aa r X i v : . [ m a t h . C V ] F e b PSEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY
DARIUSH EHSANIA
BSTRACT . We study properties of pseudodifferential operators which arise intheir use in boundary value problems. Smooth domains as well as domains ofintersections of smooth domains are considered.
1. I
NTRODUCTION
We develop here properties of pseudodifferential operators when acting on dis-tributions supported on domains with boundary, as well as when acting on distri-butions supported on the boundary of a domain. We also look at the restrictionsof pseudodifferential operators to the boundaries of domains.The analysis presented here arises naturally in the study of boundary valueproblems and the aim here is to provide the groundwork for such applications asin [2] and [3]. The calculations should be valuable in any general use of pseudodif-ferential operators within the context of boundary value problems. Of particularimportance are the estimates which can be obtained when considered pseudodif-ferential operators which arise in connection with Poisson and Green’s operators.The case of (transversal) intersections of domains is also considered, with themain results relating to extending estimates obtained in the case of smooth do-mains via weighted estimates.Due to the local nature of pseudodifferential operators we can (in a neighbor-hood of a boundary point) assume coordinates ( x , ρ ) ∈ R n + for ρ < R n . In the case of intersections,the coordinates will be chosen so that near a point on the intersection of severalboundaries, the domain looks like the intersection of several lower-half planes.2. A NALYSIS ON THE LOWER HALF - SPACE
In this section we develop some of the properties of pseudodifferential oper-ators on half-spaces. Of particular importance for the reduction to the boundarytechniques are the boundary values of pseudodifferential operators on half-spaces,
Mathematics Subject Classification. pseudodifferential operators acting on distributions supported on the boundary,as well as pseudodifferential operators on the boundary itself.We first fix some notation to be used throughout this paper. We use coordi-nates ( x , ρ ) on R n + with x = ( x , . . . , x n ) ∈ R n . The full Fourier Transform of afunction, f ( x , ρ ) , will be written b f ( ξ , η ) = ( π ) n + Z f ( x , ρ ) e ix ξ e i ρ dxd ρ where ξ = ( ξ , . . . , ξ n ) . For f defined on a subset of R n + we define its FourierTransform as the transform of the function extended by zero to all of R n + . Thus,for instance for f defined on { ( x , ρ ) ∈ R n + : ρ < } , we write b f ( ξ , η ) = ( π ) n + Z − ∞ Z R n f ( x , ρ ) e ix ξ e i ρ dxd ρ .A partial Fourier Transform in the x variables will be denoted by e f ( ξ , ρ ) = ( π ) n Z R n f ( x , ρ ) e ix ξ dx .We define the half-space H n + − : = { ( x , ρ ) ∈ R n + : ρ < } . The space ofdistributions, E ′ ( H n + − ) is defined as the distributions in E ′ ( R n + ) with supportin H n + − . The topology of E ′ ( H n + − ) is inherited from that of E ′ ( R n + ) . We endow C ∞ ( R n + ) with the topology defined in terms of the semi-norms p l , K ( φ ) = max ( x , ρ ) ∈ K ⊂⊂ R m + ∑ | α |≤ l | ∂ α φ ( x , ρ ) | .A regularizing operator, Ψ − ∞ ( R n + ) , is a continuous linear map(2.1) Ψ − ∞ : E ′ ( R n + ) → C ∞ ( R n + ) .Continuity of (2.1) can be shown, for instance, by proving the continuity on therestriction to each W s ( K ) for K ⊂⊂ R n + so that continuity is read from estimatesof the form max ( x , ρ ) ∈ KK ⊂⊂ R n + ρ ≥ ∑ | α |≤ l | ∂ α Ψ − ∞ φ ( x , ρ ) | . k φ k W s ( K ) for each l ≥ s ≥ ( x , . . . , x n , ρ ) , ρ <
0, we will have the need to show that by multiplying symbolsof inverses of elliptic operators with smooth cutoffs with compact support, whichare functions of transform variables corresponding to tangential coordinates, weproduce operators which are smoothing.
SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 3
Lemma 2.1.
Let A ∈ Ψ k ( R n + ) for k ≤ − be such that the symbol, σ ( A )( x , ρ , ξ , η ) ,is meromorphic (in η ) with poles at η = q ( x , ρ , ξ ) , . . . , q k ( x , ρ , ξ ) with q i ( x , ρ , ξ ) themselves symbols of pseudodifferential operators of order 1 (restricted to η = ) such that for each ρ , Res η = q i σ ( A ) ∈ S k + ( R n ) with symbol estimates uniform inthe ρ parameter.Let A χ denote the operator with symbol χ ( ξ ) σ ( A ) , where χ ( ξ ) ∈ C ∞ ( R n ) . Then A χ is regularizing on distributions supported on the bound-ary: A χ : E ′ ( R n ) × δ ( ρ ) → C ∞ ( H n + − ) . Proof.
Without loss of generality we suppose φ b ( x ) ∈ L c ( R n ) , and let φ = φ b × δ ( ρ ) . We estimate derivatives of A χ ( φ ) . We let a ( x , ρ , ξ , η ) denote the symbol of A ,and a χ ( x , ρ , ξ , η ) that of A χ .We assume without loss of generality that A is the inverse to an elliptic operatorwith non-vanishing symbol. The general case is handled in the same manner byusing a cutoff χ as above which vanishes at the zeros of the symbol (of the ellipticoperator to which A is an inverse).We first note that derivatives with respect to the x variables pose no difficultydue to the χ term in the symbol of A χ : from A χ φ = ( π ) n + Z a χ ( x , ρ , ξ , η ) b φ ( ξ , η ) e ix ξ e i ρη d ξ d η = ( π ) n + Z a χ ( x , ρ , ξ , η ) e φ b ( ξ ) e ix ξ e i ρη d ξ d η ,we calculate | ∂ α x A χ φ | . Z | ∂ α x a χ ( x , ρ , ξ , η ) || ξ | | α | (cid:12)(cid:12) e φ b ( ξ ) (cid:12)(cid:12) d ξ d η . k φ b k L (cid:18) Z | ξ | | α | χ ( ξ ) | ∂ α x a ( x , ρ , ξ , η ) | d ξ d η (cid:19) . k φ b k L (cid:18) Z | ξ | | α | χ ( ξ ) ( + ξ + η ) k d ξ d η (cid:19) . k φ b k L (cid:18) Z | ξ | | α | χ ( ξ ) d ξ (cid:19) . k φ b k L ,where α + α = α . For any mixed derivative ∂ α x ∂ βρ , the x derivatives can be han-dled in the manner above and so we turn to derivatives of the type ∂ βρ A χ φ . DARIUSH EHSANI
We use the residue calculus to integrate over the η variable in A χ φ = ( π ) n + Z a χ ( x , ρ , ξ , η ) e φ b ( ξ ) e ix ξ e i ρη d ξ d η .For ρ <
0, we integrate over a contour in the lower-half plane and analyze a typicalterm resulting from a simple pole at η = q − ( x , ρ , ξ ) of a χ ( x , ρ , ξ , η ) . Let a q − ( x , ρ , ξ ) = i Res η = q − a χ ( x , ρ , ξ , η ) and A q − χ φ = ( π ) n Z a q − ( x , ρ , ξ ) e φ b ( ξ ) e ix ξ e i ρ q − d ξ .Note that the factor of χ ( ξ ) is contained in a q − . We can now estimate ∂ βρ A q − χ φ bydifferentiating under the integral. For ρ < (cid:12)(cid:12)(cid:12) ∂ βρ A q − χ φ (cid:12)(cid:12)(cid:12) . Z | ∂ β ρ a q − ( x , ρ , ξ ) || q − | | β | | ∂ β ρ q − | (cid:12)(cid:12) e φ b ( ξ ) (cid:12)(cid:12) d ξ . k φ b k L ,where β + β + β = β . The integral over ξ converges due to the factor of χ ( ξ ) contained in a q − .As poles of other orders are handled similarly, we conclude the proof of thelemma. (cid:3) For example, if A is a finite sum of the first terms of the expansion of the inverseto an elliptic operator of order k ≥
1, then A satisfies the hypothesis of Lemma 2.1.In this case we would also have that the poles q i ( x , ρ , ξ ) are elliptic (uniformlyin the ρ parameter). For such operators, without the assumption of the cutofffunction χ ( ξ ) in the symbol of A χ we can still prove Theorem 2.2.
Let g ∈ D ( R n + ) of the form g ( x , ρ ) = g b ( x ) δ ( ρ ) for g b ∈ W s ( R n ) .Let A ∈ Ψ k ( R n + ) , k ≤ − be as in Lemma 2.1 with the additional assumption that theimaginary parts of the poles, q i ( x , ρ , ξ ) are elliptic operators (restricted to η = ). Thenfor all s ≥ k Ag k W sloc ( H n + − ) . k g b k W s + k + ( R n ) . Proof.
We follow and use the notation of the proof of Lemma 2.1, and analyze atypical term resulting from a simple pole at η = q − ( x , ρ , ξ ) of a ( x , ρ , ξ , η ) . With a q − ( x , ρ , ξ ) = i Res η = q − a ( x , ρ , ξ , η ) and A q − g = ( π ) n Z a q − ( x , ρ , ξ ) e g b ( ξ ) e ix ξ e i ρ q − d ξ , SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 5 we estimate ∂ α x ∂ βρ A q − g by differentiating under the integral for the case s ≥
0. For ρ < (cid:12)(cid:12)(cid:12) ∂ α x ∂ βρ A q − g (cid:12)(cid:12)(cid:12) . Z | ∂ α x ∂ β ρ a q − ( x , ρ , ξ ) | | ∂ α x q − | | β | | ∂ α x ∂ β ρ q − | | ξ | | α | | e g b ( ξ ) | e ρ | Im q − | d ξ ,where β + β + β = β and α + · · · α = α and | α | + β = s .From the assumption that Im q − is a symbol of an elliptic operator, we have theproperty 1 | Im q − | ∼ | ξ | .Thus, when we integrate over ρ we get a factor on the order of | ξ | which lowersthe order of the norm in the tangential directions by 1/2: (cid:13)(cid:13)(cid:13) ∂ α x ∂ βρ A q − g (cid:13)(cid:13)(cid:13) L loc . Z | + ξ | k + | ξ | | β | | ξ | | β | | ξ | | α | | e g b ( ξ ) | + | ξ | d ξ . Z | + ξ | k + + β + β + | α | | e g b ( ξ ) | d ξ . Z | + ξ | k + + s | e g b ( ξ ) | d ξ . k g b k W s + k + ( R n ) .Note that we use the term 1 + | ξ | in the denominator to avoid singularities at theorigin. In general cutoffs can be used which vanish at any zeros of q − withoutchanging the results of the Theorem.This proves the Theorem for integer s ≥
0. The non-integer case follows byinterpolation. (cid:3)
The hypotheses of Lemma 2.1 and Theorem 2.2 are satisfied for instance in thecase of an inverse to an elliptic differential operator such as the Laplacian.There are analogue estimates for functions with support in the half-plane (asopposed to support on the boundary):
Theorem 2.3.
Let f ∈ W s ( H n + − ) for s ≥ . Let A ∈ Ψ k ( R n + ) , k ≤ − be as inTheorem 2.2. Then k A f k W sloc ( H n + − ) . k f k W s + k ( H n + − ) . Proof.
We follow the proof of Theorem 2.2, and again analyze a term resulting froma simple pole with positive imaginary part denoted η = q + ( x , ρ , ξ ) of a ( x , ρ , ξ , η ) ,the symbol of the operator A . Integrating through the η transform variable, we set A q + f = ( π ) n Z a q + ( x , ρ , ξ ) G ( f )( x , ρ , ξ , q + ) e ix ξ d ξ , DARIUSH EHSANI where a q + ( x , ρ , ξ ) = i Res η = q + a ( x , ρ , ξ , η ) .and G ( f )( x , ρ , ξ , q + ) = π Z η − q + b f ( ξ , η ) e i ρη d η = i Z ρ − ∞ e f ( ξ , t ) e q + ( t − ρ ) dt .We note that ∂ α x ∂ βρ A q + f is of the form ∑ ≤ j ≤ β Z γ j ∂ j − ρ e f ( ξ , ρ ) e ix ξ d ξ + ∑ j ≤ α + β Z γ j G ( t j f ) e ix ξ d ξ ,where each γ j and γ j can be estimated by | γ j | . | ξ | k + + α | γ j | . | ξ | k + + α + β .For α + β ≤ s , the L -norm of the first term is immediately bounded by k f k W k + s ( H n + − ) .To bound a term R γ j G ( t j f ) e ix ξ d ξ , we note Z γ j G ( t j f ) e ix ξ d ξ = π Z γ j η − q + c ρ j f ( ξ , η ) e ix ξ e i ρη d ξ d η ,an L -bound of which is given by Z | γ j | | η − q + | (cid:12)(cid:12)(cid:12) c ρ j f ( ξ , η ) (cid:12)(cid:12)(cid:12) d ξ d η . Z | + ξ | k + + s | η − q + | (cid:12)(cid:12)(cid:12) c ρ j f ( ξ , η ) (cid:12)(cid:12)(cid:12) d ξ d η . ∑ j Z | + ξ | k + + s η + ξ (cid:12)(cid:12)(cid:12) c ρ j f ( ξ , η ) (cid:12)(cid:12)(cid:12) d ξ d η . k ρ j f k W k + s ( H n + − ) . k f k W k + s ( H n + − ) .Thus k ∂ α x ∂ βρ A q + f k W sloc ( H n + − ) . k f k W s + k ( H n + − ) for α + β ≤ s and the theorem is proved for integer s ≥
0. The general case followsby Sobolev interpolation. (cid:3)
In the case A ∈ Ψ k ( R n + ) for k ≤ − σ ( A )( x , ρ , ξ , η ) , which for instance arise as error terms in a symbol expan-sion, we can still derive estimates, up to certain order. SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 7
Theorem 2.4.
Let g ∈ D ( R n + ) of the form g ( x , ρ ) = g b ( x ) δ ( ρ ) for g b ∈ W s ( R n ) .Let A ∈ Ψ k ( R n + ) , k ≤ − Then for s ≤ | k | − k Ag k W sloc ( H n + − ) . k g b k W s + k + ( R n ) . Proof.
As in the proof of Theorem 2.2, we write Ag = ( π ) n Z a ( x , ρ , ξ , η ) e g b ( ξ ) e ix ξ e i ρη d ξ d η ,and estimate (cid:13)(cid:13)(cid:13) ∂ α x ∂ βρ Ag (cid:13)(cid:13)(cid:13) L loc . Z ξ | α | η β ( η + ξ ) | k | | e g b ( ξ ) | d η d ξ . Z ξ ( | α | + β ) | ξ | | k |− | e g b ( ξ ) | d ξ . k g b k W s + k + ( R n ) .This handles the case of s ≥
0. Negative values of s can be handled by writing k Ag k W sloc ( H n + − ) ≃ k Λ −| s | ◦ Ag k L loc ( H n + − ) where Λ −| s | is a pseudodifferential oper-ator with symbol σ (cid:16) Λ −| s | (cid:17) = ( + ξ + η ) | s | /2 and applying the theorem to Λ −| s | ◦ A . (cid:3) In the case of a distribution supported on the half-space, we have
Theorem 2.5.
Let f ∈ L ( H n + − ) . Let A ∈ Ψ k ( R n + ) , k ≤ − Then for ≤ s ≤ | k |k A f k W sloc ( H n + − ) . k f k L ( H n + − ) . Proof. we write
A f = ( π ) n Z a ( x , ρ , ξ , η ) b f ( ξ , η ) e ix ξ e i ρη d ξ d η ,and estimate (cid:13)(cid:13)(cid:13) ∂ α x ∂ βρ Ag (cid:13)(cid:13)(cid:13) L loc . Z ξ | α | η β ( η + ξ ) | k | | b f ( ξ , η ) | d η d ξ . k f k L ( H n + − ) . (cid:3) We can combine Theorems 2.2 and 2.4 (respectively Theorems 2.3 and 2.5) andapply them to operators which can be decomposed into an operator satisfying thehypothesis of Theorem 2.2 and a remainder term.
DARIUSH EHSANI
Definition 2.6.
We say an operator B ∈ Ψ − k for k ≥ decomposable if for any N ≥ k it can be written in the form B = A − k + Ψ − N ,where A − k ∈ Ψ − k is an operator satisfying the hypothesis of Theorem 2.2 .Then, with a similar use of operators, Λ −| s | , as in Theorem 2.4 we can show theconclusions of Theorems 2.2 and 2.3 hold for all s (including negative values of s ,using only a slight meromorphic modification in the case s is an odd number) fordecomposable operators.We use the notation Ψ kb ( R n ) , respectively Ψ kb ( ∂ Ω ) in the case of pseudodiffer-ential operators on a domain Ω ⊂ R n + , to denote the space of pseudodifferentialoperators of order k on R n = ∂ H n + − , respectively ∂ Ω . Further following our useof the notation Ψ k to denote any operator belonging to the family Ψ k ( H n + − ) (re-spectively Ψ k ( Ω ) ) when acting on distributions φ ∈ E ′ ( H n + − ) (respectively in E ′ ( Ω ) ) we write for φ b ∈ E ′ ( R n ) (respectively in E ′ ( ∂ Ω ) ) Ψ kb φ b , Ψ kb denoting anypseudodifferential operator of order k on the appropriate (boundary of a) domain.With coordinates ( x , . . . , x n , ρ ) in R n + , let R denote the restriction operator, R : D ( R n + ) → D ( R n ) , given by R φ = φ | ρ = . Lemma 2.7.
Let g ∈ D ( R n + ) of the form g ( x , ρ ) = g b ( x ) δ ( ρ ) for g b ∈ D ( R n ) .Let A ∈ Ψ k ( R n + ) , be an operator of order k, for k ≤ − . Then R ◦ A induces apseudodifferential operator in Ψ k + b ( R n ) acting on g b viaR ◦ Ag = Ψ k + b g b . Proof.
Denote the symbol of A with a ( x , ρ , ξ , η ) . The symbol α ( x , ρ , ξ ) = π Z ∞ − ∞ a ( x , ρ , ξ , η ) d η (for any fixed ρ ) belongs to the class S k + ( R n ) , which follows from the propertiesof a ( x , ρ , ξ , η ) as a member of S k ( R n + ) and differentiating under the integral. Thecomposition R ◦ Ag is given by1 ( π ) n + Z a ( x , 0, ξ , η ) e g b ( ξ ) e i x · ξ d ξ d η = ( π ) n Z (cid:20) π Z ∞ − ∞ a ( x , 0, ξ , η ) d η (cid:21) e g b ( ξ ) e i x · ξ d ξ = ( π ) n Z α ( x , 0, ξ ) e g b ( ξ ) e i x · ξ d ξ = Ψ k + b g b . (cid:3) SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 9
Remark . We can generalize Lemma 2.7 to decomposable operators of order k = − η variable. Remark . If we choose coordinates ( x , . . . , x n − , ρ ) on a domain Ω ⊂ R n + ina neighborhood, U , around a boundary point, p ∈ ∂ Ω , such that Ω T U = { z : ρ ( z ) < } and ρ | ∂ Ω T U = Ω replacing H n + − ⊂ R n + and ∂ Ω replacing R n . Lemma 2.10.
Let g ∈ D ( R n + ) of the form g ( x , ρ ) = g b ( x ) δ ( ρ ) for g b ∈ D ( R n ) . LetA ∈ Ψ k ( R n + ) , be a pseudodifferential operator of order k. Let ρ denote the operator ofmultiplication with ρ . Then ρ ◦ A induces a pseudodifferential operator of order k − ong: ρ Ag = Ψ k − g . Proof.
We write the symbol of the operator A symbol as a ( x , ρ , ξ , η ) : A = Op ( a ) .Since a ( x , ρ , ξ , η ) is of order k , ρ · a ( x , ρ , ξ , η ) is also of order k , and ρ ◦ A ( g ) = Z ρ a ( x , ρ , ξ , η ) e g b ( ξ ) e ix ξ e i ρη d ξ d η = − i Z a ( x , ρ , ξ , η ) e g b ( ξ ) e ix ξ ∂∂η e i ρη d ξ d η = i Z ∂∂η (cid:16) a ( x , ρ , ξ , η ) (cid:17)e g b ( ξ ) e ix ξ e i ρη d ξ d η = Ψ k − g since ∂∂η (cid:16) a ( x , ρ , ξ , η ) (cid:17) is a symbol of class S k − ( R n + ) . (cid:3) Lemma 2.7 concerned itself with the restrictions of pseudodifferential operators(applied to distributions supported on the boundary) to the boundary, while The-orem 2.2 allows us to consider pseudodifferential operators applied to restrictionsof distributions. A special case of Theorem 2.2 is
Lemma 2.11.
Let A ∈ Ψ k ( R n + ) , for k ≤ − , be a decomposable operator. ThenA ◦ R ◦ Ψ − ∞ : E ′ ( H n + − ) → C ∞ ( H n + − ) i.e., A ◦ R ◦ Ψ − ∞ = Ψ − ∞ .Proof. Let f ∈ E ′ ( H n + − ) and apply Theorem 2.2 (for decomposable operators)with g b = R ◦ Ψ − ∞ f . Then for all s k A ◦ R ◦ Ψ − ∞ f k W s ( H n + − ) . k R ◦ Ψ − ∞ f k W s + k + ( R n ) . k Ψ − ∞ f k W ( R n + ) . k f k W − ∞ ( H n + − ) .The lemma thus follows from the Sobolev Embedding Theorem. (cid:3) Similarly proven is the
Lemma 2.12.
Let A ∈ Ψ k ( R n + ) , for k ≤ − , be a decomposable operator. ThenA ◦ Ψ − ∞ b : E ′ ( R n ) → C ∞ ( R n + ) .3. A PPLICATIONS
We end this section with illustrations of how our analysis lends itself to theproof of useful theorems on Dirichlet’s problem and on harmonic functions. Ourapplications will include results of elliptic operators acting distributions supportedon a domain with boundary, Ω , such as in an inhomogeneous Dirichlet problem:let Γ be a second order elliptic differential operator, let g ∈ W s ( Ω ) for some s ≥ u , to the boundary value problem(3.1) Γ u = f in Ω u = ∂ Ω .We obtain estimates for the solution to (3.1) via Poisson’s operator, giving thesolution, v , to(3.2) Γ v = Ω v = g on ∂ Ω .We will be concerned with establishing regularity of the solution, taking forgranted classical results of the existence and uniqueness of solutions (see for in-stance [6]). Theorem 3.1.
Denote by P ( g ) the solution operator to (3.2) . Then the estimates k P ( g ) k W s + ( Ω ) . k g k W s ( ∂ Ω ) hold for all s ≥ .Proof. We suppose Γ = − ∂ ρ − n ∑ j = ∂ x j + ∑ ij c ij ( x ) ∂ x i ∂ x j + O ( ρ ) Ψ + s ( x , ρ ) ∂ ρ ,modulo tangential first order terms, where Ψ is a second order differential oper-ator, and the c ij ( x ) = O ( x ) . Such representations, for example, arise in the use oflocal coordinates near a boundary point of a domain. The principal symbol of Γ istherefore(3.3) σ ( Γ ) = η + Ξ ( x , ξ ) + O ( ρ ) SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 11 where(3.4) Ξ ( x , ξ ) = ∑ j ξ j − ∑ ij c ij ( x ) ξ i ξ j .The expression Γ v = Ω can thus be written (locally)1 ( π ) n + Z (cid:16) η + Ξ ( x , ξ ) + O ( ρ ) + i η s ( x , ρ ) (cid:17)b v ( ξ , η ) e ix ξ e i ρη d ξ d η − ( π ) n + Z (cid:16)g ∂ ρ v ( ξ , 0 ) + ( i η + s ( x , 0 )) e g ( ξ ) (cid:17) e ix ξ e i ρη d ξ d η = Γ thus gives v = ( π ) n + Z g ∂ ρ v ( ξ , 0 ) + i η e g ( ξ ) η + Ξ ( x , ξ ) e ix ξ e i ρη d ξ d η + Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − ( g × δ ( ρ )) + Ψ − ∞ v ,(3.5)where we use Lemma 2.10 to group any term of the form O ( ρ ) ◦ Op (cid:18) η + Ξ ( x , ξ ) (cid:19) (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) with the Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) term, with similar simplifications for O ( ρ ) termscombined with operators on g × δ .Consistent with the calculus of pseudodifferential operators, in writing FourierTransforms, we will assume compact support (in a neighborhood of the boundarypoint near which the equation is being studied), as well as assume any cutoffs (intransform space) are to be applied at singularities in the integrands. We can useLemma 2.1 to show such cutoffs, which vanish on compact sets (in a neighborhoodof a point singularity), introduce smoothing terms into the equations. We shallmake these assumptions without explicitly writing the cutoffs or stating that theexpressions hold locally.Performing a contour integration in the η variable in the first integral on theright of (3.5) yields v = ( π ) n Z g ∂ ρ v ( ξ , 0 ) + | Ξ ( x , ξ ) | e g ( ξ ) | Ξ ( x , ξ ) | e ix ξ e ρ | Ξ ( x , ξ ) | d ξ + Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − (cid:0) g × δ ( ρ ) (cid:1) modulo smoothing terms. Letting ρ → − and using Lemma 2.7 for the last twoterms on the right gives g ( x ) = ( π ) n Z g ∂ ρ v ( ξ , 0 ) | Ξ ( x , ξ ) | e ix ξ e ρ | Ξ ( x , ξ ) | d ξ + g ( x )+ Ψ − b (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − b (cid:0) g × δ ( ρ ) (cid:1) modulo smooth terms.We can now solve for ∂ ρ v ( x , 0 ) to write(3.6) ∂ ρ v ( x , 0 ) = | D | g ( x ) + Ψ b (cid:0) g × δ ( ρ ) (cid:1) where | D | is the operator with symbol | Ξ ( x , ξ ) | .(3.6) is valid modulo smoothing operators on ∂ ρ v ( x , 0 ) as well as smooth termsof the form R ◦ Ψ − ∞ v , and imply the Sobolev estimates k ∂ ρ v ( x , 0 ) k W s ( ∂ Ω ) . k g k W s + ( ∂ Ω ) + k v k − ∞ for all s .We return to (3.5) to get estimates for the Poisson operator. Inserting (3.6) into(3.5) yields a principal Poisson operator. Define Θ + ∈ Ψ − ( Ω ) by the symbol σ ( Θ + ) = i η + i | Ξ ( x , ξ ) | .Then we have from (3.5)(3.7) v = Θ + g + Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − ( g × δ ( ρ )) ,modulo smoothing terms. Estimates for v = P ( g ) now follow from L estimatesfor the Poisson operator, k P ( g ) k L ( Ω ) . k g k L ( ∂ Ω ) [6], and from Theorems 2.2 and2.4 (note that the operators written as Ψ − and Ψ − are decomposable). From (3.7)we then have k P ( g ) k W s + ( Ω ) . k g k W s ( ∂ Ω ) + (cid:13)(cid:13)(cid:13) ∂ ρ P ( g ) (cid:12)(cid:12) ρ = (cid:13)(cid:13)(cid:13) W s − ( ∂ Ω ) + k P ( g ) k − ∞ . k g k W s ( ∂ Ω ) . (cid:3) We remark that (3.6) is known as the relation for the Dirichlet to Neumannoperator, giving the normal derivatives of the solution to the Dirichlet problem(3.2) in terms of the boundary data. This operator is the object of study in [1],where we further use the techniques of Section 2 to obtain an expression for zeroorder terms of the operator.Estimates for the problem (3.1) now follow from the Poisson operator. To solve Γ u = f in Ω u = ∂ Ω .with f ∈ W s ( Ω ) , we first take an extension F ∈ W s ( R n + ) such that F | Ω = f (see [4]). We can then take a solution, U ∈ W s + ( R n + ) to Γ U = F . For instancesuch a solution can be found by taking F to have compact support and invertingthe Γ operator using the pseudodifferential calculus. Now let g b = U | ∂ Ω . Then SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 13 u = U − P ( g b ) solves the boundary value problem with estimates given in thefollowing theorem: Theorem 3.2.
Let G ( f ) denote the solution, u, to the boundary value problem (3.1) . Then k G ( f ) k W s + ( Ω ) . k f k W s ( Ω ) , for s ≥ .Proof. Using the notation above, we have k u k W s + ( Ω ) . k F k W s ( Ω ) + k P ( U | ∂ Ω ) k W s ( Ω ) . k f k W s ( Ω ) + k U | ∂ Ω k W s + ( ∂ Ω ) . k f k W s ( Ω ) + k U k W s + ( Ω ) . k f k W s ( Ω ) ,where we use the Sobolev Trace Theorem in the last step and Theorem 3.1 in thesecond. (cid:3) The operator G in Theorem 3.2 is known as Green’s operator. An alternativeproof to Theorem 3.2 relies on Theorem 2.3 to estimate some principal terms forthe solution operator, and Theorem 2.5 for error terms.The Dirichlet to Neumann operator gives the boundary values of the normalderivative applied to the Poisson operator. We now apply our methods to calcu-late the principal term of the boundary values of the normal derivative applied toGreen’s operator. Theorem 3.3.
Let Θ − ∈ Ψ − ( Ω ) be the operator with symbol σ ( Θ − ) = i η − i | Ξ ( x , ξ ) | . Then (3.8) R ◦ ∂∂ρ ◦ G ( g ) = R ◦ Θ − g + Ψ − b ◦ R ◦ Ψ − g + R ◦ Ψ − g , modulo smoothing terms.Proof. The expression Γ v = g in Ω can be written (locally)1 ( π ) n + Z (cid:16) η + Ξ ( x , ξ ) + O ( ρ ) (cid:17)b v ( ξ , η ) e ix ξ e i ρη d ξ d η − ( π ) n + Z ∂ ρ e v ( ξ , 0 ) e ix ξ e i ρη d ξ d η = g , modulo smoothing terms. Inverting the Γ operator thus gives v = ( π ) n + Z ( + O ( ρ )) b g ( ξ , η ) η + Ξ ( x , ξ ) e ix ξ e i ρη d ξ d η + ( π ) n + Z ∂ ρ e v ( ξ , 0 ) η + Ξ ( x , ξ ) e ix ξ e i ρη d ξ d η + Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − g + Ψ − ∞ v ,(3.9)where we use Lemma 2.10 to group any term of the form O ( ρ ) ◦ Op (cid:18) η + Ξ ( x , ξ ) (cid:19) (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) with the Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) term.Note the O ( ρ ) term in (3.9) is a zero order operator. We apply a normal deriva-tive to (3.9): ∂ v ∂ρ = ( π ) n + Z ( + O ( ρ )) i ηη + Ξ ( x , ξ ) b g ( ξ , η ) e ix ξ e i ρη d ξ d η + ( π ) n + Z i ηη + Ξ ( x , ξ ) ∂ ρ e v ( ξ , 0 ) e ix ξ e i ρη d ξ d η + Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + Ψ − g ,(3.10)modulo smoothing terms. We now calculate1 ( π ) n + Z ( + O ( ρ )) i ηη + Ξ ( x , ξ ) b g ( ξ , η ) e ix ξ e i ρη d ξ d η = ( π ) n + Z ( + O ( ρ )) i ηη + Ξ ( x , ξ ) Z − ∞ e g ( ξ , t ) e − it η dte ix ξ e i ρη d ξ d η = ( π ) n + Z ( + O ( ρ )) Z − ∞ e g ( ξ , t ) i η e i ( ρ − t ) η η + Ξ ( x , ξ ) d η dte ix ξ d ξ = ( π ) n Z ( + O ( ρ )) Z − ∞ (cid:0) sgn ( t − ρ ) (cid:1)e g ( ξ , t ) e −| ρ − t || Ξ ( x , ξ ) | dte ix ξ d ξ ,(3.11)which, in the limit ρ →
0, tends to − ( π ) n Z Z − ∞ e g ( ξ , t ) e t | Ξ ( x , ξ ) | dte ix ξ d ξ . SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 15
Similarly, using the residue calculus, we have, for ρ < ( π ) n + Z i ηη + Ξ ( x , ξ ) ∂ ρ e v ( ξ , 0 ) e ix ξ e i ρη d ξ d η = ( π ) n Z ∂ ρ e v ( ξ , 0 ) e ix ξ e ρ | Ξ ( x , ξ ) | d ξ → ( π ) n Z ∂ ρ e v ( ξ , 0 ) e ix ξ d ξ = ∂ ρ v ( x , 0 ) ,and thus, letting ρ → − in (3.10), we have ∂ ρ v ( x , 0 ) = − ( π ) n Z b g ( ξ , i | Ξ ( x , ξ ) | ) e ix ξ d ξ + R ◦ Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) + R ◦ Ψ − g ,(3.12)modulo smoothing terms of the form R ◦ Ψ − ∞ v . We note that − ( π ) n Z b g ( ξ , i | Ξ ( x , ξ ) | ) e ix ξ d ξ = i ( π ) n + Z η − i | Ξ ( x , ξ ) | b g ( ξ , η ) e ix ξ d ξ d η .Thus, the first term on the right-hand side of (3.12) can be written as R ◦ Θ − g .Furthermore, using R ◦ Ψ − (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = × δ ( ρ ) (cid:17) = Ψ − b (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = (cid:17) from Lemma 2.7, we have ∂ ρ v ( x , 0 ) = R ◦ Θ − g + Ψ − b (cid:16) ∂ ρ v (cid:12)(cid:12) ρ = (cid:17) + R ◦ Ψ − g ,modulo smoothing terms. (cid:3) Theorem 3.3 and in particular, the relation (3.8) is essential in relating a solutionto the ¯ ∂ -problem on weakly pseudoconvex domains to a solution of the boundarycomplex in [2]. Furthermore, on domains determined by intersections of smoothdomains the analogue(s) of the operator Θ − , one for each domain Ω j , will play acrucial role in allowing the application of weighted estimates of pseudodifferentialoperators in the solution of the ¯ ∂ -Neumann problem. We refer the reader to [3] fordetails.For our last application, we return to harmonic functions on Ω with prescribedboundary data:(3.13) Γ u = Ω u = g on ∂ Ω . We look at a result of Ligocka, which states that multiplication with the definingfunction of the solution to a Dirichlet problem as in (3.13) leads to an increase insmoothness:
Theorem 3.4 (Ligocka, see Theorem 1 [5]) . Let Ω be a smoothly bounded domain withdefining function ρ , and H s ( Ω ) be the space of harmonic functions belonging to W s ( Ω ) .Let T k u = ρ k u, where k is a positive integer. Then for each integer s, the operator T k mapsH s ( Ω ) continuously into H s + k ( Ω ) .Proof. We will prove the Theorem in the case s ≥
1; the proof in [5] was also brokeninto cases, s ≥ Γ u = Γ whose parametrix Γ − isdecomposable, and we provide here a sketch of the proof of Theorem 3.4 usingthe techniques from this article. We make a change of coordinates so that locally Ω = { ( x , ρ ) | ρ < } , and ∂ Ω = { ( x , 0 ) } . We denote g jb = ∂ j u ∂ρ j (cid:12)(cid:12)(cid:12)(cid:12) ∂ Ω g j = g jb × δ ( ρ ) j =
0, 1.Let Γ − denote a parametrix for Γ . We will use the fact that, modulo smooth termsof the form R ◦ Ψ − ∞ u , g b = Ψ b g b where the operator Ψ b is the Dirichlet to Neumann operator.The solution u can be written locally as Γ − ∑ j = Ψ jt g − j ! = Γ − ◦ Ψ t g ,modulo smoothing terms, where Ψ jt is used to denote a tangential differential op-erator of order j . This representation of the solution was worked out in (3.5). If the Ψ t on the right-hand side depends on ρ we can apply an expansion in powers of ρ , to obtain a sum of terms of the form(3.14) ∑ l ≥ (cid:16) Ψ − − l ◦ Ψ (cid:17) g ,where Ψ − − l are decomposable operators, the sum over l coming from the ex-pansion in powers of ρ by Lemma 2.10, as well as an expansion of the symbol ofthe inverse operator Γ − . The subscript 0 is to denote that the operators Ψ have SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 17 symbols which are independent of ρ . A finite sum over l may be taken with a re-mainder term of the form Ψ − N g for arbitrarily large N . In particular, if N is chosento be larger than k + s +
2, such a remainder term can be estimated by (cid:12)(cid:12)(cid:12) ∂ k x ∂ k ρ Ψ − N g (cid:12)(cid:12)(cid:12) . Z | ξ | | k | | η | k ( + ξ + η ) N /2 | e u b ( ξ ) | d ξ d η . k u b k L ( ∂ Ω ) . k u k W ( Ω ) . k u k W s ( Ω ) for | k | + k = k + s by the Sobolev Trace Theorem. We use the notation u b : = u | ∂ Ω above.Furthermore, ρ applied to a term in the sum in (3.14) gives ρ k Ψ − − l ◦ Ψ g = Ψ − − l − k ◦ Ψ g by Lemma 2.10. The operators Ψ − − l − k are also decomposable, and as such Theo-rems 2.2 and Theorem 2.4 apply. We thus estimate (cid:13)(cid:13)(cid:13) Ψ − + l − k Ψ g (cid:13)(cid:13)(cid:13) W k + s ( Ω ) . (cid:13)(cid:13)(cid:13) Ψ g (cid:13)(cid:13)(cid:13) W − + l + s + ( ∂ Ω ) . k u b k W l + s − ( ∂ Ω ) . k u b k W s − ( ∂ Ω ) . k u k W s ( Ω ) ,where we use the Trace Theorem in the last step.The smooth terms are handled as in Theorem 3.1. (cid:3)
4. A
NALYSIS ON INTERSECTIONS OF HALF - PLANES
Another situation in which the theory of elliptic operators can be applied ison an (non-degenerate) intersection of smooth domains. Localizing the problemin analogy with what was done in Section 3 leads to each domain composing theintersection as a seperate half-space. With appropriate choice of metric the domaincan be modeled by the intersection of several half-spaces. In this section we studysome properties of pseudodifferential operators on such spaces. The goal for thisstudy is the application the results to the study of elliptic operators on intersectiondomains, such as in [3], and in particular to be able to obtain weighted estimatesfor solutions to elliptic problems on the intersection of smooth domains.We define the half-spaces H nj = { ( x , ρ ) ∈ R n : ρ j < } . With a multi-index I = ( i , . . . , i k ) , we also denote the intersection of half-spaces H nI = \ j ∈ I H nj .The convention used here is H nI = R n when I = ∅ . For this section, we will fix I and denote m = | I | . Without loss of generality, I = (
1, . . . , m ) .We use the multi-index notation: ρ α J = ∏ j ∈ J ρ α j j .To indicate a missing index, j , we use the notation ˆ j . Thus we write I ˆ j : = I \ { j } .For ease of notation, in place of ρ α I ˆ j , we write ρ α ˆ j = ρ α · · · ρ α j − j − ρ α j + j + · · · ρ α m m .Similarly, we write ρ α ˆ k ˆ j ∏ i ∈ Ji = j , k ρ α i i .In the case we have equal powers, α = · · · = α j − = α j + = · · · = α m = r ,we write ρ r × ( m − ) ˆ j : = ρ α ˆ j .We now define the weighted Sobolev norms on the half-spaces, for α ∈ R , and s , k ∈ N : W α , s ( H nI , ρ , k ) = n f ∈ W α ( H nI ) (cid:12)(cid:12)(cid:12) ρ ( sk − rk ) × m f ∈ W α + s − r ( H nI ) for each 0 ≤ r ≤ s o with norm k f k W α , s ( H nI , ρ , k ) = ∑ ≤ j ≤ s (cid:13)(cid:13)(cid:13) ρ ( sk − jk ) × m f (cid:13)(cid:13)(cid:13) W α + s − j ( H nI ) .In the case k = W α , s ( H nI , ρ ) : = W α , s ( H nI , ρ , 1 ) which has norm k f k W α , s ( H nI , ρ ) = s ∑ r = k ρ r × ( m − ) f k W α + r ( H nI ) . SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 19
To denote extensions by 0 across ρ j = ρ j >
0, we use the superscript E j : let g ∈ L ( H nI ) then g E j ∈ L (cid:18) H nI ˆ j (cid:19) is defined by g E j = g if ρ j <
00 if ρ j ≥ J , we define g E J ∈ L (cid:16) H nI \ J (cid:17) by g E J = g on H nI and 0elsewhere (that is for any ( x , ρ ) ∈ H nI \ J for which any ρ j ≥ j ∈ J ).We establish Lemma 4.1.
Let g ∈ W s (cid:0) H nI (cid:1) for some integer s ≥ . Then ρ sj g E j ∈ W s (cid:18) H nI ˆ j (cid:19) .Proof. We only need to check the derivatives with respect to ρ j . We have(4.1) ∂ s ρ j (cid:16) ρ sj g E j (cid:17) = ∑ c k ρ s − kj ∂ s − k ρ j g E j . ∂ s − k ρ j g E j itself is a sum of terms of (derivatives of) delta functions, δ ( i ) , i ≤ s − k − (cid:16) ∂ s − k − i − ρ j g E j (cid:17) (cid:12)(cid:12)(cid:12) H nI : ∂ s − k ρ j g E j = s − k ∑ i = d i δ ( i − ) ( ρ j ) (cid:16) ∂ s − k − i ρ j g (cid:17) E j ,where we consider δ − ≡
1, and d =
1. Inserting this into (4.1), the delta functionscombine with the powers of ρ j to yield zero, and we have ∂ s ρ j (cid:16) ρ sj g E j (cid:17) = ∑ c k ρ s − kj (cid:16) ∂ s − k ρ j g (cid:17) E j .The lemma now follows (in the case s is an integer) by the assumption on theregularity of g in H I . (cid:3) A similar proof shows
Lemma 4.2.
Let s ≥ be an integer, k ∈ N , and ρ rkj g ∈ W r − α (cid:0) H nI (cid:1) for integer r ≤ sand α ≥ . Then ρ skj g E j ∈ W s − α (cid:18) H nI ˆ j (cid:19) . For a mutli-index, J , let us denote H n − J , bk : = ∂ H nk \ H nJ ˆ k ,with the convention J ˆ k = J in the case k / ∈ J .We present the following Theorem which is a weighted analogue of Theorem2.2. In the following Theorem we use the notation δ j : = δ ( ρ j ) . We also allow fornegative Sobolev spaces. We say φ ∈ W s (cid:16) H n − I , bj (cid:17) for s < φ E I ˆ j ∈ W s (cid:16) ∂ H n − j (cid:17) . Theorem 4.3.
Let A − α be an elliptic operator (of order − α ≤ − ) satisfying the hypothe-ses of Theorem 2.2. Then, for − ≤ γ ≤ , g b ∈ W γ , s (cid:16) H n − I , bj , ρ ˆ j , k (cid:17) , and β ≥ with β − α ≤ γ , ρ rk × m A − α (cid:18) g E I ˆ j b × δ j (cid:19) ∈ W r + β ( R n ) for all r ≤ s, and (cid:13)(cid:13)(cid:13)(cid:13) A − α (cid:18) g E I ˆ j b × δ j (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) W β − s ( R n , ρ , k ) . k g b k W β − α , s (cid:16) H n − I , bj , ρ ˆ j , k (cid:17) . Proof.
We prove the Theorem in the case β = α + g E I ˆ j b ∈ W (cid:16) ∂ H n − j (cid:17) ,or Λ g E I ˆ j b ∈ L (cid:16) ∂ H n − j (cid:17) .By assumption ρ rk × ( m − ) ˆ j Λ g b ∈ W r (cid:16) H n − I , bj (cid:17) for 0 ≤ r ≤ s . Thus, by repeated application of Lemma 4.2, we have ρ rk × ( m − ) ˆ j Λ g E ˆ j b ∈ W r (cid:16) ∂ H nj (cid:17) for r ≤ s , or ρ rk × ( m − ) ˆ j g E ˆ j b ∈ W r + (cid:16) ∂ H nj (cid:17) .Write g j = g E I ˆ j b × δ j . We use Lemma 2.10 to write ρ rkj A − α g j = Ψ − rk − α g j .We have ρ rk × m A − α g j = ρ rk × ( m − ) ˆ j ρ rkj A − α g j = ρ rk × ( m − ) ˆ j Ψ − rk − α g j = r ∑ l = Ψ − α − rk − ( r − l ) (cid:16) ρ lk × ( m − ) ˆ j g j (cid:17) .The Ψ − α − rk − ( r − l ) operator above satisfies the hypotheses of Theorem 2.2, while ρ lk × ( m − ) ˆ j g E I ˆ j b ∈ W ( ∂ H nj ) . SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 21
Therefore, Theorem 2.2 applies to give the estimates ∑ l (cid:13)(cid:13)(cid:13) Ψ − α − rk − ( r − l ) (cid:16) ρ lk × ( m − ) ˆ j g j (cid:17) (cid:13)(cid:13)(cid:13) W r + α ( R n ) . ∑ l (cid:13)(cid:13)(cid:13) ρ lk × ( m − ) ˆ j g b (cid:13)(cid:13)(cid:13) W − rk − ( r − l ) (cid:16) H n − I , bj (cid:17) . k g b k W r (cid:16) H n − I , bj , ρ ˆ j , k (cid:17) .We note that for negative values of β − α ≥ − W − s ( Ω ) : = (cid:0) W s ( Ω ) (cid:1) ′ = ( W s ( Ω )) ′ and write A − α (cid:18) g E I ˆ j b × δ j (cid:19) = A − α ◦ Λ α − β bj (cid:18) Λ β − α bj g E I ˆ j b × δ j (cid:19) = A − α ◦ Λ α − β bj (cid:18)(cid:16) Λ β − α bj g b (cid:17) E I ˆ j × δ j (cid:19) ,where Λ bj is defined in analogy to the operator Λ : σ (cid:16) Λ −| β − α | bj (cid:17) = ( + ξ + η j ) | β − α | /2 .The proof above can then be applied to the operator A − α ◦ Λ α − β bj ∈ Ψ − β ( R n ) and Λ β − α bj g b ∈ W s (cid:16) H n − I , bj , ρ ˆ j , k (cid:17) . (cid:3) For operators which do not satisfy the hypotheses of Theorem 2.2 we can deriveestimates in a similar manner to the method of Theorem 2.4.
Theorem 4.4.
Let A − α ∈ Ψ − α ( R n ) , − α ≤ − . Then for β ≤ α − , and g b ∈ W s (cid:16) H n − I , bj , ρ ˆ j , k (cid:17) , (cid:13)(cid:13)(cid:13)(cid:13) A − α (cid:18) g E I ˆ j b × δ j (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) W β − s ( R n , ρ , k ) . k g b k W β − α , s (cid:16) H n − I , bj , ρ ˆ j , k (cid:17) . Proof.
The proof of Theorem 4.3 applies up to the last estimate where Theorem2.4 is to be applied as opposed to Theorem 2.2. For this case we need to ensure β ≤ α − (cid:3) In the case of operators acting on functions supported on all of H nI we have thefollowing weighted estimates Theorem 4.5.
Let A ∈ Ψ − α ( R n ) for α ≥ . Let f ∈ W s ( H nI , ρ , k ) . Then k A f k W α , s ( R n , ρ , k ) . k f k W s ( H nI , ρ , k ) . Proof.
We have for r ≤ s , ρ rk × m A f = r ∑ l = Ψ − α − ( r − l ) (cid:16) ρ lk × m f (cid:17) . By Lemma 4.2 we have ρ lk × m f E I ∈ W l ( R n ) from which the estimates (cid:13)(cid:13)(cid:13) ρ rk × m A f (cid:13)(cid:13)(cid:13) W α + r ( R n ) . r ∑ l = (cid:13)(cid:13)(cid:13) ρ lk × m f (cid:13)(cid:13)(cid:13) W l ( H nI ) . k f k W r ( H nI , ρ , k ) follow. Summing over all r ≤ s finishes the proof. (cid:3) We can improve the above Theorem by removing one of the ρ components andusing Theorem 2.5. Theorem 4.6.
Let A ∈ Ψ − α ( R n ) for α ≥ . Let f ∈ W s ( H nI , ρ ˆ j , k ) for some j ∈ I.Then k A f k W α , s ( R n , ρ ˆ j , k ) . k f k W s ( H nI , ρ ˆ j , k ) . Proof.
The proof is almost the same as that of Theorem 4.5. We have for r ≤ s , ρ rk × m ˆ j A f = r ∑ l = Ψ − α − ( r − l ) (cid:16) ρ lk × m ˆ j f (cid:17) .By Lemma 4.2 we have ρ lk × m ˆ j f E I ˆ j ∈ W l ( H nj ) . Thus, an application of Theorem 2.5yields (cid:13)(cid:13)(cid:13) ρ rk × m ˆ j A f (cid:13)(cid:13)(cid:13) W α + r ( R n ) . r ∑ l = (cid:13)(cid:13)(cid:13) ρ lk × m ˆ j f (cid:13)(cid:13)(cid:13) W l ( H nj ) . k f k W r ( H nI , ρ ˆ j , k ) .Summing over all r ≤ s finishes the proof. (cid:3) When working with boundary value problems on intersection domains, or in-tersections of half-planes, restrictions to one boundary of an operator applied to adistribution with support on a different boundary arise. To deal with such terms,we introduce some notation: for α + ∈ N , α ≥ j = k , E jk − α : W s (cid:16) H nI , bj (cid:17) → W s + α (cid:16) H nI , bk (cid:17) ,where E jk − α is of the form(4.2) E jk − α g b = R k ◦ B − α − g j ,where, as above g j : = g E I ˆ j b × δ j , and where B − α ∈ Ψ − α − ( R n ) is decomposable. SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 23
For some crude estimates in the case 1/2 ≤ β ≤ α −
1, which we can alwaysapply to the error terms, Ψ − N g j , from the decomposition of B − α , we could write (cid:13)(cid:13)(cid:13) ρ r λ × ( m − ) ˆ k E jk − α g b (cid:13)(cid:13)(cid:13) W r + β − (cid:16) H nI , bk (cid:17) . (cid:13)(cid:13)(cid:13) ρ r λ × ( m − ) ˆ k R k ◦ Ψ − α − g j (cid:13)(cid:13)(cid:13) W r + β − ( ∂ H nk ) . (cid:13)(cid:13)(cid:13) ρ r λ × ( m − ) ˆ k Ψ − α − g j (cid:13)(cid:13)(cid:13) W r + β ( H nk ) . k g b k W β − α , r (cid:16) H n − I , bj , ρ ˆ k ˆ j , λ (cid:17) ,where in the last step we use the estimates from Theorem 4.4. And after summingover r ≤ s we would have the estimates (cid:13)(cid:13)(cid:13) E jk − α g b (cid:13)(cid:13)(cid:13) W β − s (cid:16) H nI , bk , ρ ˆ k (cid:17) . k g b k W β − α , s (cid:16) H n − I , bj , ρ ˆ k ˆ j (cid:17) for β ≤ α − ρ in the estimateabove; as g b × δ j is supported on H nI , bj , a weighted estimate, using ρ ˆ j is desired onthe right (as opposed to ρ ˆ k ˆ j ). These improvements will be made in the followingCorollary of Theorem 4.3 and 4.4. Corollary 4.7.
With E jk − α as above, − ≤ γ ≤ , and g b ∈ W γ , s (cid:16) H n − I , bj , ρ ˆ j (cid:17) , thenfor β ≥ with β − α ≤ γ , (cid:13)(cid:13)(cid:13) E jk − α g b (cid:13)(cid:13)(cid:13) W β , s ( ∂ H nk , ρ ˆ k , λ ) . k g b k W β − α , s (cid:16) H n − I , bj , ρ ˆ j , λ (cid:17) . Proof.
For given α , s , we choose N large, α + s + ≪ N , and we write E jk − α g b = R k ◦ A − α − g j + R k ◦ Ψ − N g j ,where A − α − satisfies the hypotheses of Theorem 2.2. The estimates precedingthe statement of the Corollary apply to the R k ◦ Ψ − N g j term, and we just have toestimate R k ◦ A − α − g j = ( π ) n Z a ( x , ρ ˆ k , ξ , η ) e g b ( ξ , η ˆ j ) e i ρ ˆ k · η ˆ k e ix ξ d ξ d η .By defintion of the E jk − α operator, α + ∈ N , and the symbol, a , of A − α − ismeromorphic with poles as in Theorem 2.2.We first integrate over the η j variable. Let ς j be a pole (with negative imaginarypart denoted Im ( ς j ) = − ν j ) with respect to η j of the symbol a . We obtain a sum ofterms of the form Z ρ α j γ α ( x , ρ ˆ k , ξ , η ˆ j ) e g b ( ξ , η ˆ j ) e i ρ j ς j e i ρ ˆ j ˆ k · η ˆ j ˆ k e ix ξ d ξ d η ˆ j , where γ α ∈ S − α ( R n − ) with α + α = α − L estimates, using | γ α | . ( η j + ξ ) α and Z − ∞ ρ α j e ρ j ν j d ρ j ≃ ν α + j . ( η j + ξ ) α + we can estimate (cid:13)(cid:13)(cid:13)(cid:13) Z ρ α j γ α ( x , ρ ˆ k , ξ , η ˆ j ) e g b ( ξ , η ˆ j ) e i ρ j ς j e i ρ ˆ j ˆ k η ˆ j ˆ k e ix ξ d ξ d η ˆ j (cid:13)(cid:13)(cid:13)(cid:13) L . Z ( η j + ξ ) α + α + | e g b ( ξ , η ˆ j ) | d ξ d η ˆ j . k g b k W − α (cid:16) H n − I , bj (cid:17) .In general, we have(4.3) (cid:13)(cid:13)(cid:13) R k ◦ A − α − g j (cid:13)(cid:13)(cid:13) W β ( R n − ) . k g b k W β − α ( H n − I , bj ) ,for 0 ≤ β ≤ α + γ = H I , bk , with weights which do notinclude factors of ρ k , we use the identity R k ◦ Ψ − α g j ≃ R k ◦ ∂ s ∂ρ sk ρ sk Ψ − α g j ,with α ≥
1. Note that the regularity provided by the order, − α ≤ − ρ k produce no δ (or derivatives of δ ) terms. Thus, we write ρ r λ × ( m − ) ˆ k R k ◦ A − α − g j ≃ R k ◦ ∂ r λ ∂ρ r λ k ρ r λ × m A − α − g j ,and further, we use the relation ρ r λ × m A − α − g j = ρ r λ × ( m − ) ˆ j Ψ − α − − r λ g j = r ∑ l = Ψ − α − − r λ − ( r − l ) (cid:16) ρ l λ × ( m − ) ˆ j g j (cid:17) to show ∂ r λ ∂ρ r λ k ρ r λ × m A − α − g j = r ∑ l = Ψ − α − − ( r − l ) (cid:16) ρ l λ × ( m − ) ˆ j g j (cid:17) . SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 25
Therefore, we have ρ r × ( m − ) ˆ k R k ◦ A − α − (cid:0) g b × δ j (cid:1) = R k ◦ r ∑ l = Ψ − α − − ( r − l ) (cid:16) ρ l λ × ( m − ) ˆ j g j (cid:17) .The calculation of the estimates then come from (4.3), and give (cid:13)(cid:13)(cid:13) ρ r λ × ( m − ) ˆ k R k ◦ A − α − (cid:0) g b × δ j (cid:1)(cid:13)(cid:13)(cid:13) W β + r ( R n − ) . r ∑ l = (cid:13)(cid:13)(cid:13) ρ l λ × ( m − ) ˆ j g b (cid:13)(cid:13)(cid:13) W β − α − ( r − l )+ r ( H n − I , bj ) = ∑ l ≤ r (cid:13)(cid:13)(cid:13) ρ l λ × ( m − ) ˆ j g b (cid:13)(cid:13)(cid:13) W β − α + l ( H n − I , bj ) .Lastly, summing over r ≤ s yields the Lemma. (cid:3) For boundary operators mapping H nI , bj to itself we use the notation E jj − α to de-note either operators of the form E jj − α = R j ◦ B − α − ,where B − α − ∈ Ψ − α − ( R n ) is decomposable, for α ≥
1, or of the form E jj − α = Ψ − α bj ,for α ≥
1, or to denote compositions: E jj − α = E kj − α ◦ E jk − α ,where α = α + α and α , α ≥ (cid:13)(cid:13) R j ◦ B − α − g j (cid:13)(cid:13) W β , s (cid:16) ∂ H nj , ρ ˆ j , λ (cid:17) . k g b k W β − α , s (cid:16) H n − I , bj , ρ ˆ j , λ (cid:17) .The estimates for E jj − α = Ψ − α bj b are obvious, while Corollary 4.7 applied (twice) to E jj − α = E kj − α ◦ E jk − α yield(4.4) (cid:13)(cid:13)(cid:13) E kj − α ◦ E jk − α g b (cid:13)(cid:13)(cid:13) W β , s (cid:16) ∂ H nj , ρ ˆ j , λ (cid:17) . k g b k W β − α , s (cid:16) H n − I , bj , ρ ˆ j , λ (cid:17) ,for large enough β , e.g. in the case E jj − α = E kj − α ◦ E jk − α , where the two operatorscomposed on the right have the form (4.2), we have the estimates in (4.4) for β ≥ α .In each case, we have the estimates (cid:13)(cid:13)(cid:13) E jj − α g b (cid:13)(cid:13)(cid:13) W β , s (cid:16) ∂ H nj , ρ ˆ j , λ (cid:17) . k g b k W β − α , s (cid:16) H n − I , bj , ρ ˆ j , λ (cid:17) ,again for sufficiently large β .
5. P
OISSON AND G REEN OPERATORS
The motivation of the weighted estimates in Section 4 is the study of estimatesfor operators of boundary value problems such as the Poisson and Green oper-ators. Let Ω , . . . , Ω m ⊂ R n be smoothly bounded domains which intersect realtransversely. That is to say, if ρ j is a smooth defining function for Ω j , | d ρ j | 6 = ∂ Ω j , then d ρ ∧ · · · ∧ d ρ m = Ω is a piecewise smooth domain. Then using a suitable metriclocally near a point on ∂ Ω the intersection can be modeled by the intersection of m half-planes.Weighted estimates for the Poisson and Green operators follow as in Theorems3.1 and 3.2, however the calculations for the Poisson operator, specifically thoseregarding the Dirichlet to Neumann operator (DNO), which we recall gives theboundary values of the normal derivatives of the solution to a Dirichlet’s problem,are considerably more involved. Such calculations have been worked out in [3],and so here we just state the result: Theorem 5.1.
Let Ω be a piecewise smooth domain. Let g = ∑ mj = g bj × δ j with g bj ∈ W s (cid:16) ∂ Ω j T ∂ Ω , ρ ˆ j , k (cid:17) for integer s ≥ . Denote by P ( g ) the solution operator to (3.2) .Then for k ∈ N , the estimates k P ( g b ) k W s ( Ω , ρ , k ) . ∑ j (cid:13)(cid:13)(cid:13) g bj (cid:13)(cid:13)(cid:13) W s (cid:16) ∂ Ω j T ∂ Ω , ρ ˆ j , k (cid:17) . hold. Estimates for Green’s operator follow as a consequence of those for the Poissonoperator just as in Section 3 above. We now consider the problem (3.1) on a piece-wise smooth domain, Ω . We let f ∈ L ( Ω ) and extend f by zero to all of R n toa function, F ∈ L ( R n ) (with the Extension Theorem; see Theorem 1.4.2.4 in [4]).We let U ∈ W ( R n ) be a solution to Γ U = F in R n .Using Lemma 4.1, we have that U | Ω j ∈ W s ( Ω j , ρ ˆ j ) for all integer s ≥
0. Usingthe Trace Theorem to restrict to ∂ Ω j (see Theorem 1.5.1.1 in [4]), we have U | ∂ Ω j ∈ W s ( ∂ Ω j ∩ ∂ Ω , ρ ˆ j ) From above, we then have k P ( U | ∂ Ω ) k W s ( Ω , ρ , k ) . ∑ j k U | ∂ Ω k W s ( ∂ Ω j ∩ ∂ Ω , ρ ˆ j , k ) . SEUDODIFFERENTIAL ANALYSIS ON DOMAINS WITH BOUNDARY 27
Using the Trace Theorem on the domain, Ω , by taking extensions by 0 from ∂ Ω j ∩ ∂ Ω to all of ∂ Ω , we have k U | ∂ Ω k W s ( ∂ Ω j ∩ ∂ Ω , ρ ˆ j , k ) . k U | Ω k W s ( Ω , ρ ˆ j , k ) .We now set G ( f ) to U − P ( U | ∂ Ω ) restricted to Ω . Note from the form U = Ψ − F and from (the proof of) Theorem 4.5, we also have the estimates k U k W s ( R n , ρ ˆ j , k ) . k F k W s ( R n , ρ ˆ j , k ) .We can now prove Theorem 5.2.
Let G ( f ) denote the solution, u, to the boundary value problem (3.1) . Thenfor any j, k G ( f ) k W s ( Ω , ρ ˆ j , k ) . k f k W s ( Ω , ρ ˆ j , k ) , for integer s ≥ and k ∈ N .Proof. From above, we estimate k G ( f ) k W s ( Ω , ρ ˆ j , k ) . k U k W s ( R n , ρ ˆ j , k ) + k P ( U | ∂ Ω ) k W s ( Ω , ρ ˆ j , k ) . k U k W s ( Ω , ρ ˆ j , k ) . k F k W s ( Ω , ρ ˆ j , k ) . (cid:3) R EFERENCES[1] D. Ehsani. Dirichlet to Neumann operators and the ¯ ∂ -Neumann problem. Preprint.[2] D. Ehsani. Exact regularity of the ¯ ∂ -problem with dependence on the ¯ ∂ b -problem on weakly pseu-doconvex domains in C . Preprint.[3] D. Ehsani. Weighted estimates for the ¯ ∂ -Neumann problem on intersections of strictly pseudocon-vex domains in C . Preprint.[4] P. Grisvard. Elliptic problems in nonsmooth domains . Number 24 in Monographs and studies in math-ematics. Pitman Advanced Pub. Program, 1985.[5] Ewa Ligocka. The Sobolev spaces of harmonic functions.
Studia Math. , 84:79–87, 1986.[6] J.-L. Lions and E. Magenes.
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OCHSCHULE M ERSEBURG , E
BERHARD -L EIBNITZ -S TR . 2, D-06217 M ERSEBURG , G
ERMANY
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