Uniform Convolution and Fourier Restriction estimates for complex polynomial curves in C 3
aa r X i v : . [ m a t h . C V ] D ec Uniform Convolution and Fourier Restriction estimates forcomplex polynomial curves in C Abstract
We establish optimal ( p, q ) ranges for two types of estimates associated to threedimensional complex polynomial curves. These are the estimates for the weightedrestriction of the Fourier Transform to a complex polynomial curve, and the weightedConvolution Operator associated to a complex polynomial curve. Establishing theseestimates comes down to establishing a lower bound for the Jacobian of a mappingassociated to the complex curve in question.
Consider the complex polynomial curve Γ( z ) := ( P ( z ) , P ( z ) , P ( z )), for complex polyno-mials P i ( z ). If we let L Γ ( z ) := | det(Γ ′ ( z ) , Γ ′′ ( z ) , Γ ′′′ ( z )) | , and λ Γ ( z ) := | L Γ ( z ) | , we definethe weighted Complex Convolution Operator associated to Γ as: T Γ f ( z ) := Z D f ( z − Γ( w )) λ Γ ( w ) dw, where D is some disk of radius r . In this paper, we establish the following family of ( p, q )bounds for T Γ : Theorem 1.1.
Let ( p θ , q θ ) = ( θ , θ ) . Suppose f ∈ L p θ , and Γ : C → C is a degree N polynomial curve. Then there exists a constant C = C ( N, θ ) such that: || T Γ f || L qθ ( C ) ≤ C || f || L pθ ( C ) , for all θ ∈ (0 , .Furthermore, T Γ is of restricted weak-type at ( p , q ) , and at ( p , q ) , i.e. for measurablesubsets of C , E and F , there exists C = C ( N ) such that h T χ E , χ F i ≤ C | E | p i | F | − qi , for i = 0 or . Utilising this theorem, some easy to obtain estimates, and the Marcinkiewicz Interpo-lation Theorem, we can obtain a larger range of local estimates.1
Corollary 1.2.
With ( p θ , q θ ) as in Theorem 1.1, let R be the convex hull of the points (1 , , (0 , , (cid:0) p − , q − (cid:1) , and (cid:0) p − , q − (cid:1) .If ( p − , q − ) ∈ R \ (cid:8)(cid:0) p − , q − (cid:1) , (cid:0) p − , q − (cid:1)(cid:9) , then for all complex polynomial curves, Γ ,there exists a constant C = C (Γ , r, p, q ) , where r is the radius of the disk associated to T Γ ,such that for all functions f ∈ L p ( C ) : || T Γ f || L q ( C ) ≤ C || f || L p ( C ) . Much work has been done on estimates of this sort are known when T Γ is replacedwith it’s real analogue. In the real case, in arbitrary dimension, d , Γ is a real polynomialcurve, the integration is over some interval, and the weight function λ Γ is replaced with λ Γ ( t ) := L d ( d +1) Γ ( t ). Furthermore, our constant C must depend on the dimension we are infor these estimates to hold.Usually, these estimates have been for a particular family of polynomial curves, or ina particular dimension. For example, in 1998, in [4] one of the early papers on this topic,Christ established this result for any moment curve in any dimension. That is to say heestablished it for Γ( t ) = (cid:0) t, t · · · t n (cid:1) in R n . It should be noted that for curves like this,we have λ Γ ( t ) = n !, which simplifies the problem. This can be considered the most non-degenerate case, as all derivatives of Γ are linearly independent. Later papers introducedthe weight λ Γ in order to deal with problematic singular points of Γ, where the derivativesbecome linearly dependent. At such points, λ Γ vanishes, allowing us to mitigate the effectof such points during the integration.Further estimates were established for other families of curves, but it was in 2002, in [10],that Oberlin established the real estimate for general polynomial curves in two dimensions,and in three dimensions, established it for curves of the form γ ( t ) = ( t, P ( t ) , P ( t )). In [7],this three dimensional case was extended to general polynomial curves, and in [15], this wasextended to general higher dimensions.The first such estimates for surfaces instead of curves can be found in [9], in which Druryand Guo proved a class of estimates of this type, but instead of convolving with a measurearising from a polynomial curve, their estimates were pertaining to the case where Γ wasinstead a two dimensional surface in R , of the form ( x, y, φ ( x, y ) , φ ( x, y )), with certainconditions on φ and φ .In [6], Chung and Ham established similar results, considering surfaces that could berepresented as complex curves in C d . They established these bounds for curves of the form (cid:0) z, z · · · z N (cid:1) in C d , and (cid:0) z, z , φ ( z ) (cid:1) in C , for analytic φ . They do this by using a morepowerful analogue to our Lemma 3.1, established in the complex Fourier Restriction paper[1], involving a lower bound in terms of the arithmetic means of { L Γ ( z i ) } , as opposed tothe geometric mean. This method suffices for the complex curves in question in [6], butin general does not hold. In this paper, we do not utilise a lower bound in terms of thearithmetic mean, but instead adapt the geometric mean bound deriving from [8] into thecomplex case, and achieve the overall estimate using that.Note that while the two-dimensional estimate has not been established for general com-plex polynomial curves, these estimates can be derived from the procedure described in ourproof of Theorem 1.1, with simple modifications. The explicit description of this is omittedto aid the presentation of the three dimensional case.The second types of estimates we wish to establish are pertaining to the restriction ofthe Fourier transform to Γ, and are formulated as follows: Theorem 1.3.
Let L be the line segment joining the points (1 , and ( , ) , excluding theendpoint ( , ) . If ( p − , q − ) ∈ L , then for all complex polynomial curves, Γ , of degreee N ,there exists a constant C = C ( p, N ) , such that for all functions f ∈ L q : || ˆ f (Γ( z )) || L q ( λ Γ ( z ) dz ) ≤ C || f || L p ( dz ) , where ˆ f denotes the Fourier Transform of f . Here, we should note that ˆ f (Γ( z )) is understood as ˆ f (Γ( x, y )), where z = x + iy , andwe regard Γ as a real surface, in order to ensure a sensible notion of convergence for it’sFourier transform. This detail will be omitted, unless in the case of potential confusion.Note that in [1], Bak and Ham established the optimality of the weight function λ Γ inthe complex case, in the sense that it is the largest possible weight such that we can obtainthis range of ( p, q ) values. Furthermore, in [16], we see this this is the largest possible ( p, q )range for estimates of this type.With regard to the history of this problem, initial investigations pertained to the realcase, with γ being a real curve instead of a surface . In [12], Sj¨oln demonstrated that thedifficulty in acquiring such bounds was in large part due to potential oscillations of thequantity L Γ . Considering the curve Γ( t ) = ( t, e − t sin t ) for t close to 0 demonstrates thisissue. In this paper, Sj¨oln avoided issues like this by restricting Γ to convex curves, ensuringthe single signedness of L Γ .There are several results that could be cited here, such as [2], which establishes the esti-mate for γ ( t ) being an abitrary monomial curve in R d , that is to say each of the componentsof Γ is some monomial. Furthermore, in the same paper, these estimates are established forΓ( t ), being of what is called “simple type”, which is to Γ( t ) = (cid:0) t, t · · · P ( t ) (cid:1) for polynomial P . Usually, attempts to obtain these restriction estimates aim for one of two differentranges, which each depend on the dimenson. There is the full range, in which we have( p, q ) satisfying p ∈ [1 , d + d +2 d + d ), and p ′ = d + d +22 q , and the shorter range where we still have p ′ = d + d +22 q , but we only have p ∈ [1 , d + d +2 d +2 d − ). Note that here, p ′ denotes the H¨oldersconjugate of p . In [8], the short range was established for arbitrary real polynomial curves,in any dimension, being done by decomposing R into intervals on which the quantity L γ behaved as a monomial on any interval arising from this decomposition, and this method C was utilised and improved in [14] to cover the full range, giving a complete answer to thistype of estimate in the real case, in the context of boundedness between L p spaces.The complex case also has some results established. In [11] Oberlin established thesebounds over the full range for surfaces of the form ( z, φ ( z )) with some non-degeneracyconditions on φ .In [1], this estimate was established on the full range for complex curves of the formΓ( z ) = (cid:0) z, z · · · z N (cid:1) in C d , for d ≥
3, and for Γ( z ) = (cid:0) z, z , φ ( z ) (cid:1) in C , for arbitraryanalytic φ .Before moving onto the process of proving Theorem 1.1 and 1.3, we first introducesome commonly accepted notation for these kinds of problems. We say that, for positivequantities A and B , we have A . B if A ≤ CB , for some constant C . Here, C , the implicitconstant, can depend on whatever parameters are appropriate for the problem in question.As Theorem 1.1 follows from a single restricted weak-type estimate at ( p , q ), whenproving this theorem, we only want our implicit C to be dependant on N , the degree of Γ.In Theorem 1.3, we prove the estimate via many iterative estimates, so we would expect C to depend on p in this context. We also say A ∼ B if A . B . A . C To achieve the desired results, we first partition C into various convex sets. This is done byutilising two decompositions as described in [8]. In [8], these decompositions are describedin relation to R , but can be extended to C . The first of these decompositions will be referredto as D1.D1: Given a polynomial Q ( z ), we can decompose C into a bounded number of convexsets, B i . On each of these B i , we have | Q ( z ) | ∼ c ( Q ) | z − b i | k i , for b i some root of Q ( z ).We will frequently omit the subscript of k and b for the sake of convenience.To see how this decomposition functions, we first write Q ( z ) = A d ′ Y j =1 ( z − η j ) α j , with d ′ being the amount of distinct roots of Q . Here, these roots η j are written such that | η | ≤ | η | ≤ · · · ≤ | η d ′ | . We now decompose C into a union of d ′ convex sets, givenby the Voronoi diagram associated to { η , η , · · · , η d ′ } . On these sets, we will have ourcomplex centers b be the root associated to this set. Formally these sets are denoted S ( η j ) = { z ∈ C | | z − η j | ≤ | z − η i | for all i = j } .Each of these sets are further divided into finitely many sectors, centered at b . For n ≥ nε ( η j ) = { z − η j = re iθ ∈ S ( η j ) | ( n − ε ≤ θ ≤ nε } . Here, ε is chosen such that πε is an integer. Specifically we need to take ε ≤ π , which is necessary to decompose theintersection of some annuli and these sectors into convex sets. While this is sufficient for D ε to be dependent on d , the degree of our polynomial curve. Thus the choice of ε is currently not specified.Fixing one such ∆ nε ( η j ) = ∆( η j ), we relabel η j = ˆ η , and the remaining roots as ˆ η k sothat we have | ˆ η − ˆ η | ≤ | ˆ η − ˆ η | ≤ · · · ≤ | ˆ η − ˆ η d ′ | . That is to say we order them accordingto their distance from η j We let, for i ≥ T ni,j = { z ∈ ∆( η j ) | | z − ˆ η | ≤ | ˆ η − ˆ η i |} .Labelling T ,j = ∅ , and T d ′ +1 ,j = ∆( η j ), we see that T .j ⊆ T .j ⊆ · · · ⊆ T d ′ .j ⊆ T d ′ +1 .j =∆( η j ). Also note that if z ∈ T ni,j , then | ˆ η − ˆ η k | ∼ | z − ˆ η k | for all k ≥ i , and if z / ∈ T ni,j , then | z − ˆ η | ∼ | z − ˆ η k | for all k ≤ i .Letting, for 1 ≤ i ≤ d ′ , I ni,j = T ni +1 ,j \ T ni,j , we can see that ∆ nε ( η j ) = S d ′ k =1 I nk,j , and onany fixed I k,j , we have: | Q ( z ) | = | A | d ′ Y l =1 | z − η l | α l ∼ | z − ˆ η | ˆ α +ˆ α + ··· +ˆ α k | A | d ′ Y l = k +1 | ˆ η − ˆ η l | ˆ α l , for ˆ α j being the multiplicity of ˆ η j .So indeed, writing C = πε d ′ [ l =1 I l = d ′ [ j =1 πε [ n =1 d ′ [ i =1 I ni,j , we have on B l = I ni,j that: | Q ( z ) | ∼ c ( Q ) | z − b l | k l , (2.1)where b l = b i,j,n = η j , and k l = k i,j,n = ˆ α + ˆ α · · · + ˆ α i .Note that these sets are intersections of some convex sets, and annuli centered at b l , andtherefore are not necessarily convex. This can be remedied by “convexifying” our annuli.Note that we have ε < π , and we can scale down the inner radius and scaling up theouter radius of each annuli. As long as these scalings are given by a constant multiplicativefactor, B ( ε ), the new sets retain the relevant properties of our original sets, but are nolonger disjoint from their ’neighbouring’ annuli. This means that the estimate (2.1) foreach of the original annuli will hold on the intersection of the two scaled annuli.Consider the bisector of ∆ nε ( η j ), and its intersections with the innermost boundary ofthe intersection of these annuli. Note that this boundary arises from the annulus that comes later in our sequence of annuli, because of our thickening. By choosing B suitably, we cancontrol the thickness of the two annulis’ intersection, and ensure that the tangent to theinnermost boundary divides the intersection into two sets, without intersecting the outerboundary. This then allows us to replace the our annuli with convex sets by replacing thecurved boundaries of these sets with these tangent lines.For the next decomposition, referred to as D2, we require a polynomial Q ( z ), and acentre, which is some complex number b .D2: Given a polynomial Q ( z ), and complex number b , and some convex set J , then J can be decomposed into disjoint convex sets. These convex sets are of two types, eitherdyadic or gap. On Dyadic sets, | z − b | ∼ c ( Q ), and on Gap sets, Q ( z ) ∼ c ( Q ) | z − b | k , forsome positive integer k . Also, for Q ( z + b ) = Σ c i z i , if c j = 0, then there are no gap intervalssuch that | Q ( z ) | ∼ c ( Q ) | z − b | j .Note that, if we label the collection of Dyadic sets as { D j } , and the gaps as { G j } wehave: C • D j = { z ∈ C | A − | z j | < | z | < A | z j |} , • G j = { z ∈ C | A | z j | < | z | < A − | z j +1 |} ,for A ∈ R , and z j being the roots of Q ( z ), ordered by magnitude.D2 originates from [3], and describes a decomposition of R . However, if we restrictourselves to sectors like ∆ nε ( η j ), where η j are the roots of Q , and Voronoi sets S ( η j ), thenthe method can be extended to the complex case. Note again however that this will giveus disjoint annuli in the complex case, so we repeat the thickening of these annuli, andconverting them into corresponding convex sets as in D1.For now, we will assume that C ( Q ) = c ( Q ) = c ( Q ) = 1, with the justification for thisassumption being apparent whe we introduce the main inequality we seek to establish.We also make use of the integral representation of the Jacobian in [8]. This states that,in 3-dimensions, for L ( z ) = L Γ ( z ), L ( x ) = det P ′ ( z ) P ′′ ( z ) P ′ ( z ) P ′′ ( z ) ! , and L ( z ) = P ′ ( z ),then if we consider the mapping Φ Γ ( z , z , z ) = Γ( z ) + Γ( z ) + Γ( z ), we have: J Φ Γ ( z , z , z ) = Y s =1 L ( z s ) Z z z Z z z Y s =1 L ( w s ) L ( w s ) Z w w L ( y ) L ( y ) L ( y ) dydw dw , (2.2)Where J Φ Γ is the real jacobian of the mapping Φ Γ , and the complex bounds of integra-tion, u, v in each of the above integrals are to be interpreted as the line integral betweenthose two complex numbers along the curve given by Γ( t ) = ut + ( v − u ) t for t ∈ [0 , C into convex sets so that L , L , and L can be treated as either monomials or constants. This is done as follows.First, we use D1 with respect to L , so that we get a finite amount of convex sets suchthat | L | ∼ | z − b | k on each of these sets, with k and b depending on the sets. Next, wedivide this family of sets into two types of sets, referred to as T or T sets. To do this,we first apply D2 to each set already obtained with respect to our center from the firstdecomposition, and the polynomial L . T sets are those sets are gap annuli, on which | L ( z ) | ∼ | z − b | k , and | L ( z ) | ∼ | z − b | k . T sets are dyadic annuli those on which | z − b | ∼
1, so that | L ( z ) | ∼
1. On these setswe apply D1 with respect to L to decompose these sets into a family of convex sets onwhich | L ( z ) | ∼ | z − b ′ | k .Finally, we apply D2 again to the T and T sets, but this time, with respect to L ,and the center b , and b ′ respectively. This divides the T sets into two new families of sets,referred to as T sets and T sets. T sets are those on which | L ( z ) | ∼ | z − b | k , while we still have | L ( z ) | ∼ | z − b | k ,and | L ( z ) | ∼ | z − b | k T sets are those on which | z − b | ∼
1, so that | L ( z ) | ∼ | L ( z ) | ∼
1. Like previously,we apply D1 with respect to L to decompose these sets into a family of convex sets onwhich | L ( z ) | ∼ | z − b ′′ | k .We decompose the T sets precisely the same way to get families of sets named T and T .On T sets, we have | L | ∼ | L | ∼ | z − b ′ | k and | L ( z ) | ∼ | z − b ′ | k .On T sets, we have | L ( z ) | ∼ | L ( z ) | ∼
1, and | L ( z ) | ∼ | z − b ′′ | k .Note that the various b and k values depend on the specific set within these families thatthey belong to, as opposed to every set of type T having the same k value for example.Furthermore, we shall write b ′ and b ′′ as just b for simplicity for the rest of this discussion.Note to make use of these estimates, we need the following lemma. Lemma 2.1.
For all z , z , z in one of the sets in our decomposition, we have that | J Φ Γ ( z , z , z ) | ∼ Y s =1 | L ( z s ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z z z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z z z Y s =1 | L ( w s ) || L ( w s ) | (cid:12)(cid:12)(cid:12)(cid:12)Z w w | L ( y ) || L ( y ) || L ( y ) | dy (cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Proof:
To see this, we introduce the notion of a sector contained function.
Definition 2.2. An ε -sector contained function of B is a function, f , such that for all z ∈ B , we can write f ( z ) = r ( z ) e iθ ( z ) , with θ ( z ) ∈ [ θ , θ + ε ]. That is to say, the image of B under f is contained within a sector of aperture ε .We will omit the reference to the set B when we refer to function that satisfy thisdefinition, understanding that a ε -sector contained function refers to a ε -sector containedfunction of D , where D is any set in our decomposition.Note that the reason we introduce the notion of these functions is because for f a ε -sector contained function, we have (cid:12)(cid:12)R vu f ( z ) dz (cid:12)(cid:12) ∼ (cid:12)(cid:12)R vu | f ( z ) | dz (cid:12)(cid:12) for u, v in any set of ourdecomposition, for ε = ε ( d, N ) < π .To see this, note that we immediately have that (cid:12)(cid:12)R vu f ( z ) dz (cid:12)(cid:12) ≤ (cid:12)(cid:12)R vu | f ( z ) | dz (cid:12)(cid:12) , as (cid:12)(cid:12)(cid:12)(cid:12)Z vu f ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ( v − u ) Z f ( z ( t )) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ( v − u ) Z | f ( z ( t )) | dt (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z vu | f ( z ) | dz (cid:12)(cid:12)(cid:12)(cid:12) . To establish the second inequality, notice that by factoring out some complex numberof unit length, we can assume the sector f ( z ) is contained in a sector which has one ray inthe direction of the positive real axis, and the second ray being being given by is the linecontaining the complex numbers with argument ε .We therefore have: (cid:12)(cid:12)(cid:12)(cid:12)Z vu f ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ( v − u ) Z Re( f ( z ( t ))) + i Im( f ( z ( t ))) dt (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) ( v − u ) Z Re( f ( z ( t ))) dt (cid:12)(cid:12)(cid:12)(cid:12) ∼ (cid:12)(cid:12)(cid:12)(cid:12)Z vu | f ( z ) | dz (cid:12)(cid:12)(cid:12)(cid:12) . C Therefore, to establish our lemma, we would like to show that each integrand is an ε -sectorcontained function. However, because the argument of w − w cannot be controlled, thisis not quite true, so we instead show that the integrand is the product of some ε containedfunction, and ( w − w ), which we will see is sufficient to establish our lemma.In [6], it was shown in step 2 of lemma 4.2 that for a polynomial P ( z ) = Q dj =1 ( z − η j ), if we are in ∆( η j ) ε T S ( η j ), as in the terminology of D1, then wehave, by factoring P ( z ) = g j ( z )( − d − j z j Q dl = j +1 η l , that on the j -th gap or dyadic annulus,we can decompose these sets into a bounded number of sets so that g j ( z ) is contained ina sector of aperture less that ε . As ( − d − j z j Q dl = j +1 η l will be contained in a sector ofaperture jε , we have that P ( z ) will be in a sector of aperture at most ( d + 1) ε for z inany of our sets. Notice that both decomposition techniques D D η j ) ε T S ( η j ), and on each of our sets we have applied D D L , L and L . Therefore, we have no concern about utilising this result for L , L and L simultaneously.So therefore, for any set in our decomposition, we can decompose it further to ensureeach of our L i are ( d i + 1) ε -sector contained functions, for d i the degree of L i . Therefore,we have that the rational function L L L is a 4( d + 1) ε -sector contained function, and L L isa 3( d + 1) ε -sector contained function where d is the maximum of the d i values.Lastly, we wish to observe that R w w f ( y ) dy is the product of a ε -sector contained functionand ( w − w ) if f is a ε -sector contained function, and w , w are contained within one ofour decomposed sets. This follows from the convexity, and closedness of sectors.Letting R w w f ( y ) dy = ( w − w )lim n →∞ n Σ nj =1 f ( y ( jn )) , for y ( t ) = w + ( w − w ) t .Now, as f ( y ) is in a sector for all y on this line segment, as the sector is convex, each n Σ nj =1 f ( y ( jn )) is in the sector, so therefore the limit of this sequence is in the sector.Therefore, we have that R w w f ( y ) dy = ( w − w ) F ( z ) , where F ( z ) = R ( f ( y ( t )) dt is a ε -sectorcontained function. Now note that we can write: J Φ Γ ( z , z , z ) = Y s =1 L ( z s ) Z z z Z z z Y s =1 L ( w s ) L ( w s ) Z w w L ( y ) L ( y ) L ( y ) dydw dw = Y s =1 L ( z s ) ! ( z − z )( z − z ) I where: I := Z Z
10 2 Y s =1 L ( w s ) L ( w s ) ( w − w )) Z L ( y ) L ( y ) L ( y ) dt dt dt . Here, note that w = w ( t ) , w = w ( t ) , and y = y ( t , t , t ) .We now wish to bound the size of the remaining integral I . Let α ( t , t ) and β ( t , t ) bereal valued functions such that w − w = α + iβ . Furthermore, as these polynomials, afterour decomposition, are section-contained functions, and the integral along the line from w to w of a sector contained function is a sector contained function multiplied by ( w − w ) .Therefore, the above integral can be written as: I = Z Z ( α + iβ ) g ( t , t ) dt dt , where g is a (7( d +1)) ε -sector contained function. That is to say g ( t , t ) = ξ ( t , t )+ iη ( t , t ) , forreal functions ξ and η with | g ( t , t ) | ∼ ξ ( t , t ) . Note that we make the following assumptionsabout the points z , z , z . These assumptions can be made without loss of generality viarenaming these points for the first two, and factoring out a unit complex number for thelast. First, we assume that in the triangle z , z , z , the angle at z , which we label θ , is thelargest in the triangle. Next, we assume | z − z | ≥ | z − z | . Lastly we assume z − z is apositive real number.Note that this immediately gives that β is single signed, and θ ∈ [ π , π ]. We computeour estimate by splitting into cases when θ is acute, and when it is obtuse.Case i): If θ ∈ [ π , π β ≥
0. The case where β < | β | instead of β when applying the triangle inequality in the following argument: | I | ≥ (cid:12)(cid:12)(cid:12)(cid:12) Im (cid:18)Z Z ( α + iβ ) g ( t , t ) dt dt (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z β> | α | ( βξ + ηα ) dt dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z β> | α | ( βξ − | η || α | ) dt dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ Z Z β> | α | βξdt dt . Now, on the set we restrict our integral to, β ∼ | w − w | , and we have ξ ∼ | g ( t , t ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y s =1 L ( w s ( t s )) L ( w s ( t s )) Z L ( y ( t )) L ( y ( t )) L ( y ( t )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Y s =1 | L ( w s ( t s )) || L ( w s ( t s )) | (cid:12)(cid:12)(cid:12)(cid:12)Z L ( y ( t )) L ( y ( t )) L ( y ( t )) dt (cid:12)(cid:12)(cid:12)(cid:12) . Now, as the integrand here is a sector contained function, we can move the absolutevalue signs inside the integral, yielding | g ( t , t ) | ∼ Y s =1 | L ( w s ( t s )) || L ( w s ( t s )) | (cid:12)(cid:12)(cid:12)(cid:12)Z | L ( y ( t )) || L ( y ( t )) || L ( y ( t )) | dt (cid:12)(cid:12)(cid:12)(cid:12) ∼ Y s =1 | w s ( t s ) | σ (cid:12)(cid:12)(cid:12)(cid:12)Z | y ( t ) | σ dt (cid:12)(cid:12)(cid:12)(cid:12) . (2.3)Subbing this into (2 .
4) gives | I | & (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z t ( t ) | w − w | Y s =1 | w s ( t s ) | σ (cid:12)(cid:12)(cid:12)(cid:12)Z | y ( t ) | σ dt (cid:12)(cid:12)(cid:12)(cid:12) dt dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , C where t ( t ) is the smallest value of t for which the complex number w − w hasimaginary part greater than half the absolute value of it’s real part. That is to say, t ( t ) isthe smallest value such that β > | α | θ and the relativelengths of our line segments, we have that t ( t ) < for all t .From here we will proceed to estimate the t integrand, by considering G ( t ) = Z t ( t ) | w − w | Y s =1 | w s ( t s ) | σ (cid:12)(cid:12)(cid:12)(cid:12)Z | y ( t ) | σ dt (cid:12)(cid:12)(cid:12)(cid:12) dt = Z t ( t ) | w − w | Y s =1 | w s ( t s ) | σ (cid:12)(cid:12)(cid:12)(cid:12)Z | y ( t ) | σ dt (cid:12)(cid:12)(cid:12)(cid:12) dt + Z t ( t ) t ( t ) | w − w | Y s =1 | w s ( t s ) | σ (cid:12)(cid:12)(cid:12)(cid:12)Z | y ( t ) | σ dt (cid:12)(cid:12)(cid:12)(cid:12) dt := G + G . Now note that L = z − w is a positive real number, and for all the t in the G integral, we have L ≤ | w − w | ≤ L , and also that this relation holds for t < t ( t ).Furthermore, we also note that the triangle with vertices z , z , w (2 t (0)) is containedin the circle centered at z , with radius ε | z | . This can be seen by considering the casewith z , z are positive real numbers, z is on the other ray of our sector, θ = π , and | z − z | = | z − z | , as this arrangement maximises the distance from z and w (2 t (0)).Note for all points z in this circle, we have | z | ∼ | z | . We therefore have: G ∼ Z t ( t ) t ( t ) | w − w || z | σ + σ (cid:12)(cid:12)(cid:12)(cid:12)Z dt (cid:12)(cid:12)(cid:12)(cid:12) dt ∼ Z t ( t )0 | w − w || z | σ + σ (cid:12)(cid:12)(cid:12)(cid:12)Z dt (cid:12)(cid:12)(cid:12)(cid:12) dt ∼ Z t ( t )0 | w − w | Y s =1 | w s ( t s ) | σ (cid:12)(cid:12)(cid:12)(cid:12)Z | y ( t ) | σ dt (cid:12)(cid:12)(cid:12)(cid:12) dt := G . Thus, we can write: G ( t ) ∼ G + G + G = Z | w − w | Y s =1 | w s ( t s ) | σ (cid:12)(cid:12)(cid:12)(cid:12)Z | y ( t ) | σ dt (cid:12)(cid:12)(cid:12)(cid:12) dt . Integrating both sides with respect to t as t goes from 0 to 1 yields | I | & (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z | w ( t ) − w ( t ) | Y s =1 | w s ( t s ) | σ (cid:12)(cid:12)(cid:12)(cid:12)Z | y ( t ) | σ dt (cid:12)(cid:12)(cid:12)(cid:12) dt dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which is our desired bound.Case ii) θ ∈ [ π , π ].1This case is simpler, as now we always have α >
0. Using the same notation as earlier,we have | I | = (cid:12)(cid:12)(cid:12)(cid:12)Z Z ( α + iβ )( ξ + iη ) dt dt (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Z αξ − βη + i ( βξ + αη ) dt dt (cid:12)(cid:12)(cid:12)(cid:12) . (2.4)Note that we also have, as β is single signed, and α is positive, Z Z αξdt dt + (cid:12)(cid:12)(cid:12)(cid:12)Z Z βξdt dt (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Z ( α + | β | ) ξdt dt (cid:12)(cid:12)(cid:12)(cid:12) , ∼ (cid:12)(cid:12)(cid:12)(cid:12)Z Z | w ( t ) − w ( t ) || g ( t , t ) | ξdt dt (cid:12)(cid:12)(cid:12)(cid:12) := G. If (cid:12)(cid:12)(cid:12)R R βξdt dt (cid:12)(cid:12)(cid:12) ≥ R R αξdt dt , then we have: | Im( I ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z Z βξ + αηdt dt (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12)Z Z βξdt dt (cid:12)(cid:12)(cid:12)(cid:12) − ε Z Z αξdt dt ∼ (cid:12)(cid:12)(cid:12)(cid:12)Z Z βξdt dt (cid:12)(cid:12)(cid:12)(cid:12) & G. While if R R αξdt dt ≥ (cid:12)(cid:12)(cid:12)R R βξdt dt (cid:12)(cid:12)(cid:12) , then by the same line of reasoning we have | Re( I ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z Z αξ − βηdt dt (cid:12)(cid:12)(cid:12)(cid:12) & G. So indeed this gives us in either case that | I | & | Z Z | w − w | Y s =1 | w s ( t s ) | σ (cid:12)(cid:12)(cid:12)(cid:12)Z | y ( t ) | σ dt (cid:12)(cid:12)(cid:12)(cid:12) dt dt | . as again can move the absolute value around the inner integral into it, as the integrand thea sector contained function.This completes the lemma, as in all cases, we indeed obtain: | J Φ Γ ( z , z , z ) | ∼ Y s =1 | L ( z s ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z z z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z z z Y s =1 | L ( w s ) || L ( w s ) | (cid:12)(cid:12)(cid:12)(cid:12)Z w w | L ( y ) || L ( y ) || L ( y ) | dy (cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Note that we can place absolute values around the w integral as it is now clearly anintegral of a real, positive integrand, and so we are done.So therefore, we have, on any set in our decomposition: | J Φ Γ ( z , z , z ) | ∼ Y s =1 | z s − b | σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z z z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z z z Y s =1 | w s − b | σ (cid:12)(cid:12)(cid:12)(cid:12)Z w w | y − b | σ dy (cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.5)for some values of σ , σ , σ , depending on the type of set we are on. The explicit valuesare given here.2 σ σ σ T k k − k k + k − k T k − k T k k − k k − k T k − k In this section, we establish an important inequality on the sets in our decomposition, whichshall be used in both of our main theorems. This lemma is:
Lemma 3.1.
For | J Φ Γ | yielding σ = ( σ , σ , σ ) values as above after the described decom-position | J Φ Γ ( z , z , z ) | & Y i =1 | L Γ ( z i ) | Y ≤ i 2. This is done to ensures that we avoid certain σ values, as described in[8]. Specifically, we ensure there is no k value such that k = − − k on T intervals, andno k value such that k = k + k +12 on T intervals. This ensures σ = − 1. Similarly, weavoid k = 1 on T and and any k values on T such that k = k + i for i = 1 , , , 4. Notethat as k is an integer, this involves excluding at most 2 values of k . Proof of Lemma 3.1: To establish the lemma, we first seek to show that: | J Φ Γ ( z , z , z ) | & Y i =1 | z i | σ + σ + σ (cid:12)(cid:12)(cid:12)(cid:12)Z z z Z z z (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) dw dw (cid:12)(cid:12)(cid:12)(cid:12) . (3.2)Again, calculation will yield that | z i | σ + σ + σ ∼ | L Γ ( z i ) | .To establish this, we first wish to see that: I := (cid:12)(cid:12)(cid:12)(cid:12)Z w w | y − b | σ dy (cid:12)(cid:12)(cid:12)(cid:12) & | w − b | σ | w − b | σ (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) Firstly, we may assume | w | > | w | , as if not, we can notice that (cid:12)(cid:12)(cid:12)R w w | y − b | σ dy (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) − R w w | y − b | σ dy (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)R w w | y − b | σ dy (cid:12)(cid:12)(cid:12) .3So we may relabel to ensure that the larger of the two occurs at the upper bound of theintegral.Also notice that if | w − b | < | w − b | < | w − b | , we have | w − b | ∼ | y − b | ∼ | w − b | throughout this line integral, and our result immediately follows. We therefore only haveto deal with the case in which 9 | w − b | < | w − b | .Let: • L := { y | | w − b | < | y − b | < | w − b |} . • U := { y | | w − b | < | y − b | < | w − b |} . . Note that after parameterizing our line integral, we can see that it is the integral of apurely positive quantity over a real interval, so we can delete portions of the line segmentand ensure that this results in a lower bound.So I ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:20)Z L + Z U (cid:21) | y − b | σ dy (cid:12)(cid:12)(cid:12)(cid:12) ∼ | w − b | σ +1 + | w − b | σ +1 = | w − b | | w − b | σ + | w − b | σ +1 | w − b | ! . Note that | w − b | ∼ | w − w | = (cid:12)(cid:12)(cid:12)R w w dy (cid:12)(cid:12)(cid:12) , as 9 | w − b | < | w − b | So I & (cid:12)(cid:12)(cid:12)R w w dy (cid:12)(cid:12)(cid:12) max (cid:16) | w − b | σ , | w − b | σ | w − b | (cid:17) If σ is positive, we have I & (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) | w − b | σ > (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) | w − b | σ | w − b | σ . If σ is negative, then, as σ is an integer which is not equal to -1, σ + 1 is negativealso, so I & (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) | w − b | σ +1 | w − b | > (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) | w − b | σ | w − b | σ . As this holds for all σ = − 1, we are done. With this, we can conclude that | J Φ Γ ( z , z , z ) | & Y s =1 | z s − b | σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z z z Z z z Y s =1 | w s − b | σ + σ (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) dw dw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We now seek to establish a similar inequality for I = (cid:12)(cid:12)(cid:12)(cid:12)Z z z | w − b | σ + σ (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12) . We want to see that I & max (cid:18) | z − b | σ + σ , | z − b | σ σ 32 +2 | z − b | (cid:19) (cid:12)(cid:12)(cid:12)R z z (cid:12)(cid:12)(cid:12)R w w dy (cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12) ,for σ + σ < − ≥ We assume that | z − b | < | z − b | , and relabel if we are not in such a case. We also noticethat the inequality is immediate if we are in the case that | z − b | < | z − b | < | z − b | , sowe therefore assume that 9 | z − b | < | z − b | , and again restrict the integral to L and U ,which now represent the sets: • L := { y | | z − b | < | y − b | < | z − b |} . • U := { y | | z − b | < | y − b | < | z − b |} .So we have I ≥ (cid:12)(cid:12)(cid:12)(cid:2)R L + R U (cid:3) | w − b | σ + σ | w − w | dw (cid:12)(cid:12)(cid:12) . Identically to last time, we have | w − w | ∼ | w − b | , so we can write: I & (cid:12)(cid:12)(cid:12)(cid:12)(cid:20)Z L + Z U (cid:21) | w − b | σ + σ +1 dw (cid:12)(cid:12)(cid:12)(cid:12) ∼ | z − b | σ + σ +2 + | z − b | σ + σ +2 = | z − b | | z − b | σ + σ +2 | z − b | + | z − b | σ + σ ! & max | z − b | σ + σ , | z − b | σ + σ +2 | z − b | ! (cid:12)(cid:12)(cid:12)(cid:12)Z z z (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12) . Now, if σ + σ > 0, we have: I = | z − b | σ + σ (cid:12)(cid:12)(cid:12)(cid:12)Z z z (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12) ≥ | z − b | σ + σ | z − b | σ + σ (cid:12)(cid:12)(cid:12)(cid:12)Z z z (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12) . if instead, σ + σ < − 2, we have I & | z − b | σ + σ +2 | z − b | (cid:12)(cid:12)(cid:12)(cid:12)Z z z | w − w | dw (cid:12)(cid:12)(cid:12)(cid:12) > | z − b | σ + σ | z − b | σ + σ (cid:12)(cid:12)(cid:12)(cid:12)Z z z (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12) , as σ + σ + 2 < 0A similar analysis yields that for I = (cid:12)(cid:12)(cid:12)(cid:12)Z z z | w − b | σ + σ (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12) , we have I & | z − b | σ + σ | z − b | σ + σ (cid:12)(cid:12)(cid:12)(cid:12)Z z z (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12) , for the same constraints on our σ values as previously outlined. So we therefore indeed have(3.2).5From here we seek to resolve the triple integral which appears in (3.2), and establishthat: (cid:12)(cid:12)(cid:12)(cid:12)Z z z (cid:12)(cid:12)(cid:12)(cid:12)Z z z (cid:12)(cid:12)(cid:12)(cid:12)Z w w dy (cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12) dw (cid:12)(cid:12)(cid:12)(cid:12) & | z − z | | z − z | | z − z | . (3.3)Clearly the inner most integral is given by (cid:12)(cid:12)(cid:12)R w w dy (cid:12)(cid:12)(cid:12) = | w − w | .Consider J = (cid:12)(cid:12)(cid:12)R z z | w − w | dw (cid:12)(cid:12)(cid:12) . If z − z = | z − z | e iφ , and t = e − iφ ( w − z ) ∈ R ,we have J = Z | z − z | (cid:12)(cid:12) t − w ′ (cid:12)(cid:12) = Z | z − z | q ( t − Re( w ′ )) + Im( w ′ ) dt := Z | z − z | g ( t ) dt, where w ′ = e − iφ ( w − z ).To bound this from below, we introduce a new function f ( t ) = (cid:12)(cid:12)(cid:12) t − c w ′ ( t ) (cid:12)(cid:12)(cid:12) , where c w ′ ( t ) = w ′ − i Im( w ′ )Re( w ′ ) t if t ≤ Re( w ′ ) w ′ − i Im( w ′ ) | z − z | − t | z − z | − Re( w ′ ) , t > Re( w ′ ) . Note that, for all values of t ∈ [0 , | z − z | ], we have that Re( w ′ ) = Re( c w ′ ( t )), and also that | Im( w ′ ) | > | Im( c w ′ ( t ) | . We also have c w ′ (0) = c w ′ ( | z − z | ) = w ′ .Clearly g ( t ) ≥ f ( t ) for all t in the interval [0 , | z − z | ], as t is real, and c w ′ ( t ) is w ′ with a smaller imaginary part. Note that computation shows that f ( t ) is given by : f ( t ) = | z − w | Re( w ′ ) (Re( w ′ ) − t ) if t ≤ Re( w ′ ) | z − w || z − z | − Re( w ′ ) ( t − Re( w ′ )) t > Re( w ′ ) . So J = R | z − z | g ( t ) dt > R | z − z | f ( t ) dt . Now note that g ( t ) increases as | t − Re( w ′ ) | increases. Also g (0) = | z − w | = f (0) and g ( | z − z | ) = | z − w | = f ( | z − z | ).Therefore if Re( w ′ ) < | z − z | , we have | z − w | > | z − w | .Alternatively if Re( w ′ ) > | z − z | , then we have | z − w | > | z − w | .Now, suppose Re( w ′ ) ≤ 0. Then | z − w | > | z − w | , and we have: J > Z | z − z | f ( t ) dt = Z | z − z | | z − w || z − z | − Re( w ′ ) ( t − Re( w ′ )) dt = | z − w || z − z | − Re( w ′ ) (cid:20) t ( t − Re( w ′ ))2 (cid:21) | z − z | ∼ | z − z | | z − w | . A symmetric argument gives that if Re( w ′ ) > | z − z | then | z − w | > | z − w | , and J & | z − z | | z − w | .6 Lastly, if Re( w ′ ) ∈ [0 , | z − z | ], then: J > Z Re( w ′ )0 | z − w | Re( w ′ ) (Re( w ′ ) − t ) dt + Z | z − z | Re( w ′ ) | z − w || z − z | − Re( w ′ ) ( t − Re( w ′ )) dt = 12 (cid:0) | z − w | Re( w ′ ) + | z − w | ( | z − z | − Re( w ′ )) (cid:1) ∼ | z − z | max ( | z − w | , | z − w | ) . So indeed J & ∼ | z − z | max( | z − w | , | z − w | ) in each of these cases. Applying thisto the initial double integral, we get (cid:12)(cid:12)(cid:12)(cid:12)Z z z Z z z | w − w | dw dw (cid:12)(cid:12)(cid:12)(cid:12) & | z − z | (cid:12)(cid:12)(cid:12)(cid:12)Z z z | z − w | dw (cid:12)(cid:12)(cid:12)(cid:12) & | z − z | | z − z | | z − z | . As we can go through an identical procedure for the remaining single integral.Therefore, we have established (3.3). This in conjunction with (3.2) yields the proof of(3.1). We will now use the work from Sections 2 and 3 to establish Theorem 1.1. Let us denotethe measure dσ ( z ) := λ γ ( z ) dz. In order to prove Theorem 1.1, and Corollary 1.2, we need only establish that theoperator T is of restricted weak-type at ( p, q ) = (2 , , 0) follows triviallyfrom the fact that T Γ f is bounded whenever f is bounded. Furthermore, as the operator is“almost” self dual, we can use duality properties to conclude that we also have the boundcorresponding to the vertex (1 , (cid:0) , (cid:1) , as this will give us the bound at (cid:0) , (cid:1) , by duality. From here, we can use theMarcinkiewicz Interpolation Theorem between these four vertices to establish our desiredrange, the convex hull of these points, R . The bound we establish at (cid:0) , (cid:1) is uniform, asopposed to the trivial bound at (0 , (cid:0) , (cid:1) and (cid:0) , (cid:1) , and local estimates, with implicit constants dependenton the radius, at every other ( p, q ) pair in R .To establish the restricted weak-type, we introduce two quantities, α and β , where, formeasurable sets E and F , they are given by the equations: α | F | = h T Γ χ E , χ F i = β | E | . (4.1)For χ S being the characteristic function of the set S . Our estimate at (2 , 3) is equivalentto the statement that for all measurable sets, E and F , we have: h T Γ χ E , χ F i . | E | | F | . | E | & α β . (4.2)which is the statement we will prove. To do this, we construct subsets of E and F asfollows.Let E := { y ∈ E | T ∗ χ F ( y ) ≥ β } . Note that: h T χ E , χ F i = h T χ E , χ F i − h T χ E \ E , χ F i ≥ α | F | − β | E | = α | F | . That is to say that the average value of T χ E on F is greater than α . We therefore canintroduce the set F := { x ∈ F | T χ E > α } , and this will be non-empty. Take z ∈ F , andlet P := { z | z − Γ( z ) ∈ E } . Then we have σ ( P ) = T χ E ( z ) > α .Let Q z = { z | z − Γ( z ) + Γ( z ) ∈ F } . Again, σ ( Q z ) = T ∗ χ F ( z − Γ( z )) ≥ β Let R z ,z = { z | z − Γ( z ) + Γ( z ) − Γ( z ) ∈ E } , and identically to the other sets, wehave that σ ( R z ,z ) = T χ E ( z − Γ( z ) + Γ( z )) ≥ α .We let S = P × Q z × R z ,z .We have z + Φ Γ ( S ) ⊆ E ⊆ E , where Φ Γ ( z , z , z ) = − Γ( z ) + Γ( z ) − Γ( z ). Further-more, we restrict ourselves to the subset of S on which | z | ≤ | z | ≤ | z | to ensure Φ Γ isc(N) to 1 on the region, and we may apply B´ezout’s Theorem. By B´ezout’s Theorem, wehave, for J R , and J C being the real and complex jacobian of the map of Φ Γ , the followingestimates: | E | & Z S | J R ( z , z , z ) | dz dz dz = Z S | J C ( z , z , z ) | dz dz dz & Z S Y i =1 | z i | σ + σ + σ Y i =1 ,j | z i | − | z j | > | z i | (1 − ν ).Furthermore, we have | z i | > (16 πν ) − ν α ν . It follows | z i | k ′ +66 > (16 πν ) − α , which gives us | z i | & α | z i | − k ′ . So therefore, | z i − z j | & β ( | z i | | z j | ) − k ′ , as | z i | ∼ | z j | .8 In the case where z j / ∈ B α , we have | z j | > (16 πν ) − ν (cid:0) α (cid:1) ν . It follows, similar to theother case, that | z j | > (32 πν ) − α | z j | − k ′ . For S i = P or = R z ,z depending on our i value, consider B α ( z j ) = { z i ∈ S i | | z i − z j | < c | z j | − k ′ α } .Now on B α ( z j ), we have | z i − z j | < c | z j | − k ′ α < c (32 πν ) | z j | . Consider the sigma-measure of this set. We have σ ( B α ( z j )) = R B α ( z j ) | z | k ′ dz . But for all z ∈ B α ( z j ), wehave: | z | ≤ | z j | + | z − z j | ≤ | z j | (1 + c (32 πν ) ) . Therefore, σ ( B α ( z j )) < | z j | k ′ (1+ c (32 πν ) ) k ′ π ( c | z j | − k ′ α ) . The right hand side of thisinequality is comparable to α , for c chosen sufficiently small. It can therefore be removedfrom S i , and still have σ ( S i ) ∼ α . Deleting this set gives us | z i − z j | & α ( | z i | | z j | ) − k ′ , asin the other case.Therefore, for i odd, j < i , if | z i | ∼ | z j | , we have | z i − z j | & α ( | z i | | z j | ) − k ′ . An identicalargument for the i = 2 case gives us | z i − z j | & β ( | z i | | z j | ) − k ′ .Also notice that if we are not in this case, that is we do not have | z i | ∼ | z j | , we necessarilyhave | z i | > | z j | , which gives | z i − z j | ∼ | z i | . Using these facts, our ordering of the size of z , z , and z , and our integral bound for | E | , we can establish | E | & α β as desired. Weshall prove as an example the case in which | z | ∼ | z | > | z | , and the other cases can bedealt with similarly. | E | & Z S | z | k ′ | z | k ′ | z | k ′ Y i =1 ,j j , we apply our bounds for | z i − z j | , this generates one α or β , and a | z i | − k ′ , which we use to remove one of the twoparts we split our initial monomial of that variable into.This may still leave us with some of the monomials that we still wish to replace with α or β . To do this, for any variables that are not comparable, z i and z j , i > j , we replace | z i − z j | with | z i | . From here we can redistribute the exponents of the remaining variables“downwards”, that is to variables with lower subscripts, so that all our remaining monomialsare now monomials of the form | z i | k ′ +2 . We can then use our lower bounds that we getfrom deleting B α or B β to replace this with quantities comparable to α or β as appropriate.Note that as the estimate here is uniform in r , we can in fact extend the restrictedweak-type estimate to the entirety of C by a simple re-scaling argument and the MonotoneConvergence Theorem. We therefore have global ( p, q ) estimates for T Γ for all ( p − , q − ) onthe line segment joining (cid:0) , (cid:1) , and (cid:0) , (cid:1) .0 For Theorem 1.3, we again rely on interpolation to achieve the desired range. Again, it istrivial to establish the estimate corresponding to vertex (1 , , ). To achieve this, we will use the dual estimate. The dual to estimatein this theorem is given by, for q > 7, and p ′ = q , where p ′ is the H¨older Conjugate of p : ||E Γ f || L q ≤ C || f || L p ( λ Γ ( z ) dz . (5.1)Here, we define the Extension Operator associated to Γ as: E Γ f ( z ) = Z C e i z · Γ( w ) f ( w ) λ Γ ( w ) dw. (5.2)Now to establish this, we will use the method developed by Stovall in [14], which es-tablishes the analogous result in the real case. This is more involved that the convolutionestimate, but mostly follows Stovall’s real method closely. As in that paper, we first estab-lish a bound in the case where we are on a set on which our torsion L Γ ( z ) is bounded. Thisappears as Theorem 2.1 in [14]. It translates into the complex case as follows: Lemma 5.1. Suppose we have ( p, q ) satisfying q > , and p ′ = 6 q , a degree N curve γ ( z ) ,and a convex set B ⊂ C such that for all z ∈ B , we have that < C < | L γ ( z ) | < C , then ||E γ χ B f || L q . || f || L p ( λ γ ( z ) dz ) , with implicit constants depending on N, p and the ratio C C Note, as L γ ( z ) is a polynomial of degree at most 3 N , that we can decompose B = S C C ) Nj =1 B j where on each B j we have C < | L γ ( z ) | < C , and by the triangle inequality,we may assume that our initial B is replaced with one of these B j .Furthermore, as our estimate is affine invariant, we may apply an affine transformationto assume that C = 1, and that B contains some fixed complex number, say 0 in it’s interior,and | L γ (0) | 6 = 0.To prove this theorem, we must first prove the analogue to Lemma 2.2 in [14], whichtranslates directly into the Complex case without notable changes, and is only included forcompletion, and reads as: Lemma 5.2. Suppose we have a curve γ , and a the set B satisfying the previous theo-rems conditions. Then there exists an affine transformation A , such that det A = 1 , and || Aγ || C N ( B ) . , with implicit constants depending only on N . Proof of Lemma: As in [14], we take A such that Az = (cid:0) γ ′ (0) , γ ′′ (0) , γ ′′′ (0) − (cid:1) ( z − γ (0)).As 0 ∈ B , and understanding det A refers to the determinant of the matrix component of A , we have det A ∈ [ , A with A A if necessary,we must just show || Aγ || C N ( B ) . Aγ : • Aγ (0) = 0. • | L Aγ ( z ) | ∈ [ , 4] on B . • For j ∈ { , , } , we have Aγ ( j ) (0) = e j .From here, we will use the notation that γ = Aγ . Note that by Taylor’s Theorem,we need only estimate the size of the remaining γ ( j ) (0) for j > 3, or, sufficiently, by theproperties of γ , that for all ( n , n , n ), with n < n < n < N , we havedet (cid:16) γ ( n ) (0) , γ ( n ) (0) , γ ( n ) (0) (cid:17) . . To see this, we assume a contradiction. That is to say there is a sequence { γ n } ∞ n =1 withthe properties listed above, and thatlim n →∞ max n 6, for all sufficientlylarge n , as otherwise the above determinant is 1.Now introduce a sequence of real numbers { δ n } ∞ n =1 with δ n ∈ (0 , γ n , letΓ n be the curve such that its components are given by Γ n,j ( z ) = δ − jn γ n,j ( δ n z ).Note that Γ n satisfies the properties listed above on the set B δ n , which is the set B scaled from 0 by a factor of δ Furthermore, calculation yields for ( n , n , n ) = (1 , , (cid:16) Γ ( n ) n (0) , Γ ( n ) n (0) , Γ ( n ) n (0) (cid:17) = δ n det (cid:16) γ ( n ) n (0) , γ ( n ) n (0) , γ ( n ) n (0) (cid:17) Therefore, for sufficiently large n , we can choose δ n such thatmax n 1, for any k . Therefore each coefficient ofΓ n is bounded for all n , so therefore the limit, Γ, of the sequence Γ n ∞ n =1 does exist.Furthermore, this limit retains the listed properties, on the set B dilated about 0 bya factor of n →∞ δ n , which is the entirety of C . This means that | L γ ( z ) | ∈ [ , 4] for all z ∈ C . As L γ is a polynomial, it must in fact be constant. Evaluating it at 0 yieldsthat it must identically equal to 1, and again, as it is a polynomial curve, it must begiven Γ( z ) = ( z, z , z ), up to some redistribution of the coefficients. However this clearlycontradicts the fact that det (cid:16) Γ ( n ) n (0) , Γ ( n ) n (0) , Γ ( n ) n (0) (cid:17) = 5 , should hold for all n < n < n , thus completing the proof,The second step to obtaining bounds on Dyadic Annulli involves obtaining bounds foroff spring curves Lemma 5.3. For any convex set containing , B there exists a decomposition of B = S M d,n j =1 B j such that, if γ is a degree N curve satisfying | L γ ( z ) | ∈ [ 12 , , z ∈ B, then for K ∈ Z , and ¯ h ∈ C K , then the curve γ h ( z ) = K P Kj =1 γ ( z + h j ) satisfies the followingon B h = T Kj =1 ( B − h j ) : | J γ h | & Y j =1 | L γ h ( z j ) | Y i Suppose that, for some p ∈ [1 , we have the estimate, for some offspringcurve γ h , and set B h , as described in the previous lemma, that ||E γ h ( χ B h f ) || L p ′ . || f || L p , Then we have the same estimate for p replaced with some p > satisfying the inequality p > + p . Note that this will obviously let us, starting from p = 1, continuously get estimatesfor larger and larger p -values, with the limiting behaviour of it tending towards the desired p = 7. Again, the proof does not change much from [14], but is described here for completion. Proof: First, denote v ( h ) = Q j =1 ,i 0, real numbers A j , B j and somecomplex number b j . Also recall that we have the Jacobian bound: | J γ ( z , z , z ) | & Y j =1 | L γ ( z j ) | Y i For each q ∈ (7 , ∞ ) , we have the following bound, in terms of the above B j and B j,n : (cid:12)(cid:12)(cid:12)(cid:12) E γ χ B j f (cid:12)(cid:12)(cid:12)(cid:12) L q . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n |E γ χ B j,n f | ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L q . Proof: First note that if the amount of non-empty B j,n , say M , is O (1), then the lemmaimmediately follows from the triangle inequality and the equivalence of l and l norms in C M .We may therefore assume than M = O (1). Furthermore, if we let a j be the complexnumber with smallest modulus in B j , and n j = ⌈ log | a j |⌉ + 1, then we can replace B j with B ′ j = B j T { z ∈ C | | z | > n j } . As this excludes at most 2 of the B j,n , we, as above,immediately have the estimate on these two sets. Therefore, if we prove the case with B j = B ′ j , we have (cid:12)(cid:12)(cid:12)(cid:12) E γ χ B j f (cid:12)(cid:12)(cid:12)(cid:12) L q ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E γ χ B j \ B ′ j f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L q + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E γ χ B ′ j f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L q . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n j − X n = n j − |E γ χ B j,n f | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L q + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≥ n j |E γ χ B j,n f | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L q . Now the quantity on the last line is less than the twice the maximum of these twoterms, and the maximum of these two terms is less than the right hand side of our initialinequality, so proving this assumption is indeed sufficient. To see this notice that we canwrite γ ( z ) = R za j γ ′ ( w ) dw , where we recall that this notation refers to the path integral of γ ′ along the straight line joining a j to z .As γ ′ is a polynomial, and our C decomposition was with respect to γ ′ , we can consider γ ′ as a sector contained function, as in the convolution section. We therefore have | γ ( z ) | ∼| z | l j +1 − | a j | l j +1 ∼ | z | l j +1 , due to the separation ensured by restricting ourselves to B ′ j .Now, we make the claim that has its support contained within Im ( γ ) × C . That is tosay it is contained within (cid:8) ξ ∈ C || ξ | ∈ [2 nl j , nl j +1 ] (cid:9) . To see this, consider h \ E γ χ B n,j f , φ i where φ is a Schwarz functions with support contained entirely outside of the image of γ .We have: h \ E γ χ B n,j f , φ i = hE γ χ B n,j f, b φ i = Z C E γ χ B n,j f ( z ) b φ ( z )dz= Z C Z B n,j e i ( z · γ ( w ) f ( w ) λ γ ( w )dw b φ ( z )dz= Z B n,j f ( w ) λ γ ( w ) Z C e i ( z · γ ( w )) b φ ( z )dzdw= Z B n,j f ( w ) λ γ ( w ) φ ( γ ( w ))dw , by the Fourier inversion theorem. As we chose φ to not have support in the image of γ ,this evaluate to zero. As this holds for all φ satisfying this condition, this means the supportof \ E γ χ B n,j f is contained in the image of γ . Therefore we can use the following theorem whichappears in [13]. Here, the operator S ρ denotes the operator such that d S ρ f = χ ρ ˆ f , and ∆ isthe collection of rectangles from the dyadic decomposition of C , or equivalently R , whichare just rectangles with sides which are products of 6 intervals of the form [2 k i , k i +1 ] forsome various integer k values . Theorem 5. in Chapter IV of [13] is restated here. Theorem 5.6. If f ∈ L p ( R n ) , then X ρ ∈ ∆ | S ρ f | ∈ L p ( R n ) and || f || L p ∼ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ρ ∈ ∆ | S ρ f | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p . Notice now our lemma immediately follows from letting E γ χ B j,n f serve the role of f inthis theorem, and further notice that S ρ E γ χ B j f = 0 if the rectangle ρ does not have it’sedge parallel to the first axis project onto the interval [2 nl j , nl j +1 ] for some n correspondingto a non-empty B j , n .Furthermore, if ρ does satisfy the above stated condition, we have S ρ E γ χ B j f = E γ χ B j,n f ,for the n value as stated in the condition, which proves the lemma. Proof of Theorem 1.3: To prove Theorem 1.3, from here, we first restrict ourselves toconsidering q < 12. If we establish the theorem for q < 12, we can interpolate between anyestimate corresponding to some arbitrary q < 12 and the trivial estimate at q = ∞ to getthe full range.Note furthermore that by the triangle inequality, we need only establish the estimatefor f being a function which is only supported in a single B j . As this j is now fixed, wechange our notation to label B j = B , and B j,n = B n We now have, by our previous lemma: ||E γ f || qL q . Z X n |E γ χ B n f ( x ) | ! q dx = Z Y j =1 X n j |E γ χ B nj f ( x ) | q dx . Z Y j =1 X n j |E γ χ B nj f ( x ) | q dx ∼ X n Lemma 5.7. There exists a real, positive number ε depending only on N and p such thatunder the assumptions made of γ , B , q and p . Then for n < n · · · < n , and f j ∈ L p aresupported on n j , we have that: || Y j =1 E γ f j || qL q . − ε ( n − n ) 6 Y j =1 || f j || L p ( λ γ ) . ||E γ f || qL q . X n Proof of Lemma 6.1: By H¨older’s inequality, we have that: || Y j =1 E γ f j || L q ≤ Y j =4 ||E γ f i j || L q || Y j =1 E γ f i j || L q , where the i j values are some enumeration of { , · · · } . Notice now that the lemmawill be proven if we can establish that: || Y j =1 E γ f j || L q . − ε ( n − n ) 3 Y j =1 || f j || L p ( λ γ ) . Now, notice that we have, by H¨older’s inequality, and by the fact that the support ofeach f j is in B, and Theorem 4.3, that: || Y j =1 E γ f j || L q ≤ Y j =1 ||E γ f j || L q . Y j =1 || f j || L p ( λ γ ) . Therefore the lemma follows immediately if n − n is bounded by some arbitrary con-stant. That is to say, if n − n ≤ n − n ≥ Furthermore, we can prove this for q = 12, and p = 2, as if this is established wecan interpolate with between this ( p, q ) value and some ( p, q ) value arbitrarily close to theendpoint. So the problem is now to prove that: || Y j =1 E γ f j || L . − ε ( n − n ) 3 Y j =1 || f j || L ( λ γ ) . Now, we define the measure dµ j such that: dµ j ( φ ) := Z B nj φ ( γ ( z )) f j ( z ) λ γ ( z ) dz. Then, by the Haussdorff-Young theorem, we have: || Y j =1 E γ f j || L ≤ || dµ ∗ dµ ∗ dµ || L . Now, we can compute this 3-fold convolution explicitly as: dµ ∗ dµ ∗ dµ ( φ ) = Z B n Z B n Z B n φ X j =1 γ ( z j ) Y j =1 f j ( z j ) λ γ ( z j ) dt dt dt . We now rewrite this integral in terms of the permutation group S , and the sets P σ = { z ∈ B n × B n × B n | | z σ (1) | < | z σ (2) | < | z σ (3) |} , on which the functionΦ γ ( z , z , z ) = P j =1 γ ( z j ) is injective. So we have: dµ ∗ dµ ∗ dµ ( φ ) = X σ ∈ S Z P σ φ X j =1 γ ( z j ) Y j =1 f j ( z j ) λ γ ( z j ) . Therefore, by making the substitution ν = φ γ ( z , z , z ) we can notice that this 3-foldconvolution of measure is actually a function of C given by: dµ ∗ dµ ∗ dµ ( ν ) = X σ ∈ S F σ ( ν ) , where F σ ( ν ) := χ P σ ( z ) Y j =1 f j ( z j ) λ γ ( z j ) | J γ ( z ) | − . Note here that we proceed with understanding that z is just the preimage of ν underthe mapping Φ γ restricted to P σ . We therefore have: || F σ || L = Z P σ Y j =1 | f j ( z j ( ν )) λ γ ( z j ( ν )) || J γ ( z ( ν )) | − dν = Z P σ | Y j =1 f j ( z j ) λ γ ( z j ) || J γ ( z ) | − | J γ ( z ) | dz Z P σ Y j =1 | f j ( z j ) λ γ ( z j ) || J γ ( z ) | − dz = || χ P σ Y j =1 f j ( z j ) λ γ ( z j ) J γ ( z ) − || L . || χ P σ Y j =1 f j ( z j ) λ γ ( z j ) Y ≤ i 2. We can do both casessimultaneously, understanding an empty product represents 1.We have Y j =1 , i 2, this completes the proof of the lemma. [1] J. Bak and S. Ham: Restriction of the Fourier transform to some complex curves, J. Math.Anal. Appl. (2014), 1107–1127.[2] J. Bak, D. Oberlin, and A. Seeger: Restriction of Fourier transforms to curves: An endpointestimate with affine arclength measure , Journal f¨ur die reine und angewandte Mathematik (2013), 167-205.[3] A. Carbery, F. Ricci, and J. Wright: Maximal functions and Hilbert transforms associated topolynomials, Rev. Mat. Iberoamericana (1998), no. 1, 117–144.[4] M. 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