of this lemma is also similar to that of Lemma 6.8.Proof. Let Π( F ) = { p j } Nj =1 = { q j } (cid:96)j =1 ∪ {± n i } n − mi =1 where { q j } (cid:96)j =1 = Π a ( F ) ⊂ Π a and {± n i } n − mi =1 = Π b so that F ∗ | P = CoSp( { q j } (cid:96)j =1 } ∪ {± n i } n − mi =1 ) . (7.21)By (4.19), F = (cid:96) (cid:92) j =1 F j with F j = π q j ∩ P . Since ˜ m ∈ F and m ∈ P \ F , m ∈ P \ F k for some k ∈ { , · · · , (cid:96) } and ˜ m ∈ F ⊂ F k . (7.22)Thus by (7.22) and (2.9) in Definition 2.10, q k · ( m − ˜ m ) > η > q j · ( m − ˜ m ) ≥ j = 1 , · · · , (cid:96). (7.23)where η depends on m , ˜ m . Let p ∈ D r ( F ∗ ). Then p = (cid:80) (cid:96)j =1 c j q j + r where c j ≥ r and r ∈ V ⊥ ( P ) according to Definition 7.3 and (7.21). Thus, by using (7.23) and r · ( m − ˜ m ) = 0, p · ( m − ˜ m ) = (cid:96) (cid:88) j =1 c j q j · ( m − ˜ m )= c k q k · ( m − ˜ m ) + (cid:96) (cid:88) j =1 ,j (cid:54) = k c j q j · ( m − ˜ m ) ≥ c k q k · ( m − ˜ m ) + 0 ≥ rη > c = rη is depending on r, m , ˜ m and independent of p . (cid:3) ULTIPLE HILBERT TRANSFORMS 57 Lemma 7.6. Let F = ( F ν ) ∈ F ( (cid:126) N (Λ , S )) and let F ∗ ν ( s ) where s = 1 , · · · , N be defined as inDefinition 7.2. Then for each s = 1 , · · · , N , every vector p = (cid:80) Nj =1 α j p j ∈ Cap( F ∗ )(id) = (cid:110)(cid:80) Nj=1 α j p j : α ≥ · · · ≥ α N ≥ (cid:111) defined in (7.6) is expressed as p = r ( s ) + r ( s ) + r ( s ) where for each ν = 1 , · · · , d , (1) r ( s ) ∈ F ∗ ν ( s − , (2) r ( s ) = α s u ( s ) where u ( s ) ∈ D r ( F ∗ ν ( s )) for r > independent of α , · · · , α N , (3) r ( s ) = α s +1 p s +1 + · · · + α N p N ∈ F ∗ ν ( N ) .Proof. We express p = (cid:80) Nj =1 α j p j as r ( s ) + r ( s ) + r ( s ) where r ( s ) = ( α − α s ) p + · · · + ( α s − − α s ) p s − , (7.24) r ( s ) = α s ( p + · · · + p s ) , (7.25) r ( s ) = α s +1 p s +1 + · · · + α N p N . (7.26)By (7.10) and (7.24), r ( s ) ∈ F ∗ ν ( s − C ν ( s ) = CoSp (cid:16) {± n νi } n − m − νi =1 ∪ { p j } sj =1 (cid:17) in (7.10), p + · · · + p s = s (cid:88) j =1 p j + (cid:32) n − m ν (cid:88) i =1 n νi + n − m (cid:88) i =1 − n νi (cid:33) ∈ D t ( C ν ( s )) for t = 1 . (7.27)By (7.16), F ∗ ν ( s ) = F ( C ν ( s ) | F ∗ ) = CoSp( { p νj } j ∈ B νs ) . (7.28)By Lemmas 4.10, C ν ( s ) ◦ ⊂ F ( C ν ( s ) | F ∗ ν ) ◦ = CoSp( { p νj } j ∈ B νs ) ◦ . (7.29)By using (7.28),(7.29) and Lemmas 7.4, D t ( C ν ( s )) ⊂ D ct (CoSp( { p νj } j ∈ B νs )) = D ct ( F ∗ ν ( s ))for some c > p νj ’s. By this and (7.27), put u ( s ) = p + · · · + p s ∈ D ct ( F ∗ ν ( s )) . Set r = ct > 0. Note (2) follows from r ( s ) = α s u ( s ) in (7.25). Finally (3) follows from(7.26),(7.10) and (7.13). (cid:3) Using Lemmas 7.5 and 7.6, we obtain Proposition 7.2. Let F ν ( s ) and F ∗ ν ( s ) where ν = 1 , · · · , d and s = 1 , · · · , N be definedas in Definition 7.2. Suppose that p = N (cid:88) j =1 α j p j ∈ Cap( F ∗ )(id) = N (cid:88) j=1 α j p j : α ≥ · · · ≥ α N ≥ , and ˜ m ∈ F ν ( N ) = F ν . Then for s = 1 , · · · , N , there exist C , C > such that p · ( m − ˜ m ) ≥ C α s for m ∈ F ν ( s − \ F ν ( s ) , (7.30) p · ( n − ˜ m ) ≤ C α s for n ∈ F ν ( s − where C , C > are independent of p ∈ Cap( F ∗ )(id) , but may depend on m , ˜ m and n .Proof. By Lemma 7.6, we have p = r ( s ) + r ( s ) + r ( s ) satisfying (1),(2) and (3). Since m ∈ F ν ( s − \ F ν ( s ) and ˜ m ∈ F ν ( N ) ⊂ F ν ( s ), the property (2) of Lemma 7.6 combinedwith Lemma 7.5 yields that u ( s ) · ( m − ˜ m ) > c > , that is r ( s ) · ( m − ˜ m ) ≥ cα s where c is independent of p . (7.32)Since r ( s ) + r ( s ) ∈ F ∗ ν ( N ) where F ∗ ν (1) (cid:22) F ∗ ν ( N ) and ˜ m ∈ F ν ( N ),( r ( s ) + r ( s )) · ( m − ˜ m ) ≥ . (7.33)Thus (7.30) follows from (7.32) and (7.33). Finally, the property (1) of Lemma 7.6 togetherwith the fact n ∈ F ν ( s − 1) and ˜ m ∈ F ν ( N ) ⊂ F ν ( s − 1) yields r ( s ) · ( n − ˜ m ) = 0 . Using this and α s ≥ α s +1 ≥ · · · ≥ α N in (7.25) and (7.26), p · ( n − ˜ m ) = ( r ( s ) + r ( s )) · ( n − ˜ m ) (cid:46) α s which proves (7.31). (cid:3) ULTIPLE HILBERT TRANSFORMS 59 Proof of Sufficiency In this section, we shall finish the proof of Theorem 6.1. Remind (6.17) and write theFourier multiplier for the operator f → H F ( s ) J ∗ f with F ν ( s ) (cid:22) F ν (0) = P ν = N (Λ ν , S ) as I J ( P F ( s ) , ξ ) = (cid:90) e i (cid:80) dν =1 ( (cid:80) m ∈ F ( s,ν ) ∩ Λ ν c ν m − J · m t m ) ξ ν (cid:89) h ( t ν ) dt. Lemma 8.1. For s = 0 , , · · · , N and J ∈ (cid:84) dν =1 F ∗ ν , (cid:12)(cid:12) I J ( P F ( s ) , ξ ) (cid:12)(cid:12) ≤ CK min (cid:110) | − J · m ν ξ ν | − δ : m ν ∈ P ν = N (Λ ν , S ) where ν = 1 , · · · , d (cid:111) (8.1) where K = (cid:89) ν (cid:89) m ∈ Λ ν ( | c ν m | + 1 / | c ν m | ) /δ . Proof. By (6.18) in Theorem 6.1 and F ν ( s ) (cid:23) F ν ( N ) = F ν , for all s = 0 , , · · · , N , (cid:12)(cid:12) I J ( P F ( s ) , ξ ) (cid:12)(cid:12) ≤ C min (cid:110) | c ν ˜ m − J · ˜ m ν ξ ν | − δ : ˜ m ν ∈ F ν ∩ Λ ν (cid:111) ≤ C min (cid:110) | c ν ˜ m − J · ˜ m ν ξ ν | − δ : ˜ m ν ∈ F ν (cid:111) where the second inequality follows from (4.24) in Lemma 4.7. This combined with thefact that for J ∈ F ∗ ν for each ν = 1 , · · · , d ,2 − J · m ν ≤ − J · ˜ m ν where m ν ∈ F ν (0) = P ν and ˜ m ν ∈ F ν ( N ) = F ν ,(8.2)yields (8.1). (cid:3) Choose d vectors˜ m ν ∈ F ν ( N ) ∩ Λ ν = F ν ∩ Λ ν for each ν = 1 , · · · , d. According to (5.2), define for each { α, β } ⊂ { , · · · , d } with α > β , (cid:92) A ( α,β ) J ( ξ ) = ψ (cid:18) − J · ˜ m α ξ α − J · ˜ m β ξ β (cid:19) and (cid:92) A ( β,α ) J ( ξ ) = 1 − (cid:92) A ( α,β ) J ( ξ ) . There are M = (cid:0) d (cid:1) collections of ( α, β ) with α > β in { , · · · , d } . Then1 = (cid:89) ( α,β ) ⊂{ , ··· ,d } , α<β (cid:18) (cid:92) A ( α,β ) J ( ξ ) + (cid:92) A ( β,α ) J ( ξ ) (cid:19) = (cid:88) γ (cid:99) A γJ ( ξ ) where (cid:99) A γJ ( ξ ) = (cid:81) Mk =1 (cid:92) A ( α k ,β k ) J ( ξ ) with γ = (( α k , β k )) Mk =1 and the summation above is overall possible 2 M choices of γ having α k < β k or α k > β k for each k ∈ { , · · · , M } . In orderto show (6.20), we prove that for each γ = (( α j , β j )) Mj =1 , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Cap( F ∗ )(id) H N (Λ ,S ) J ∗ A γJ ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) . (8.3)Note that there exists an n -permutation σ such thatsupp (cid:16) (cid:99) A γJ (cid:17) ⊂ (cid:110) ξ ∈ R d : (cid:12)(cid:12)(cid:12) − J · ˜ m σ (1) ξ σ (1) (cid:12)(cid:12)(cid:12) (cid:46) · · · (cid:46) (cid:12)(cid:12)(cid:12) − J · ˜ m σ ( d ) ξ σ ( d ) (cid:12)(cid:12)(cid:12)(cid:111) . Without loss of generality,supp (cid:16) (cid:99) A γJ (cid:17) ⊂ (cid:110) ξ ∈ R d : | − J · ˜ m ξ | (cid:46) · · · (cid:46) | − J · ˜ m d ξ d | (cid:111) . (8.4)In proving (8.3) it suffices to show that for A J = A γJ satisfying (8.4), (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Cap( F ∗ )(id) H F (0) J ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) (8.5)where F (0) = ( F ν (0)) dν =1 = ( N (Λ ν , S )) = N (Λ , S ). Remark 8.1. From now on, we write A (cid:46) B when A ≤ CB where C is a constantmultiple of the constant K in Lemma 8.1. By Proposition 6.2 and N (Λ ν , S ) = F ν (0), it suffices to assume thatrank (cid:32) d (cid:91) ν =1 F ν (0) (cid:33) = n. (8.6)To show (8.5), by Proposition 7.1, it suffices to prove (8.7) and (8.8) below:If rank (cid:32) d (cid:91) ν =1 F ν ( s − (cid:33) = n , then(8.7) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Cap( F ∗ )(id) (cid:16) H F ( s − J − H F ( s ) J (cid:17) ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) . ULTIPLE HILBERT TRANSFORMS 61 Also, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Cap( F ∗ )(id) H F ( N ) J ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) → L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) . (8.8)We claim that (8.7) and (8.8) imply (8.5). Assume that (8.7) and (8.8) are true. Letrank (cid:16)(cid:83) dν =1 F ν ( s − (cid:17) = n for all s = 1 , · · · , N . Then (8.7) and (8.8) yield that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Cap( F ∗ )(id) H F (0) J ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) . Let rank (cid:16)(cid:83) dν =1 F ν ( s − (cid:17) = n for s = 1 , · · · , r, and rank (cid:16)(cid:83) dν =1 F ν ( r ) (cid:17) ≤ n − 1. Then(8.7) yields that(8.9) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Cap( F ∗ )(id) H F (0) J ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Cap( F ∗ )(id) H F ( r ) J ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) . By (8.6), r ≥ 1. Thus, by Lemma 7.2, we have an overlapping condition d (cid:91) ν =1 ( F ∗ ν ( r )) ◦ (cid:54) = ∅ . From the hypothesis of Theorem 6.1 and the rank conditionrank (cid:32) d (cid:91) ν =1 F ν ( r ) (cid:33) ≤ n − , it follows that (cid:83) dν =1 ( F ν ( r ) ∩ Λ ν ) is an even set. Thus the convolution kernel H F ( r ) J vanishesin the right hand side of (8.9) as its Fourier multiplier I J ( P F ( r ) , ξ ) ≡ Proof of (8.7). Let s ∈ { , · · · , N } fixed. Choose µ ∈ { , · · · , d } such thatrank (cid:32) d (cid:91) ν = µ F ν ( s − (cid:33) = n, (8.10) rank d (cid:91) ν = µ +1 F ν ( s − ≤ n − where (cid:83) dν = µ +1 F ( s − , ν ) ∩ Λ ν = ∅ for the case µ = d . For each s , set F (cid:48) ( s − 1) = ( ∅ , · · · , ∅ , F µ +1 ( s − , · · · , F d ( s − F (cid:48) ( s ) = ( ∅ , · · · , ∅ , F µ +1 ( s ) , · · · , F d ( s )) . In order to show (8.7), we shall prove (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Cap( F ∗ )(id) (cid:16) H F (cid:48) ( s − J − H F (cid:48) ( s ) J (cid:17) ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) , (8.12)and(8.13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Cap( F ∗ )(id) (cid:16) H F ( s − J − H F (cid:48) ( s − J − H F ( s ) J + H F (cid:48) ( s ) J (cid:17) ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) . Proof of (8.12). Note that from (8.11) and F ν ( s ) (cid:22) F ν ( s − d (cid:91) ν = µ +1 F ν ( s − ≤ n − , and rank d (cid:91) ν = µ +1 F ν ( s ) ≤ n − . By Lemma 7.2, for s = 2 , · · · , N , d (cid:92) ν = µ +1 ( F ∗ ν ( s − ◦ (cid:54) = ∅ and d (cid:92) ν = µ +1 ( F ∗ ν ( s )) ◦ (cid:54) = ∅ . Thus d (cid:91) ν = µ +1 ( F ν ( s − ∩ Λ ν ) and d (cid:91) ν = µ +1 ( F ν ( s ) ∩ Λ ν ) are even sets.Thus I J ( F (cid:48) ( s − , ξ ) = I J ( F (cid:48) ( s ) , ξ ) ≡ . We next consider the case for s = 1, that is, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Cap( F ∗ )(id) (cid:16) H F (cid:48) (0) J − H F (cid:48) (1) J (cid:17) ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (8.14) ULTIPLE HILBERT TRANSFORMS 63 where I J ( F (cid:48) (1) , ξ ) ≡ d (cid:91) ν = µ +1 F ν (0) ≤ n − F ν (0) = N (Λ ν , S ) , we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Cap( F ∗ )(id) (cid:16) H F (cid:48) (0) J (cid:17) ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) . Therefore we proved (8.12). (cid:3) Proof of (8.13). We denote by I J ( F ( s − , F ( s ) , ξ ) the Fourier multiplier of (cid:16) H F ( s − J − H F (cid:48) ( s − J − H F ( s ) J + H F (cid:48) ( s ) J (cid:17) ∗ A J . Then I J ( F ( s − , F ( s ) , ξ ) is(8.15) (cid:18)(cid:0) I J ( F ( s − , ξ ) − I J ( F (cid:48) ( s − , ξ ) (cid:1) − (cid:0) I J ( F ( s ) , ξ ) − I J ( F (cid:48) ( s ) , ξ ) (cid:1) (cid:19) (cid:99) A J ( ξ ) . We shall show that for all J ∈ Cap( F ∗ )(id), |I J ( F ( s − , F ( s ) , ξ ) | (cid:46) min (cid:8) | − J · n ν ξ ν | ± (cid:15) : n ν ∈ F ν ( s − (cid:9) dν = µ . (8.16)This combined with the rank condition (8.10) and Proposition 5.1 implies (8.13). To show(8.16), by Lemma 8.1, it suffices to show that |I J ( F ( s − , F ( s ) , ξ ) | (cid:46) min (cid:8) | − J · n ν ξ ν | (cid:15) : n ν ∈ F ν ( s − 1) for (cid:9) dν = µ . (8.17)Thus, the proof of (8.13) is finished if (8.17) is proved. (cid:3) Proof of (8.17). We write I J ( F ( s − , F ( s ) , ξ ) as(8.18) (cid:18) ( I J ( F ( s − , ξ ) − I J ( F ( s ) , ξ )) − (cid:0) I J ( F (cid:48) ( s − , ξ ) − I J ( F (cid:48) ( s ) , ξ ) (cid:1) (cid:19) (cid:99) A J ( ξ ) . By using mean value theorem in (8.18), |I J ( F ( s − , F ( s ) , ξ ) | (cid:46) d (cid:88) ν =1 (cid:88) m ν ∈ ( F ν ( s − ∩ Λ ν ) \ ( F ν ( s ) ∩ Λ ν ) | c ν m − J · m ν ξ ν | . (8.19) By (7.30) of Proposition 7.2, for any m ν ∈ F ν ( s − \ F ν ( s ) and ˜ m ν ∈ F ν ( N ), there existsa constant b > J and α s such that J · ( m ν − ˜ m ν ) ≥ b α s where J = N (cid:88) j =1 α j p j ∈ Cap( F ∗ )(id) . So in (8.19), | − J · m ν ξ ν | (cid:46) − bα s | − J · ˜ m ν ξ ν | . This together with Lemma 8.1 yields that in (8.19), |I J ( F ( s − , F ( s ) , ξ ) | (cid:46) d (cid:88) ν =1 − bα s | − J · ˜ m ν ξ ν | (cid:46) − c α s . (8.20)By using the mean value theorem in (8.15) together with (8.2) and the support conditionof (cid:99) A J in (8.4), |I J ( F ( s − , F ( s ) , ξ ) | (cid:46) µ (cid:88) ν =1 (cid:88) m ν ∈ ( F ν ( s − ∪ F ν ( s )) ∩ Λ ν | ξ ν − J · m ν | (cid:46) | ξ µ − J · ˜ m µ | for any ˜ m µ ∈ F ν ( N ).(8.21)By (8.20),(8.21) and (8.4), |I J ( F ( s − , F ( s ) , ξ ) | (cid:46) min {| ξ µ − J · ˜ m µ | , − c α s : ˜ m µ ∈ F µ ( N ) } (cid:46) min {| ξ ν − J · ˜ m ν | , − c α s : ˜ m ν ∈ F ν ( N ) } dν = µ . (8.22)By (7.31) of Proposition 7.2, J · ( n ν − ˜ m ν ) (cid:46) α s where n ν ∈ F ν ( s − 1) and ˜ m ν ∈ F ν ( N ) . Hence for any n ν ∈ F ν ( s − 1) and ˜ m ν ∈ F ν ( N ) with ν = µ, µ + 1 , · · · , d in (8.22), | − J · ˜ m ν ξ ν | (cid:46) c α s | − J · n ν ξ ν | . (8.23)Then by (8.22) and (8.23), |I J ( F ( s − , F ( s ) , ξ ) | (cid:46) min (cid:8) c α s | ξ ν − J · n ν | , − c α s : n ν ∈ F ν ( s − (cid:9) dν = µ (cid:46) min (cid:8) | ξ ν − J · n ν | (cid:15) : n ν ∈ F ν ( s − (cid:9) dν = µ . This yields (8.17). (cid:3) ULTIPLE HILBERT TRANSFORMS 65 Therefore the proof of (8.13) is finished. (cid:3) Proof of (8.8). Assume that rank (cid:16)(cid:83) dν =1 F ν ( N ) (cid:17) ≤ n − 1. By this and Lemma 7.2, (cid:83) dν =1 F ν ( N ) ∩ Λ ν is an even set so that I J ( F ( N ) , ξ ) ≡ 0. Thus we suppose thatrank (cid:32) d (cid:91) ν =1 F ν ( N ) (cid:33) = n. As in (8.10) and (8.11), we choose µ ∈ { , · · · , d } such thatrank (cid:32) d (cid:91) ν = µ F ν ( N ) (cid:33) = n and rank d (cid:91) ν = µ +1 F ν ( N ) ≤ n − . (8.24)Set F (cid:48) ( N ) = ( ∅ , · · · , ∅ , F µ +1 ( N ) , · · · , F d ( N )) for µ ≤ d − , (8.25)and F (cid:48) ( N ) = ( ∅ , · · · , ∅ ) for µ = d . By Lemma 7.2 d (cid:92) ν = µ +1 ( F ∗ ν ( N )) ◦ (cid:54) = ∅ . Thus by this and (8.24), (cid:83) dν = µ +1 ( F ν ( N ) ∩ Λ ν ) is an even set. So, I J ( F (cid:48) ( N ) , ξ ) ≡ 0. Thusit suffices to show that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) J ∈ Cap( F ∗ )(id) (cid:16) H F ( N ) J − H F (cid:48) ( N ) J (cid:17) ∗ A J ∗ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) . (8.26)Let J ∈ Cap( F ∗ )(id) ⊂ (cid:84) F ∗ ν ⊂ F ∗ ν = F ∗ ν (N) . Then for every n ν ∈ F ν ( N ) = F ν , J · ( n ν − ˜ m ν ) = 0 where ˜ m ν ∈ F ν ( N ).By this and the support condition (8.4) for (cid:99) A J ( ξ ) such that (cid:12)(cid:12) − J · ˜ m ξ (cid:12)(cid:12) (cid:46) · · · (cid:46) | − J · ˜ m d ξ d | , (cid:12)(cid:12)(cid:12)(cid:0) I J ( F ( N ) , ξ ) − I J ( F (cid:48) ( N ) , ξ ) (cid:1) (cid:99) A J ( ξ ) (cid:12)(cid:12)(cid:12) (cid:46) µ (cid:88) ν =1 (cid:88) m ν ∈ F ν ( N ) ∩ Λ ν (cid:12)(cid:12) − J · m ν ξ ν (cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12) − J · ˜ m µ ξ µ (cid:12)(cid:12)(cid:12) (cid:46) min (cid:110)(cid:12)(cid:12)(cid:12) − J · ˜ m ν ξ ν (cid:12)(cid:12)(cid:12) : ν = µ, · · · , d (cid:111) (cid:46) min (cid:8)(cid:12)(cid:12) − J · n ν ξ ν (cid:12)(cid:12) : n ν ∈ F ν ( N ) (cid:9) dν = µ . (8.27) )( FV p )( GV )( GV q p F G m Figure 5. Transitivity.Thus Lemma 8.1 combined with rank (cid:16)(cid:83) dν = µ F ν ( N ) (cid:17) = n together and Proposition 5.1yields (8.26). This completes the proof of (8.8). Therefore we finish the proof of (6.20).Similarly, we also obtain (6.19) as in (5.8). (cid:3) Necessity Theorem To prove the necessity part of Main Theorem, we need more properties of cones.9.1. Transitivity Rule for Cones.Proposition 9.1. Let P ⊂ R n be a polyhedron and F , G ∈ F ( P ) such that G (cid:22) F . Supposethat q ∈ ( F ∗ ) ◦ | P . Suppose that p ∈ ( G ∗ ) ◦ | F . Then there exists (cid:15) > such that < (cid:15) < (cid:15) implies that q + (cid:15) p ∈ ( G ∗ ) ◦ | P . See Figure 5 that visualizes Proposition 9.1 and Lemma 9.1. Definition 9.1. Let V be a subspace of R n . Denote a projection from R n to V by P V . ULTIPLE HILBERT TRANSFORMS 67 Lemma 9.1. Let P be a polyhedron in R n and G (cid:22) P . Given q ∈ ( G ∗ ) ◦ | P , there exists r > depending only q such that for any n ∈ P \ G and m ∈ G , q · P V ⊥ ( G ) ( n − m ) | P V ⊥ ( G ) ( n − m ) | ≥ r > . (9.1) Proof. We start with the case that G = { m } is a vertex of P . Observe that given apolyhedron Q , there exists a small positive number (cid:15) such that (cid:26) n | n | : n ∈ Q \ { } where 0 is a vertex of Q (cid:27) = (cid:26) n | n | : n ∈ Q , | n | ≥ (cid:15) (cid:27) , which is a closed set in the sphere S n − . By this, we set a closed set in S n − : S ( P − m ) = (cid:26) n − m | n − m | : n ∈ P \ { m } (cid:27) . (9.2)By q ∈ ( G ∗ ) ◦ | P , we have for all n ∈ P \ { m } , q · n − m | n − m | > , which with (9.2) implies that q · s > s ∈ S ( P − m ) . A map s → q · s is continuous on the compact set S ( P − m ). So it has a minimum r > q · s ≥ r for all s ∈ S ( P − m ) . (9.3)From V ( G ) = { } and V ⊥ ( G ) = R n ,P V ⊥ ( G ) ( n − m ) = n − m . From this together with (9.2), S ( P − m ) = (cid:40) P V ⊥ ( G ) ( n − m ) | P V ⊥ ( G ) ( n − m ) | : n ∈ P \ { m } (cid:41) . By this and (9.3), q · P V ⊥ ( G ) ( n − m ) | P V ⊥ ( G ) ( n − m ) | ≥ r for all n ∈ P \ { m } . We next consider the general case that G is a k -dimensional face of P . We shall use thefollowing two properties: (cid:110) x : P V ⊥ ( G ) ( x ) = 0 (cid:111) = V ( G )(9.4) and If n ∈ P \ G and m ∈ G , then n − m / ∈ V ( G ) . (9.5)Choose any m ∈ G (cid:22) P . Since an image of polyhedron under any linear transform is alsoa polyhedron, we see that P V ⊥ ( G ) ( P − m ) is a polyhedron in V ⊥ ( G ). Moreover,0 is a vertex of P V ⊥ ( G ) ( P − m ).(9.6) Proof of (9.6). By Definition 2.10, we see that for q ∈ ( G ∗ ) ◦ | P and for m ∈ G and n ∈ P \ G ,0 < q · ( n − m ) . (9.7)Let P V ⊥ ( G ) ( n − m ) ∈ P V ⊥ ( G ) ( P − m ) \ { } , that is, P V ⊥ ( G ) ( n − m ) (cid:54) = 0. Then we have n − m / ∈ V ( G ) by (9.4), that is, n / ∈ G . Thus n ∈ P \ G . By (9.7), q · < q · ( n − m ) = q · ( n − m ) = q · P V ⊥ ( G ) ( n − m )where q ⊥ V ( G ) in the last equality. Thus the condition (2.3) of Definition 2.7 holds. (cid:3) In view of (9.2) and (9.6), we set a compact set K = S (P V ⊥ ( G ) ( P − m ) − 0) = (cid:26) n − | n − | : n ∈ P V ⊥ ( G ) ( P − m ) \ { } (cid:27) . (9.8)In the above, P V ⊥ ( G ) ( P − m ) \ { } = P V ⊥ ( G ) (( P \ G ) − G ) , (9.9) Proof of (9.9). Let z ∈ P V ⊥ ( G ) ( P − m ) \ { } . Then z = P V ⊥ ( G ) ( n − m ) (cid:54) = 0 with n ∈ P .From (9.4) n − m / ∈ V ( G ), which also implies that n / ∈ G . Thus z ∈ P V ⊥ ( G ) (( P \ G ) − G ).Let z ∈ P V ⊥ ( G ) (( P \ G ) − G ). Then z = P V ⊥ ( G ) ( n − m ) with n ∈ P \ G and m ∈ G . Thus z = P V ⊥ ( G ) ( n − m + m − m )= P V ⊥ ( G ) ( n − m ) + P V ⊥ ( G ) ( m − m )= P V ⊥ ( G ) ( n − m ) (cid:54) = 0where the last inequality follows from (9.4) and (9.5). Hence z ∈ P V ⊥ ( G ) ( P − m ) \ { } . (cid:3) ULTIPLE HILBERT TRANSFORMS 69 By (9.9), we rewrite the compact set in (9.8) as K = (cid:40) P V ⊥ ( G ) ( n − m ) | P V ⊥ ( G ) ( n − m ) | : n ∈ P \ G , m ∈ G (cid:41) . (9.10)By q ∈ ( G ∗ ) ◦ | P , for all n ∈ P \ G and m ∈ G , q · P V ⊥ ( G ) ( n − m ) = q · ( n − m ) > q ⊥ V ( G ) and Definition 2.10. Therefore q · s > s ∈ K .From this combined with the compactness of K , there exists r > q · s ≥ r for all s ∈ K .By (9.10), for any n ∈ P \ G and m ∈ G , q · P V ⊥ ( G ) ( n − m ) | P V ⊥ ( G ) ( n − m ) | ≥ r > . This completes the proof of Lemma 9.1. (cid:3) Proof of Proposition 9.1. It suffices to show that 0 < (cid:15) < (cid:15) implies that for all n ∈ P \ G and m ∈ G , ( q + (cid:15) p ) · ( n − m ) > . (9.11)We first observe from q ∈ ( F ∗ ) ◦ | P , q · ( n − m ) ≥ n ∈ P \ G and m ∈ G . (9.12)Moreover, combined with p ∈ ( G ∗ ) ◦ | F , q · ( n − m ) > n ∈ P \ F and m ∈ G , (9.13) p · ( n − m ) > n ∈ F \ G and m ∈ G (9.14)Let π p be a supporting plane of G (cid:22) F , G ⊂ π p and F \ G ⊂ ( π + p ) ◦ . Split P = (cid:0) P ∩ π − p (cid:1) ∪ (cid:0) P ∩ π + p (cid:1) = P − ∪ P + that are visualized in Figure 5 . Case 1 . Suppose n ∈ P + \ G . Then in view that n ∈ P + ⊂ π + p , p · ( n − m ) ≥ n ∈ P \ G and m ∈ G . (9.15)By (9.12) and (9.15), we have ≥ in (9.11). Thus either (9.13) or (9.14) yields > in (9.11). Case 2 . Suppose that n ∈ P − \ G . Note that G (cid:22) P − . Consider a hyperplane π q ( F )containing F and whose normal vector is q . Then π q ( F ) is a supporting plane of G (cid:22) P − .Thus q ∈ ( G ∗ ) ◦ | ( P − , R n ) . Hence by Lemma 9.1, for all n ∈ P − \ G and m ∈ G , q · P V ⊥ ( G ) ( n − m ) | P V ⊥ ( G ) ( n − m ) | ≥ r > V ⊥ ( G ) ( n − m ) (cid:54) = 0(9.16)where the last follows from (9.4) and (9.5). Split n − m = P V ( G ) ( n − m ) + P V ⊥ ( G ) ( n − m ) . Since q ∈ ( G ∗ ) ◦ | ( P − , R n ) and p ∈ ( G ∗ ) ◦ | F , that is q , p ⊥ V ( G ), q · P V ( G ) ( n − m ) = p · P V ( G ) ( n − m ) = 0 . So ( q + (cid:15) p ) · ( n − m ) = ( q + (cid:15) p ) · P V ⊥ ( G ) ( n − m ) . Choose (cid:15) = r | p | and 0 < (cid:15) < (cid:15) . Then, by (9.16),( q + (cid:15) p ) · ( n − m ) = ( q + (cid:15) p ) · P V ⊥ ( G ) ( n − m ) ≥ r (cid:12)(cid:12)(cid:12) P V ⊥ ( G ) ( n − m ) (cid:12)(cid:12)(cid:12) − (cid:15) | p | (cid:12)(cid:12)(cid:12) P V ⊥ ( G ) ( n − m ) (cid:12)(cid:12)(cid:12) ≥ r (cid:12)(cid:12)(cid:12) P V ⊥ ( G ) ( n − m ) (cid:12)(cid:12)(cid:12) > . This completes the proof of Proposition 9.1 (cid:3) Lemma 9.2. Let P ν ⊂ R n be a polyhedron and F ν , G ν ∈ F ( P ν ) such that G ν (cid:22) F ν .Suppose that q ∈ (cid:84) ν ( F ∗ ν ) ◦ | ( P ν , R n ) . Suppose p ∈ (cid:84) ν ( G ∗ ν ) ◦ | ( F ν , R n ) . Then there exists avector w ∈ (cid:84) ( G ∗ ν ) ◦ | ( P ν , R n ) . ULTIPLE HILBERT TRANSFORMS 71 Proof. By Proposition 9.1, there exists (cid:15) ν > < (cid:15) < (cid:15) ν , q + (cid:15) p ∈ ( G ν ∗ ) ◦ | ( P ν , R n ) . Choose (cid:15) = min { (cid:15) ν : ν = 1 , · · · , d } . Then for 0 < (cid:15) < (cid:15) , q + (cid:15) p ∈ (cid:84) dν =1 ( G ν ∗ ) ◦ | ( P ν , R n ) . (cid:3) Lemma for Necessity.Lemma 9.3. Let Λ = (Λ ν ) with Λ ν ⊂ Z n + and let P Λ ∈ P Λ . Fix S ⊂ { , · · · , n } . Given F = ( F ν ) ∈ F ( N (Λ , S )) , define I ( P F , ξ, r ) = (cid:90) (cid:81) ( − r j ,r j ) e i (cid:16) ξ (cid:80) m ∈ F ∩ Λ1 c m t m + ··· + ξ d (cid:80) m ∈ F d ∩ Λ d c d m t m (cid:17) dt t · · · dt n t n , I ( P F , ξ, a, b ) = (cid:90) (cid:81) { a j < | t j | for j ∈ S \ S and u j = 0 for j ∈ S ⊂ S by Lemma 6.3 . (9.17) Then sup ξ ∈ R d ,a,b ∈ I ( S ) |I ( P F , ξ, a, b ) | < ∞ . (9.18) Proof of (9.18). By the definition of ( F ∗ ν ) ◦ , there exists ρ ν such that u · m = ρ ν < u · n for all m ∈ F ν and n ∈ N (Λ ν , S ) \ F ν . (9.19)Let a = ( a j ) , b = ( b j ) ∈ I ( S ) where I ( S ) = (cid:81) nj =1 I j where(9.20) I j = (0 , ∞ ) for j ∈ { , · · · , n } \ S and I j = (0 , 1) for j ∈ S as in (1.3).Let ρ = ( ρ ν ) with ρ ν in (9.19). Set I ( a ( δ ) , b ( δ )) = n (cid:89) j =1 { a j δ u j < | t j | < b j δ u j } and δ − ρ ξ = ( δ − ρ ξ , · · · , δ − ρ d ξ d ) . Then I ( P Λ , δ − ρ ξ, a ( δ ) , b ( δ ))= (cid:90) I ( a ( δ ) ,b ( δ )) e i (cid:16) δ − ρ ξ (cid:80) m ∈ Λ1 c m t m + ··· + δ − ρd ξ d (cid:80) m ∈ Λ d c d m t m (cid:17) dt t · · · dt n t n . By (9.17) and (9.20), we find a sufficiently small δ such that δ u j a j , δ u j b j < j ∈ S \ S and δ u j a j = a j , δ u j b j = b j < j ∈ S ⊂ S .Thus a ( δ ) , b ( δ ) ∈ I ( S ). Hence, by our hypothesis (cid:12)(cid:12) I ( P Λ , δ − ρ ξ, a ( δ ) , b ( δ )) (cid:12)(cid:12) ≤ C uniformly in ξ and a, b, δ .(9.21)Consider the difference of two multipliers given by M ( ξ, δ, a, b ) = I ( P Λ , δ − ρ ξ, a ( δ ) , b ( δ )) − I ( P F , δ − ρ ξ, a ( δ ) , b ( δ ))= (cid:90) I ( a ( δ ) ,b ( δ )) e i (cid:16) ξ δ − ρ (cid:80) m ∈ Λ1 c m t m + ··· + ξ d δ − ρd (cid:80) m ∈ Λ d c d m t m (cid:17) dt t · · · dt n t n − (cid:90) I ( a ( δ ) ,b ( δ )) e i (cid:16) ξ δ − ρ (cid:80) m ∈ F ∩ Λ1 c m t m + ··· + ξ d δ − ρd (cid:80) m ∈ F d ∩ Λ d c d m t m (cid:17) dt t · · · dt n t n . By the mean value theorem and change of variable t (cid:48) j = δ − u j t j in the above two integrals, | M ( ξ, δ, a, b ) |≤ (cid:90) I ( a,b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ (cid:88) n ∈ Λ \ F δ u · n − ρ c n t n + · · · + ξ d (cid:88) n ∈ Λ d \ F d δ u · n − ρ d c d n t n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt | t | · · · dt n | t n |≤ | ξ | (cid:88) n ∈ Λ \ F δ u · n − ρ | c n | C n ( a, b ) + · · · + | ξ d | (cid:88) n ∈ Λ d \ F d δ u · n − ρ d | c d n | C d n ( a, b ) . The constants C m ( a, b ) , · · · , C d m ( a, b ) above are absolute value for the integral of t m | t |···| t n | on the region I ( a, b ). From u · n − ρ ν > δ > δ u · n − ρ ν above is small enough to satisfy | M ( ξ, δ, a, b ) | ≤ . By this and (9.21), (cid:12)(cid:12) I ( P F , δ − ρ ξ, a ( δ ) , b ( δ )) (cid:12)(cid:12) ≤ C. (9.22) ULTIPLE HILBERT TRANSFORMS 73 By (9.19) and the change of variables δ − u j t j = t (cid:48) j for all j = 1 , · · · , n , I ( P F , δ − ρ ξ, a ( δ ) , b ( δ ))= (cid:90) I ( a ( δ ) ,b ( δ )) e i (cid:16) ξ δ − ρ (cid:80) m ∈ F c m t m + ··· + ξ d δ − ρd (cid:80) m ∈ F d c d m t m (cid:17) dt t · · · dt n t n = (cid:90) I ( a,b ) e i (cid:16) ξ (cid:80) m ∈ F c m t m + ··· + ξ d (cid:80) m ∈ F d c d m t m (cid:17) dt t · · · dt n t n = I ( P F , ξ, a, b ) . Hence this identity combined with (9.22) yields (9.18). (cid:3) Necessity Theorem.Definition 9.2. To each subset M = { q , · · · , q N } ⊂ R n , we associate a matrix:Mtr(M) = q ... q N whose rows are vectors in M . We define a class of rank m -subsets in R n : M m,n = (cid:26) M ⊂ R n : Mtr(M) ∼ · · · a ,m +1 · · · a ,n a ,m +1 · · · a ,n ... 0 1 0 ... · · · ...0 · · · a m,m +1 · · · a m,n (cid:27) . (9.23)Here ∼ means row equivalence and ( a ij ) ≤ i ≤ m, m +1 ≤ j ≤ n a real m × ( n − m ) matrix. Theorem 9.1 (Necessity Part of Main Theorems 1 through 3) . Let Λ = (Λ ν ) where Λ ν ⊂ Z n + is a finite set for ν = 1 , · · · , d and let S ⊂ { , · · · , n } . Suppose that there existfaces F ∈ F ( N (Λ , S )) such that d (cid:91) ν =1 ( F ν ∩ Λ ν ) is not an even set , (9.24) and F ∈ F lo ( N (Λ , S )) , that is,rank (cid:32)(cid:91) ν F ν (cid:33) ≤ n − and (cid:92) ν ( F ∗ ν ) ◦ | N (Λ ν , S ) (cid:54) = ∅ . (9.25) Then there exist a vector polynomial P Λ ∈ P Λ so that sup ξ ∈ R d ,r ∈ I ( S ) |I ( P Λ , ξ, r ) | = ∞ . Proof of Theorem 9.1. Choose the integer m such that(9.26) m = min (cid:40) rank (cid:0) (cid:91) ν F ν (cid:1) : ∃ F ∈ F ( N (Λ , S )) satisfying (9.24), (9.25) (cid:41) . Then we have F = ( F ν ) with F ν ∈ F ( N (Λ ν , S )) such that d (cid:91) ν =1 ( F ν ∩ Λ ν ) is not an even set ,(9.27)and rank (cid:32)(cid:91) ν F ν (cid:33) = m ≤ n − (cid:92) ν ( F ∗ ν ) ◦ | N (Λ ν , S ) (cid:54) = ∅ . (9.28)This with (9.23) implies that we can assume that without loss of generality,Sp (cid:32)(cid:91) ν F ν (cid:33) ∈ M m,n . (9.29)By (9.28) and (9.17), we have for some S ⊂ S ⊂ N n ( u j ) ∈ (cid:92) ν ( F ∗ ν ) ◦ | N (Λ ν , S ) and u j = 0 for j ∈ S and u j > j ∈ S \ S . By Lemma 4.7,(9.30) { e j : j ∈ S } ⊂ Sp (cid:32)(cid:91) ν F ν (cid:33) . Moreover by (9.26), for any G ν (cid:22) F ν ,(9.31) (cid:91) G ν ∩ Λ ν is even whenever rank (cid:0) (cid:91) G ν (cid:1) ≤ m − (cid:92) ( G ∗ ν ) ◦ | N (Λ ν , S ) (cid:54) = ∅ . In order to show Theorem 9.1, by Lemma 9.3, it suffices to find, under the assumption(9.27)-(9.31), a polynomial P Λ ( t ) = (cid:16)(cid:80) q ∈ Λ ν c ν q t q (cid:17) ∈ P Λ with an appropriate c ν q such thatsup ξ ∈ R d ,a,b ∈ I ( S ) |I ( P F , ξ, a, b ) | = ∞ (9.32) ULTIPLE HILBERT TRANSFORMS 75 where I ( P F , ξ, a, b ) = (cid:90) (cid:81) { a j < | t j |
Let 1 ≤ m < n . Using R n = X (cid:76) Y with X = R m × { } and Y = { } × R n − m , we write a = ( a , · · · , a n ) ∈ R n as a = ( a X , a Y ) so that a X = ( a , · · · , a m ) ∈ R m and a Y = ( a m +1 , · · · , a n ) ∈ R n − m . Note that 1 X ( S ) is restricted to R m so that1 X ( S ) = ( r j ) mj =1 with r j = 1 for j ∈ S and r j = ∞ for j ∈ N m \ S ,and I X ( S ) is also restricted to R m in view of (1.3) so that I X ( S ) = m (cid:89) j =1 I j where I j = (0 , 1) for j ∈ S and I j = (0 , ∞ ) for j ∈ N m \ S .To show (9.32), we prove that there exists C ( ξ ) > (cid:12)(cid:12)(cid:12)(cid:12) lim a X → ,b X → X ( S ) I ( P F , ξ, a, b ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ C ( ξ ) n (cid:89) j = m +1 log( b j /a j ) → ∞ as a j → , where the integral I ( P F , ξ, a, b ) is evaluated so that dt t · · · dt m t m first and dt m +1 t m +1 · · · dt n t n next: I ( P F , ξ, a, b ) = (cid:90) (cid:81) nj = m +1 { a j < | t j |
Proof of Necessity Proof of (9.35) and (9.36). Let Ω = (Ω ν ) where Ω ν ⊂ R m is given byΩ ν = ( F ν ∩ Λ ν ) X = { ( q , · · · , q m ) : ( q , · · · , q n ) ∈ F ν ∩ Λ ν } . (10.1)For each t Y = ( t m +1 , · · · , t n ) ∈ (cid:81) nj = m +1 { a j < | t j | < b j } , define I ( P Ω , ξ, a X , b X , t Y ) = (cid:90) (cid:81) mj =1 { a j < | t j |
Suppose (9.26),(9.27),(9.28) and (9.29) hold. Then d (cid:91) ν =1 ( K ν ∩ Ω ν ) is an even setwhenever K = ( K ν ) ∈ F lo ( (cid:126) N (Ω , S )) where F lo ( (cid:126) N (Ω , S )) is (cid:40) K ∈ F ( (cid:126) N (Ω , S )) : rank (cid:0) d (cid:91) ν =1 K ν (cid:1) ≤ m − and d (cid:92) ν =1 ( K ∗ ν ) ◦ | N (Ω ν , S ) (cid:54) = ∅ (cid:41) . To prove Proposition 10.1, we first observe that for F = ( F ν ) satisfying (9.27) and (9.28), (cid:8) q ∈ Σ (cid:32) d (cid:91) ν =1 ( F ν ∩ Λ ν ) (cid:33) : q = ( odd, · · · , odd (cid:124) (cid:123)(cid:122) (cid:125) m components , ∗ , · · · , ∗ ) (cid:9) ⊂ q ∈ Σ (cid:32) d (cid:91) ν =1 ( F ν ∩ Λ ν ) (cid:33) : q = ( odd, · · · , odd (cid:124) (cid:123)(cid:122) (cid:125) n components ) (10.3)where Σ( A ) with A ⊂ Z n is defined below (3.1). The proof for (10.3) follows by taking U = Σ (cid:16)(cid:83) dν =1 ( F ν ∩ Λ ν ) (cid:17) in the following lemma: Lemma 10.1. Suppose Sp ( U ) = k ≤ m and U ∈ M k,n where M k,n is defined in (9.23).If there exists a vector p = ( odd, · · · , odd ) ∈ U , then (cid:8) q ∈ U : q = ( odd, · · · , odd (cid:124) (cid:123)(cid:122) (cid:125) k components , ∗ , · · · , ∗ ) (cid:9) ⊂ { q ∈ U : q = ( odd, · · · , odd ) } . Proof. Assume that there exists µ ∈ { k + 1 , · · · , n } such that q = ( odd, · · · , odd (cid:124) (cid:123)(cid:122) (cid:125) k components , ∗ , · · · , ∗ , even (cid:124) (cid:123)(cid:122) (cid:125) µ components , ∗ , · · · , ∗ ) ∈ U (10.4)where q = ( q j ) with q j = odd numbers for j = 1 , · · · , k and q µ = even number. Thus r = q + p = ( even, · · · , even (cid:124) (cid:123)(cid:122) (cid:125) k components , ∗ , · · · , ∗ , odd (cid:124) (cid:123)(cid:122) (cid:125) µ components , ∗ , · · · , ∗ ) ∈ Σ( U )where r = ( r j ) with r j = even numbers for j = 1 , · · · , k and r µ = even number. We add r as the last row to the matrix in (9.23) in view of U ∈ M k,n . Then,Mtr ( U ) ∼ · · · c ,k +1 · · · c ,µ · · · c ,n c ,k +1 · · · c ,µ · · · c ,n ... 0 1 0 ... · · · ... · · · ...0 · · · c k,k +1 · · · c k,µ · · · c k,n even · · · even even ∗ · · · odd · · · ∗ . (10.5)Consider the ( k + 1) × ( k + 1) submatrix Mtr µ ( U ) consisting of the first k columns andthe µ th column of the matrix in (10.5):Mtr µ ( U ) = · · · c ,µ c ,µ ... 0 1 0 ...0 · · · c k,µ even · · · even even odd . Then we expand the entries of the last row multiplied by their minors to computedet(Mtr µ ( U )) = odd (cid:54) = 0 . ULTIPLE HILBERT TRANSFORMS 79 Thus rank(Mtr µ ( U )) = k + 1, which is a contradiction to U ∈ M k,n . Therefore there is nosuch µ satisfying (10.4). (cid:3) Proof of Proposition 10.1. By (9.29),Sp (cid:16)(cid:91) F ν (cid:17) ∈ M mn . Thus a projection P X : Sp ( (cid:83) F ν ) → X = R m defined by P X ( q , · · · , q m , q m +1 , · · · , q n ) = ( q , · · · , q m )is an isomorphism. We shall denote P X ( q ) = q X . To show Proposition 10.1, we use theinvariance properties proved in the following three lemmas: Lemma 10.2. Let T : V → W be an isomorphism where V, W be inner product spaces in R n . Let P = P (Π) with Π = { π q ,r , · · · , π q N ,r N } be a polyhedron in V . Then (1) T ( P ) is a polyhedron P (Π T ) with Π T = { π ( T − ) t ( q ) ,r , · · · , π ( T − ) t ( q N ) ,r N } . (2) If F ∈ F k ( P ) , then T ( F ) ∈ F k ( T ( P )) for all k ≥ . (3) ( T ( F ) ∗ ) ◦ | ( T ( P ) , W ) = ( T − ) t (( F ∗ ) ◦ | ( P , V )) where T t denotes a transpose of T . (4) For any set B ⊂ V , we have T (Ch(B)) = Ch( T (B)) . Proof. Our proof is based on(10.6) (cid:104) ( T − ) t q , T ( x ) (cid:105) = (cid:104) q , T − T ( x ) (cid:105) = (cid:104) q , x (cid:105) for q , x ∈ V .By (10.6), T ( π q j ,r j ) = {T ( x ) : (cid:104) q j , x (cid:105) = r j } = π ( T − ) t ( q j ) ,r j and T ( π + q j ,r j ) = π +( T − ) t ( q j ) ,r j . Thus T ( P ) = P (Π T ) is a polyhedron because of Definition 2.3 and T ( P ) = T (cid:16)(cid:92) π + q j ,r j (cid:17) = (cid:92) T ( π + q j ,r j ) = (cid:92) π +( T − ) t ( q j ) ,r j . Hence (1) is proved. If F ∈ F ( P ), by (2.2), F = π q j ,r j ∩ P and P \ F ⊂ ( π + q j ,r j ) ◦ . So, T ( F ) = π ( T − ) t ( q j ) ,r j ∩ T ( P ) and T ( P ) \ T ( F ) = T ( P \ F ) ⊂ ( π +( T − ) t ( q j ) ,r j ) ◦ . This means T ( F ) ∈ F ( T ( P )). Moreover V ( F ) and V ( T ( F )) are isomorphic. Hence T ( F ) ∈F k ( T ( P )). So (2) is proved. Next, (10.6) yields that( T ( F ) ∗ ) ◦ | ( T ( P ) , W )= { q ∈ W : ∃ ρ such that (cid:104) q , T ( u ) (cid:105) = ρ < q · T ( y ) for all u ∈ F , y ∈ P \ F } = { ( T − ) t ( p ) : ∃ ρ such that (cid:104) p , u (cid:105) = ρ < (cid:104) p , y (cid:105) for all u ∈ F , y ∈ P \ F } = ( T − ) t (( F ∗ ) ◦ | ( P , V )) . This proves (3). Finally, T (Ch(B)) = T ( N (cid:88) j =1 c j x j ) : x j ∈ B and N (cid:88) j =1 c j = 1 with c j ≥ = N (cid:88) j =1 c j T ( x j ) : T ( x j ) ∈ T ( B ) and N (cid:88) j =1 c j = 1 with c j ≥ = Ch(T(B))which proves (4). (cid:3) Lemma 10.3. Let X = R m with S ⊂ { , · · · , m } . Then, N ( P X ( F ν ∩ Λ ν ) , S ) = P X ( N ( F ν ∩ Λ ν , S )) , (10.7) K ν ∈ F ( N ( P X ( F ν ∩ Λ ν ) , S )) if and only if P − X ( K ν ) ∈ F ( N ( F ν ∩ Λ ν ) , S )) , (10.8) (cid:0) P − X ( K ν ) ∗ (cid:1) ◦ | N ( F ν ∩ Λ ν , S ) = ( K ∗ ν ) ◦ | N ( P X ( F ν ∩ Λ ν ) , S )) . (10.9) Proof. By (4) of Lemma 10.2 with Definition 2.11 and P X ( B + R S + ) = P X ( B ) + R S + , P X ( N ( F ν ∩ Λ ν , S )) = P X (cid:16) Ch (cid:16) ( F ν ∩ Λ ν ) + R S + (cid:17)(cid:17) = Ch (cid:16) P X (cid:16) ( F ν ∩ Λ ν ) + R S + (cid:17)(cid:17) = Ch (cid:16) P X ( F ν ∩ Λ ν ) + R S + (cid:17) = N ( P X ( F ν ∩ Λ ν ) , S ) ULTIPLE HILBERT TRANSFORMS 81 which yields (10.7). Next (10.8) follows from (10.7) and (2) of Lemma 10.2. Lastly, by(10.7) and (3) of Lemma 10.2, (cid:0) P − X ( K ν ) ∗ (cid:1) ◦ | N ( F ν ∩ Λ ν , S ) = (cid:0) P − X ( K ν ) ∗ (cid:1) ◦ | P − X ( N ( P X ( F ν ∩ Λ ν ))= [( P − X ) − ] t ( K ∗ ν ) ◦ | N ( P X ( F ν ∩ Λ ν ) , S ))= ( K ∗ ν ) ◦ | N ( P X ( F ν ∩ Λ ν ) , S )) , which proves (10.9). (cid:3) We continue the roof of Proposition 10.1. Let K ν ∈ F ( N ( P X ( F ν ∩ Λ ν ) , S )), whereΩ ν = P X ( F ν ∩ Λ ν ) as in (10.1). By (10.8) of Lemma 10.3, there exists G ν = P − X ( K ν ) ∈ F ( N ( F ν ∩ Λ ν , S )) . From (cid:83) dν =1 G ν = P − X ( (cid:83) dν =1 K ν ) , rank( d (cid:91) ν =1 G ν ) = rank (cid:0) d (cid:91) ν =1 K ν (cid:1) ≤ m − P X is an isomorphism. By (4.24) and (10.9), d (cid:92) ν =1 ( G ∗ ν ) ◦ | F ν = d (cid:92) ν =1 ( G ∗ ν ) ◦ | N (Λ ν ∩ F ν , S )= d (cid:92) ν =1 (cid:0) P − X ( K ν ) ∗ (cid:1) ◦ | N ( F ν ∩ Λ ν , S )(10.11) = d (cid:92) ν =1 ( K ∗ ν ) ◦ | N (Ω ν , S ) (cid:54) = ∅ . The last line follows from the second condition defining F lo ( (cid:126) N (Ω , S )) in Proposition 10.1.By (9.28), d (cid:92) ν =1 ( F ∗ ν ) ◦ | N (Λ ν , S ) (cid:54) = ∅ . (10.12)By applying Lemma 9.2 together with (10.11) and (10.12), d (cid:92) ν =1 ( G ∗ ν ) ◦ | N (Λ ν , S ) (cid:54) = ∅ . (10.13) By (10.10),(10.13) and (9.31), d (cid:91) ν =1 G ν ∩ Λ ν is an even set having no point of ( odd, · · · , odd ) in Σ (cid:32) d (cid:91) ν =1 G ν ∩ Λ ν (cid:33) . By this together with (10.3),Σ (cid:32) d (cid:91) ν =1 G ν ∩ Λ ν (cid:33) (cid:92) q ∈ Σ (cid:32) d (cid:91) ν =1 ( F ν ∩ Λ ν ) (cid:33) : q = ( odd, · · · , odd (cid:124) (cid:123)(cid:122) (cid:125) m components , ∗ , · · · , ∗ ) ⊂ Σ (cid:32) d (cid:91) ν =1 G ν ∩ Λ ν (cid:33) (cid:92) q ∈ Σ (cid:32) d (cid:91) ν =1 ( F ν ∩ Λ ν ) (cid:33) : q = ( odd, · · · , odd (cid:124) (cid:123)(cid:122) (cid:125) n components ) = ∅ . ThereforeΣ (cid:16)(cid:91) K ν ∩ Ω ν (cid:17) = P X (cid:16) Σ (cid:16)(cid:91) G ν ∩ Λ ν (cid:17)(cid:17) contains no point of ( odd, · · · , odd (cid:124) (cid:123)(cid:122) (cid:125) m components ).Hence (cid:83) ( K ν ∩ Ω ν ) is an even set. Therefore, the proof of Proposition 10.1 is finished. (cid:3) Proof of (9.39) and (9.40). We prove the independence (9.39) and the non-vanishing property (9.40) to finish the necessity proof for Theorems 1 through 3. Proof of (9.39). Recall (9.38) J ( P F , ξ, t Y ) = lim a X → ,b X → X ( S ) J ( P F , ξ, t Y , a X , b X ) , with J ( P F , ξ, t Y , a X , b X ) = (cid:90) (cid:81) mj =1 { a j Fix t Y = ( t m +1 , · · · , t n ) and use the change of variables: x = t q , · · · , x m = t q m . (10.15)As q ∈ (cid:83) ν ( F ν ∩ Λ ν ) ⊂ Sp ( (cid:83) F ν ) is expressed as a linear combination of q , · · · , q m , thereexists a vector b ( q ) = ( b , · · · , b m ) ∈ R m such that t q = t b q + ··· + b m q m = x b · · · x b m m = x b ( q ) . This implies that the phase function P F ( ξ, σt ) is written as P F ( ξ, σ t , · · · , σ n t n ) = d (cid:88) ν =1 (cid:88) q ∈ F ν ∩ Λ ν c ν q σ q t q ξ ν (10.16) = d (cid:88) ν =1 (cid:88) q ∈ F ν ∩ Λ ν c ν q σ q x b ( q ) ξ ν = Q F ( ξ, σ, x ) . By (i) and (ii) above, the m × m matrix whose i -th row given by ( q i ) X = e i ∈ R m , is theidentity matrix: I = q , · · · , q m ... q m , · · · , q mm . Then compute for fixed t Y = ( t m +1 , · · · , t n ), ∂ ( x , · · · , x m ) ∂ ( t , · · · , t m ) = det q t q t , · · · , q m t q t m ... q m t q m t , · · · , q mm t q m t m = det( I ) x · · · x m t · · · t m . Takeing logarithms on both sides of (10.15), q , · · · , q n ... q m , · · · , q mn log t ...log t n = log x ...log x m . Then log t ...log t m = log x − ( q ,m +1 log t m +1 + · · · + q ,n log t n )...log x m − ( q m,m +1 log t m +1 + · · · + q m,n log t n ) . (10.17)Solve ( t , · · · , t m ) in (10.17) in terms of ( x , · · · , x m ) and t Y = ( t m +1 , · · · , t n ), t i = x i t q i,m +1 m +1 · · · t q i,n n for i = 1 , · · · , m. Note from this together with q = e , · · · , q k = e k in (ii) above, t = x , · · · , t k = x k , t i = x i / ( t q i,m +1 m +1 · · · t q i,n n ) for i = k + 1 , · · · , m. So the region (cid:81) mj =1 { a j < t j < b j } in (9.38) is transformed to the region U ( a X , b X , t Y ) = k (cid:89) i =1 { a i < x i < b i } m (cid:89) i = k +1 (cid:40) a i < x i t q i,m +1 m +1 · · · t q i,n n < b i (cid:41) (10.18)Thus as a X = ( a i ) mi =1 → X and b X = ( b i ) mi =1 → X ( S ) = ( 1 , · · · , (cid:124) (cid:123)(cid:122) (cid:125) k components , ∞ , · · · , ∞ (cid:124) (cid:123)(cid:122) (cid:125) m − k components ), J ( P F , ξ, ( t m +1 , · · · , t n ))= lim a X → X ,b X → X ( S ) (cid:90) U ( a X ,b X ,t Y ) (cid:88) σ ∈ O ( − | σ | exp ( iQ F ( ξ, x )) dx x · · · dx m x m , is independent of t Y ∈ (cid:81) nj = m +1 { a j < t j < b j } since t q i,m +1 m +1 · · · t q i,n n is absorbed in the limitof a X → X and b X → X ( S ) in (10.18). (cid:3) Proof of (9.40). Since J ( P F , ξ, ( t m +1 , · · · , t n )) is independent of t Y = ( t m +1 , · · · , t n ), itsuffices to show that for some choices of ξ and coefficients in P F ,(10.19) J ( P F , ξ ) = J ( P F , ξ, Y ) (cid:54) = 0 where Y = (1 , · · · , (cid:124) (cid:123)(cid:122) (cid:125) n − m components . Let Z = { , } be the additive group and let Z n = { ( v , · · · , v n ) : v i ∈ Z } . Define afunction Γ : Z n → Z n by Γ( q , · · · , q n ) = ( γ ( q ) , · · · , γ ( q n )) ULTIPLE HILBERT TRANSFORMS 85 where γ ( q i ) = q i is an even number,1 if q i is an odd number.We put Γ (cid:32) d (cid:91) ν =1 ( F ν ∩ Λ ν ) (cid:33) = { z , · · · , z L } ⊂ Z n . For (cid:96) = 1 , · · · , L, let Γ − { z (cid:96) } = (cid:40) q ∈ d (cid:91) ν =1 ( F ν ∩ Λ ν ) : Γ( q ) = z (cid:96) (cid:41) . We then have d (cid:91) ν =1 ( F ν ∩ Λ ν ) = L (cid:91) (cid:96) =1 Γ − { z (cid:96) } . (10.20)Since (cid:83) dν =1 ( F ν ∩ Λ ν ) is an odd set, there exist z , · · · , z s ∈ Γ (cid:16)(cid:83) dν =1 ( F ν ∩ Λ ν ) (cid:17) such that z ⊕ · · · ⊕ z s = (1 , · · · , 1) in Z n . (10.21)Assume the contrary to (10.19). Then from (9.38) and (9.39), for all ξ , · · · , ξ d and allchoices of coefficients c ν q ∈ R \ { } , we have in (10.14), J ( P F , ξ, Y ) = lim a X → ,b X → X ( S ) J ( P F , ξ, Y , a X , b X )= lim a X → ,b X → X ( S ) (cid:90) (cid:81) mj =1 { a j Proofs of Corollary 3.1 and Main Theorem 3 Proof of Corollary 3.1. Proof of Corollary 3.1. Sufficiency . Suppose that( F n +1 ∩ Λ n +1 ) ∪ A is an even set whenever rank( F n +1 ∪ A ) ≤ n − F n +1 ∈ F ( N (Λ n +1 , S )) and A ⊂ { e , · · · , e n } . It suffices to deduce from (11.1) thatthe hypothesis of Main Theorem 2 holds for the case Λ = ( { e } , · · · , { e n } , Λ n +1 ), since wehave already proved Main Theorem 2. Let rank (cid:16)(cid:83) n +1 ν =1 F ν (cid:17) ≤ n − (cid:84) n +1 ν =1 ( F ∗ ν ) ◦ (cid:54) = ∅ .We claim that (cid:83) n +1 ν =1 ( F ν ∩ Λ ν ) is an even set. Observe that for every nonempty face F ν ∈ F ( N ( { e ν } , S )), F ν ∩ Λ ν = { e ν } . Thus for A = { e ν : F ν (cid:54) = ∅ for ν = 1 , · · · , n } , wewrite n +1 (cid:91) ν =1 F ν ∩ Λ ν = ( F n +1 ∩ Λ n +1 ) ∪ A. By (11.1), (cid:83) n +1 ν =1 F ν ∩ Λ ν is an even set. Necessity . Suppose that (11.1) does not hold. Then there exists A = { e ν : ν ∈ I } ⊂{ e , · · · , e n } and F n +1 such thatrank ( A ∪ F n +1 ) ≤ n − A ∪ ( F n +1 ∩ Λ n +1 ) is odd.Let Sp( F n +1 ) ∩ { e ν : ν ∈ S } = { e ν , · · · , e ν k } where { ν , · · · , ν k } = S ⊂ S . Choose • For ν ∈ N n \ I , let F ν = ∅ with ( F ∗ ν ) ◦ = Z ( S ) \ { } . • For ν ∈ I , let F ν = { e ν } + R S with( F ∗ ν ) ◦ = CoSp ◦ ( { e j : j ∈ S \ S } ∪ {± e j : j ∈ N n \ S } ) . Then we can observe that ( F ∗ n +1 ) ◦ ⊂ ( F ∗ ν ) ◦ for all ν = 1 , · · · , n . Therefore,rank (cid:32) n +1 (cid:91) ν =1 F ν (cid:33) = rank ( A ∪ F n +1 ) ≤ n − n +1 (cid:92) ν =1 ( F ∗ ν ) ◦ (cid:54) = ∅ , but (cid:83) n +1 ν =1 F ν ∩ Λ ν = A ∪ ( F n +1 ∩ Λ n +1 ) is an odd set, which implies that the hypothesisof Main Theorem 2 breaks. Let q = ( q , · · · , q n ) ∈ (cid:84) dν =1 ( F ∗ ν ) ◦ with q j = 0 for j ∈ S ⊂ S . ULTIPLE HILBERT TRANSFORMS 89 We follow the same argument for the necessity proof in Section 9. Then we obtain (9.33)so that there exists P Λ ∈ P Λ such that (cid:13)(cid:13)(cid:13) H P F S (cid:13)(cid:13)(cid:13) L ( R d ) → L ( R d ) = ∞ . This implies (cid:13)(cid:13) H P S (cid:13)(cid:13) L ( R d ) = ∞ by the following standard argument: For δ > 0, define adilation f δ ( x , · · · , x n , x n +1 ) = f ( δ − q x , · · · , δ − q n x n , δ − d x n +1 )and a measure µ Sδ ( φ ) = (cid:90) I ( S ) φ ( δ − q t , · · · , δ − q n t n , δ − d P ( t , · · · , t n )) dt t · · · dt n t n satisfying H P S ( f ) = [ µ Sδ ∗ f δ − ] δ . By usinglim δ → µ Sδ ( φ ) = (cid:90) I ( S ) φ ( t , · · · , t n , P F ( t , · · · , t n )) dt t · · · dt n t n , we conclude that the boundedness of (cid:13)(cid:13) H P S (cid:13)(cid:13) L ( R d ) → L ( R d ) implies the boundedness of (cid:107)H P F S (cid:107) L ( R d ) → L ( R d ) . (cid:3) Proof of Main Theorem 3. We now develop the argument of [16] for n ≥ 3, andobtain Main Theorem 3. Definition 11.1. Let P ∈ P Λ where Λ = (Λ ν ) with Λ ν ⊂ Z n + and S ⊂ { , · · · , n } . Let A ∈ GL ( d ). We set the collection of all (cid:126) N ( AP, S ) with A ∈ GL ( d ) in Definition 3.4, N ( P, S ) = { (cid:126) N ( AP, S ) : A ∈ GL ( d ) } . Consider the collection of equivalent classes A ( P ) = { [ A ] : A ∈ GL ( d ) } with the equiva-lence relation for A, B ∈ GL ( d ), A ∼ B if and only if Λ( AP ) = Λ( BP ).We see that A ( P ) is finite and write A ( P ) as { [ A k ] : k = 1 , · · · , N } . Thus we can regard N ( P, S ) as the ordered N -tuples of (cid:126) N ( A k P, S ) (indeed, N d tuples of Newton Polyhedrons N (( A k P ) ν , S )): N ( P, S ) = (cid:16) (cid:126) N ( AP, S ) (cid:17) [ A ] ∈A ( P ) = (cid:16) (cid:126) N ( A k P, S ) (cid:17) Nk =1 . So, we define the class of all combinations of N d -tuples of faces by F ( N ( P, S )) = (cid:110) ( F [ A ] ) [ A ] ∈A ( P ) : F [ A ] ∈ F (cid:16) (cid:126) N ( AP, S ) (cid:17)(cid:111) (11.2) = (cid:110) ( F A k ) Nk =1 : F A k ∈ F (cid:16) (cid:126) N ( A k P, S ) (cid:17)(cid:111) , where F A k = (( F A k ) , · · · , ( F A k ) d ) with ( F A k ) ν ∈ F ( N (( A k P ) ν , S )) . To prove Main Theorem 3, we apply the Proposition 6.1 for every (cid:126) N ( A k P, S ) with k = 1 , · · · , N to obtain the following general form of cone decomposition. Lemma 11.1. Let P ∈ P Λ where Λ = (Λ ν ) with Λ ν ⊂ Z n + and S ⊂ { , · · · , n } . Then, (cid:91) ( F [ A ] ) [ A ] ∈A ( P ) ∈F ( N ( P,S )) (cid:92) [ A ] ∈A ( P ) Cap( F ∗ [A] ) = Z ( S ) . Given Λ , there are finitely many Newton polyhedrons in { (cid:126) N ( AP, S ) : A ∈ GL ( d ) , P ∈ P Λ } .Proof. By (11.2), the left hand side above is (cid:91) ( F Ak ) Nk =1 ∈F ( N ( P,S )) (cid:32) N (cid:92) k =1 d (cid:92) ν =1 ( F A k ) ∗ ν (cid:33) = N (cid:92) k =1 (cid:91) F Ak ∈F ( (cid:126) N ( A k P,S ) ) d (cid:92) ν =1 ( F A k ) ∗ ν . For each fixed A k , Proposition 6.1 yields that (cid:91) F Ak ∈F ( N ( A k P,S )) d (cid:92) ν =1 ( F A k ) ∗ ν = Z ( S ) , which proves Lemma 11.1. (cid:3) To each [ A ] ∈ A ( P ), we first assign a d -tuple of faces F A ∈ F (cid:16) (cid:126) N ( AP, S ) (cid:17) . Next, fix(11.3) ( F A ) [ A ] ∈A ( P ) . To show Main Theorem 3, in view of Lemma 11.1, it suffices to show that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) J ∈ Z H P Λ J (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C where Z = (cid:92) [ A ] ∈A ( P ) Cap( F ∗ A ) with F A chosen in (11.3) . (11.4) ULTIPLE HILBERT TRANSFORMS 91 To show (11.4), we can replace H P Λ J by H UP Λ J for some U ∈ GL ( d ) and prove that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) J ∈ Z H P Λ J (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) J ∈ Z H UP Λ J (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C (11.5)where the equality follows from (cid:90) f ( x − P ( t )) d (cid:89) ν =1 χ (2 j ν t ν ) t ν dt = (cid:90) f ( U − ( U x − U P ( t )) d (cid:89) ν =1 χ (2 j ν t ν ) t ν dt. Without the disjointness of Λ ν ’s, we are lack of the decay condition (5.15) in Remark 5.3and (6.18) of Theorem 6.1. In order to recover this, we shall modify the proof of [16] andfind an appropriate U to satisfy the desirable decay estimate in Lemma 11.2. We workthis process for d = 3. Let [ A ] ∈ A ( P ) with A = I . Then( A P )( t ) = P ( t ) = (cid:88) m ∈ Λ(( A P ) ν )=Λ( P ν ) c ν m t m ν =1 . Take any vector m ( A , ∈ ( F A ) ∩ Λ(( A P ) ) where F A ∈ F ( (cid:126) N ( A P, S )) was chosen in(11.3) with [ A ] ∈ A ( P ). Define A = − c m ( A , c m ( A , − c m ( A , c m ( A , so that A A P ( t ) = ( A P ) ( t )( A P ) ( t ) − c m ( A , c m ( A , ( A P ) ( t )( A P ) ( t ) − c m ( A , c m ( A , ( A P ) ( t ) where • t m ( A , does not appear in each of 2 th and 3 rd components of A A P ( t ).Next choose m ( A , ∈ ( F A A ) ∩ Λ(( A A P ) ) where F A A ∈ F ( (cid:126) N ( A A P, S )) waschosen in (11.3) with [ A A ] ∈ A ( P ). Define a matrix A = − c m ( A , c m ( A , so that A A A P ( t ) = ( A A P ) ( t ) = ( A P ) ( t )( A A P ) ( t )( A A P ) ( t ) − c m ( A , c m ( A , ( A A P ) ( t ) where t m ( A , does not appear in each of 2 th and 3 rd components of A A A P ( t ) , (11.6) t m ( A , does not appear in the 3 rd component of A A A P ( t ).(11.7)Choose m ( A , ∈ ( F A A A ) ∩ Λ(( A A A P ) ) where F A A A ∈ F ( (cid:126) N ( A A A P, S ))was chosen in (11.3) with [ A A A ] ∈ A ( P ). Since m ( A k , k ) ∈ ( F A k ··· A ) k and J ∈ (cid:92) [ A ] ∈A ( P ) Cap( F ∗ A ) ⊂ ( F A k ··· A ) ∗ k for k = 1 , · · · , , we have for each k = 1 , , − J · m ( A k ,k ) ≥ − J · m for m ∈ Λ(( A k · · · A P ) k )(11.8)where Λ(( A k · · · A P ) k ) = Λ(( A A A P ) k ) for each k by construction above. Lemma 11.2. Let U = A A A and F U = F A A A ∈ F ( (cid:126) N ( U P, S )) where A , A , A and F A A A were defined above. For G ∈ F ( (cid:126) N ( U P, S )) such that G (cid:23) F U , let I J ([ U P ] G , ξ ) = (cid:90) e i ( (cid:80) ν =1 ( (cid:80) m ∈ G ν ∩ Λ(( UP ) ν ) − J · m c ν m t m ) ξ ν ) n (cid:89) (cid:96) =1 h ( t (cid:96) ) dt. Then for J ∈ Z = (cid:84) A ∈A ( P ) Cap( F ∗ A ) ⊂ Cap ( F ∗ U ) , there exists C > and δ that areindependent of J, ξ satisfying: (11.9) |I J ([ U P ] G , ξ ) | ≤ C min (cid:110) | − J · m ξ ν | − δ : m ∈ Λ(( U P ) ν ) , ν = 1 , , (cid:111) . Proof of (11.9). By (11.8), it suffices to show that |I J ([ U P ] G , ξ ) | ≤ C | − J · m ( A k ,k ) ξ k | − δ for k = 1 , , . (11.10)The case k = 1 follows from (11.6). To show (11.10) for k = 2, it suffices to consider | − J · m ( A , ξ | (cid:29) | − J · m ( A , ξ | . This and (11.7) yield the desired result for k = 2. Since(11.10) holds for k = 1 , 2, we may assume that | − J · m ( A , ξ | (cid:29) | − J · m ( A k ,k ) ξ k | for k = 1 , k = 3 is obtained by the Van der Corput lemma. (cid:3) ULTIPLE HILBERT TRANSFORMS 93 Proof of (11.4). The first hypothesis of Theorem 6.1 is satisfied by Lemma 11.2. Thesecond hypothesis of Theorem 6.1 is also satisfied by the hypothesis (3.4) of Main Theorem3 such that d (cid:91) ν =1 [ K U ] ν ∩ [Λ( U P Λ )] ν is an even setwhenever K U ∈ (cid:40) K U ∈ F ( (cid:126) N ( U P, S )) : d (cid:92) ν =1 ([ K U ] ∗ ν ) ◦ (cid:54) = ∅ and rank (cid:32) d (cid:91) ν =1 [ K U ] ν (cid:33) ≤ n − (cid:41) . Therefore by applying Theorem 6.1 for Z = (cid:84) [ A ] ∈A ( P ) Cap( F ∗ A ) ⊂ Cap F ∗ U with U = A A A , we obtain (11.4). (cid:3) Finally, the proof of necessity part of Main Theorem 3 is the same as that of MainTheorems 1-3 once it is assumed that the evenness hypothesis (3.4) is broken with a fixedmatrix A . References [1] A. Carbery, S. Wainger, and J. Wright, Triple Hilbert transform along polynomial surfaces in R , Rev.Mat. Iberoam. (2009), 471–519.[2] A. Carbery, S. Wainger, and J. 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