Analytic characterization of high dimension weighted special atom spaces
AAnalytic characterization of high dimension weightedspecial atom spaces
Eddy Kwessi ∗ and Geraldo de Souza † Abstract
Special atom spaces have been around for quite awhile since the introduction ofatoms by R. Coifman in his seminal paper who led to another proof that the dual ofthe Hardy space H is in fact the space of functions of bounded means oscillations(BMO). Special atom spaces enjoy quite a few attributes of their own, among whichthe fact that they have an analytic extension to the unit disc. Recently, an extensionof special atom spaces to higher dimensions was proposed, making ripe the possibleexploration of the above extension in higher dimensions. In this paper we proposean analytic characterization of special atom spaces in higher dimensions. MSC Classification: 32C20, 32C37, 32K05 32K12
Let d be some positive integer. We define the unit disk as D = { z ∈ C : | z | < } , thesphere as T = { z ∈ C : | z | = 1 } , the polydisk and polysphere respectively as D d and T d .Atoms were introduced by Coifman in [5] as a tool to explicitly represent functions inthe Hardy space H p for 0 < p ≤
1. The following definition was proposed:
Definition 1.1.
Let < p ≤ and an interval J of R . An atom is a function b definedon the interval J and satisfying1. | b ( ξ ) | ≤ | J | /p .2. (cid:90) ∞−∞ ξ k b ( ξ ) dξ = 0 , for ≤ k ≤ (cid:104) p (cid:105) − , where [ x ] is the integer part of x . From this definition, functions in H p ( R ) could now be characterized via their atomicdecomposition in the following theorem: ∗ Corresponding author, Department of Mathematics, Trinity University, San Antonio, TX 78212,USA; [email protected] † Department of Mathematics, Auburn University, Auburn, AL 36849, USA. a r X i v : . [ m a t h . C V ] F e b heorem 1.2 (see [5]) . Let < p ≤ . Then f ∈ H p ( R ) if and only if there exist realnumbers a i , atoms b i for i ∈ N , and absolute constants c, C such that such that f ( ξ ) = ∞ (cid:88) i =0 a i b i ( ξ ) and c (cid:107) f (cid:107) H p ≤ ∞ (cid:88) i =0 | a i | p ≤ C (cid:107) f (cid:107) H p . Fefferman [8] observed that this result is in fact due to the duality between H ( R )and the space of functions of bounded means oscillations (BMO), therefore providinganother proof that the dual space of H ( R ) is in fact BMO. The era of the atomicdecomposition therefore started. One criticism of the atomic decomposition at the timewas that it was too general, making it difficult or not very useful for applications. Thisatomic decomposition was proved to be quite useful in harmonic analysis. However, inan attempt to answer this criticism, Richard O’Neil and Geraldo De Souza proposedan example of atoms defined on the interval I = [0 ,
1] that was latter dubbed “specialatoms”. This special atom has some very desirable properties as we will see in the sequel.
Definition 1.3.
Consider ≤ p < ∞ .(a) A special atom of type 1 is a function b : I → R such that b ( ξ ) = | J | /p [ χ R ( ξ ) − χ L ( ξ )] , if ξ ∈ J
1, if ξ ∈ I \ J , where J is a subinterval of I , L and R are the halves of J such that J = L ∪ R ,and | J | is the length of J .(b) A special atom of type 2 is a function c : J → R such that c ( ξ ) = 1 | J | /p [ χ J ( ξ )] , where J is an interval contained in I . Remark 1.4.
We observe that this definition can be extended on the unit ball of R d byusing dyadic decomposition, see [1] . From this definition, they introduced the special atom space B p (for type 1 atom) definedon J , but with a different norm from the L p -norm. Definition 1.5.
Let ≤ p < ∞ . The special atom space B p (of type 1) is defined as B p = (cid:40) f : I → R ; f ( ξ ) = ∞ (cid:88) n =0 α n b n ( ξ ); ∞ (cid:88) n =0 | α n | < ∞ (cid:41) , where the b n ’s are special atoms of type 1. The space B p is endowed with the norm (cid:107) f (cid:107) B p = inf ∞ (cid:88) n =0 | α n | , where the infimum is taken over all representations of f . Definition 1.6.
We define the weighted special atom (of type 1) on J as: b w ( ξ ) = 1 w ( J ) [ χ R ( ξ ) − χ L ( ξ )] , where w ∈ L ( I ) with w ( J ) = (cid:90) J w ( ξ ) dξ, and L, R are as in Definition 1.3 . The weighted special atom space is the space B w of functions f with atomic decomposition f ( ξ ) = ∞ (cid:88) n =0 α n b w,n ( ξ ) , endowed with the Infimum norm. The importance of this definition can not be overstated. Indeed, weighted specialatom spaces are invariant under the Hilbert transform and they contain some functionswhose Fourier series diverge, see [7]. One of their most applicable features is their con-nection to Haar wavelets, in that, a Haar wavelet function is just a special atom withweight 2 − n/ , [13]. Weighted special atom spaces are also Banach equivalent to someBergman-Besov-Lipschitz spaces (see [6]), which leads to a complete characterizationof their lacunary functions, see [12]. Moreover, functions in B w have analytic corre-spondences by integrating against analytic functions whose real parts coincide with thePoisson kernel. In particular, B is Banach equivalent to the space of analytic functions F on the complex unit disc for which F ( z ) = 12 π (cid:90) π e iξ + ze iξ − z f ( ξ ) dξ . The question thatwas latter raised by Brett Wick in 2010 (personal communication with the first author)was whether this analytic characterization could be achieved in higher dimensions. Inorder to entertain such a question, one has to, for d ≥ I d sothat its restriction to I = [0 ,
1] is the original special atom.2. second, provide a definition of the special atom space B p on I d .3. third, verify that the Banach structure of B p is preserved.4. fourth, set the conditions on the weight function w on I d and define the weightedspecial atom space B w on I d .5. fifth, define the analytic extension F ( z ) of a function f ( ξ ) ∈ B w for z = ( z , z , · · · , z d ) ∈ D d and ξ = ( ξ , ξ , · · · , ξ d ) ∈ I d .6. sixth, verify that B w and its analytic extension A w are indeed Banach-equivalentunder the above conditions. 3 emark 1.7. The first step was recently accomplished in [13]. Also the requirementthat by restricting to I = [0 , we obtain the original special atom is for simplicity sake.The argument is important in high dimensions to prove for example in the case of Haarwavelets that we obtain an orthonormal system. However, there exist numerous ways todefine atoms similar to the special atom. We end this introductory part by recalling the definition of the weighted Lipschitz classof functions.
Definition 1.8.
Let w be a weight function defined on J = [ a − h, a + h ] ⊆ I . Theweighted Lipschitz class is the class of continuous functions defined as Λ w = (cid:26) f : R → R : (cid:107) f (cid:107) Λ w = sup h> ,ξ (cid:12)(cid:12)(cid:12)(cid:12) f ( ξ + h ) + f ( ξ − h ) − f ( ξ ) w ( J ) (cid:12)(cid:12)(cid:12)(cid:12) < ∞ . (cid:27) For completeness, recall that for w ( t ) = t , Λ w is the Zygmund class and for w ( t ) = t α , <α <
2, Λ w is the Lipschitz class of order α . It was proved in [3] that the dual space B ∗ w of B w is Λ (cid:48) w = { f (cid:48) : f ∈ Λ w } , where f (cid:48) is understood in the sense of distributions.The remainder of the paper is organized as follows: In Section 2, we show how to extendweighted special atoms to high dimensions. In the last step, we state the Main Theoremin Section 3, and we will make concluding remarks in Section 4. Let z = ( z , z , . . . , z d ) ∈ C d for an integer d ≥
1. In fact, in the sequel, bold-facedsymbols will represent vectors. We start out by proposing a definition of a weightedspecial atom in higher dimensions, for a general weight function w . When w is theLebesgue measure, the interested reader can refer to [13] for a more constructive approachin the definition. Definition 2.1.
Let ξ = ( ξ , ξ , . . . , ξ d ) ∈ R d and ≤ p < ∞ . (a) Let J := d (cid:89) j =1 [ a j − h j , a j + h j ] where a j , h j are real numbers with h j > . The weightedspecial atom (of type 1) on J , a sub-interval of I d , is defined as b w ( ξ ) = 1 w ( J ) { χ R ( ξ ) − χ L ( ξ ) } , where w ( J ) = (cid:90) J w ( ξ ) d ξ and R = d − (cid:91) j =1 J i j for some ( i , i , · · · , i d − ∈ (cid:8) , , · · · , d (cid:9) ) with i < i < · · · < i d − and L = J \ R . { J , J , · · · , J d } is the collection of sub-cubes of J , cut by the hyperplanes x = a , x , · · · , x d = a d , and χ A represents thecharacteristic function of set A .(b) The weighted special atom space B w is the space of real-valued functions f definedon I d such that f ( ξ ) = ∞ (cid:88) n =0 α n b w,n ( ξ ) with ∞ (cid:88) n =0 | α n | < ∞ , ndowed with the norm (cid:107) f (cid:107) B w = inf ∞ (cid:88) n =0 | α n | , where the infimum is taken over all possible representations of f . For example, for d = 2, for real numbers a , a , h , and h such that h , h >
0, considera sub-interval J of I d defined as J = [ a − h , a + h ] × [ a − h , a + h ] . Let L = [ a − h , a ] × [ a − h , a ] , L = [ a − h , a ) × [ a , a + h ] ,R = [ a , a + h ] × [ a − h , a ) , R = ( a , a + h ] × ( a , a + h ] . Consider L = L ∪ R and R = L ∪ R . The special atom b ( ξ , ξ ) is then defined as: b w ( ξ , ξ ) = 1 w ( J ) (cid:26) χ R ( ξ , ξ ) − χ L ( ξ , ξ ) (cid:27) = 1 w ( J ) (cid:26) χ L ( ξ , ξ ) + χ R ( ξ , ξ ) − χ L ( ξ , ξ ) − χ R ( ξ , ξ ) (cid:27) . For d ≥
2, we consider J = d (cid:89) j =1 [ a j − h j , a j + h j ] where a j , h j are real numbers with h j > j = 1 , · · · , d , we define b w ( ξ ) similarly. Figure 1 below is an illustration of b w for d = 2 (a) and d = 3 (b) when w is the Lebesgue measure.5a) (b) a - h a a + h a - h a a + h L L R R y x x y z a - h a - h a a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a - h a a + h a - h a a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a - h a a + h a - h a a + h a - h a a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a a + h a - h a - h a a + h a - h a a + h a - h a a + h a - h a - h a a + h a - h a a + h a - h a a + h a a - h a - h a a + h a - h a a + h a - h a a + h a a + h a - h a - h a a + h a - h a a + h a - h a a + h a a + h a - h a - h a a + h a - h a a + h a - h a a + h a a + h a - h a - h a a + h a - h a a + h a - h a a + h a a + h a - h a - h a a + h a - h a a + h a - h a a + h a a + h a - h a - h a a + h a - h a a + h a - h a a + h a a + h a - h a - h a a + h a - h a a + h a - h a a + h a a + h a - h a - h a a + h a - h a a + h a - h a a + h a a + h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a + h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a + h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a + h a + h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a + h a + h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a + h a + h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a + h a + h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a + h a + h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a + h a + h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a + h a + h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a + h a + h a - h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a + h a + h a - h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a a + h a + h a - h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a a + h a + h a + h a - h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a a + h a + h a + h a - h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a a + h a + h a + h a - h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a a + h a + h a + h a - h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a a + h a + h a + h a - h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a a + h a + h a + h a - h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a a + h a + h a + h a - h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a a + h a + h a + h a - h a - h a - h a - h a a + h a - h a a + h a - h a a + h a a a a - h a + h a + h a + h a - h a - h a - h a a - h a a + h a - h a a + h a - h a a + h a a a a - h a + h a + h a + h a + h a - h a - h a - h a a - h a a + h a - h a a + h a - h a a + h a a a a - h a + h a + h a + h a + h a - h a - h a - h a a - h a a + h a - h a a + h a - h a a + h a a a a - h a + h a + h a + h a + h a - h a - h a - h a a - h a a + h a - h a a + h a - h a a + h a a a a - h a + h a + h a + h a + h a - h a - h a - h a a - h a a + h a - h a a + h a - h a a + h a a a a - h a + h a + h a + h a + h a - h a - h a - h a a - h a a + h a - h a a + h a - h a a + h a a a a - h a + h a + h a + h a + h a - h a - h a - h a a - h a a + h a - h a a + h a - h a a + h a a a a - h a + h a + h a + h a + h a - h a - h a - h a a - h a a + h a - h a a + h a - h a a + h a a a a - h a + h a + h a + h a + h a - h a - h a - h a a - h a a + h a - h a a + h a - h a a + h a a a a - h a + h a + h a + h a + h a - h a - h a - h a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a + h a + h a + h a + h a - h a - h a - h a a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a + h a + h a + h a + h a + h a - h a - h a - h a a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a + h a + h a + h a + h a + h a - h a - h a - h a a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a + h a + h a + h a + h a + h a - h a - h a - h a a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a + h a + h a + h a + h a + h a - h a - h a - h a a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a + h a + h a + h a + h a + h a - h a - h a - h a a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a + h a + h a + h a + h a + h a - h a - h a - h a a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a + h a + h a + h a + h a + h a - h a - h a - h a a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a + h a + h a + h a + h a + h a - h a - h a - h a a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a + h a + h a + h a + h a + h a - h a - h a - h a a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a - h a + h a + h a + h a + h a + h a - h a - h a - h a a a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a - h a + h a + h a + h a + h a + h a + h a - h a - h a - h a a a a - h a a + h a - h a a + h a - h a a + h a a a a - h a - h a - h a + h a + h a + h Figure 1: (a) represents a special Atom for d = 2 and (b) represents the special atom for d = 3. Note that for d = 3, the color areas represent the different partitions of J j intosubintervals R and L intervals.With this definition, we can prove the following theorem about the Banach structureof B w . Theorem 2.2.
For ≤ p < ∞ , (cid:0) B w , (cid:107)·(cid:107) B w (cid:1) is a Banach space.Proof. The proof can be seen in Section 3 below.
Definition 2.3.
Consider the function P : D d × I d defined as P ( z , ξ ) = d (cid:89) j =1 P j ( z j , ξ j ) where P j ( z j , ξ j ) = e iξ j + z j e iξ j − z j . We observe that for fixed 1 ≤ j ≤ d and z j ∈ D , Re( P j ( z j , ξ j )) is the Poisson Kernel.For a function F defined on D d , we define F (cid:48) ( z ) as F (cid:48) ( z ) = ( f ( z ) , · · · , f d ( z )) , where f j ( z ) = ∂F ( z ) ∂z j . In the sequel d ξ = dξ dξ · · · dξ d , and this will be the case for similar bold-faced symbols.Also for a set A ⊆ S , we will denote by A c = S \ A .Now we give the definition of special weights functions that will be necessary for theProof of the main theorem 6 efinition 2.4. Let w be a real-valued function defined on [0 , . Let m and n be positiveintegers.(a) Then w is said to be Dini of order m ≥ and we denote w ∈ D m if w ( u ) u m ∈ L (0 , ,and there exists an absolute constant C for which (cid:90) u w ( ξ ) ξ m dξ ≤ Cw ( u ) , for < u < . (b) A function w : [0 , ∞ ) → R is said to be in the class B n for some positive integer n and we denote w ∈ B n if for < u < ,1. w is increasing and w (0) = 0 .2. There exists a constant C such that (cid:90) u w ( ξ ) ξ n +1 dξ ≤ C w ( u ) u n .(c) Let ξ = ( ξ , ξ , · · · , ξ d ) ∈ R d . For ≤ j ≤ d , consider weight functions w j definedon R + . We define the product weight function w ( ξ ) as w ( ξ ) = d (cid:89) j =1 w j ( ξ j ) , (2.1) (d) A weight w is said to be in the class B p on [0 , if there exists a constant C suchthat for any interval J ⊆ [0 , with center ξ J , we have | J | p w ( J ) (cid:90) J c w ( ξ ) | ξ − ξ J | p dξ ≤ C .
Remark 2.5.
1. We observe that w ∈ B p for some p > if and only if w is a doubling measure,that is, there exists an absolute constant C such that w [ Q h ( ξ )] ≤ Cw [ Q h ( ξ )] , where Q h ( ξ ) = { t ∈ R : | ξ − t | ≤ h } . The class of doubling weights will be referred to as D .2. The Muckenhoupt class of weights, (see [14]) A p = (cid:40) w ∈ L : (cid:18) | I | (cid:90) I w ( ξ ) dξ (cid:19) (cid:18) | I | (cid:90) I w / − p ( ξ ) dξ (cid:19) p − < ∞ (cid:41) is strictly contained in the class B p . The following is an important lemma relating class D m , B n and D . Lemma 2.6.
Let w be a weight function. Then for all integer n ≥ , we have: D ∩ B n ⊆ D . roof. Fix n ≥ w ∈ D ∩ B n . We would like to show that for u such that 2 u ≤ C such that (cid:90) u w ( ξ ) ξ dξ ≤ Cw ( u ) . We have that (cid:90) u w ( ξ ) ξ dξ = (cid:90) u w ( ξ ) ξ dξ + (cid:90) uu w ( ξ ) ξ dξ = I + I . On one hand, since w ∈ D , there exists C > I ≤ C · w ( u ). On the otherhand, I = (cid:90) uu w ( ξ ) ξ dξ = u n (cid:90) uu w ( ξ ) ξu n dξ ≤ u n n +1 (cid:90) uu w ( ξ ) ξ n +1 dξ, since ξu n ≤ u n +1 ≤ n +1 ξ n +1 ≤ C · n +1 · w ( u ) , since w ∈ B n ≤ C · w ( u ) . It follows that I + I ≤ C · w ( t ) and thus w ∈ D . Definition 2.7.
A product weight w ∈ D m (respectively w ∈ B n or w ∈ B p ) if for each ≤ j ≤ d , the weight w j ∈ D m (respectively w j ∈ B n or w j ∈ B p ) for integers m, n ≥ and a real number p > . We now introduce spaces of analytic functions that will be proven to be the analyticcharacterizations of weighted special atom spaces.
Definition 2.8.
Let ≤ p < ∞ be a real number. Given a function w : D d → R + , wewill consider A pw as the space of analytic functions F defined on D d such that (cid:107) F (cid:107) A pw = | F ( ) | + 1(2 π ) d (cid:90) D d (cid:12)(cid:12) F (cid:48) ( r e i ξ ) (cid:12)(cid:12) p w ( r e i ξ ) d ξ d r < ∞ , (2.2) where (cid:12)(cid:12) F (cid:48) ( r e i ξ ) (cid:12)(cid:12) = (cid:13)(cid:13) F (cid:48) ( r e i ξ ) (cid:13)(cid:13) = (cid:113) | f | + · · · + | f d | and z = r e i ξ . (2.3) Remark 2.9.
1. We note that for d = 1 and w ( re iξ ) = 1 , A w is contained in the Hardy space H ( D d ) .2. By Holder’s inequality, A pw ⊆ A w for p > . In particular, A w contains the weightedDirichlet space D w (see [11]) of analytic functions F such that (cid:90) D (cid:12)(cid:12) F (cid:48) ( re iξ ) (cid:12)(cid:12) w ( re iξ ) drdξ < ∞ . . When r = | z | and w ( re iξ ) = (1 − r ) α = (1 − | z | ) α for α > , then A w containsthe Bloch space B α (see [16]) of analytic functions F such that sup D (cid:8) (1 − | z | ) α | F (cid:48) ( z ) | (cid:9) < ∞ . Definition 2.10.
Let f ∈ B w . We define the analytic extension F of f as F ( z ) = 1(2 π ) d (cid:90) I P ( z , ξ ) f ( ξ ) d ξ , (2.4) in the sense that we can recover f by taking the radial limit f ( ξ ) = lim r → Re F ( r e i ξ ) , where the limit is taken element-wise. The space A w defined in equation (2.2) for some f ∈ B w will be referred to as the analytic extension of B w . Remark 2.11.
1. We observe that in the definition of the analytic extension F in (2.4) , the function log( F ( z )) can be viewed as the outer function whereas f ( ξ ) can be viewed as theinner function, similarly to a Beurling factorization, see for example Definition17.14 in [15].2. Additionally, in this definition, if one choose f ( ξ ) = log(1 − | b ( ξ ) | ) , then we candefine the function a ( z ) , the so called-Pythagorean mate of b ∈ H ( b ) (The interestedreader can refer for example to [10] for more on these spaces) as a ( z ) = exp (cid:18)(cid:90) I ξ + zξ − z log(1 − | b ( ξ ) | ) d ξ (cid:19) This suggests that the space A pw is closely related to the theory of H ( b ) spaces, butmore importantly, can be used as a gateway to their study in higher dimensions. Weknow that there is an extensive literature on these spaces H ( b ) in one dimension,see [2, 10, 4].3. Moreover, if f ∈ L p ( T d ) for < p < ∞ , then by Theorem 17.26 in [15], the analyticextension F ∈ H ( D d ) , the Hardy’s space on the polydisk. We will prove in the sequel that when the weight function w satisfies certain condi-tions, then the spaces B w and A w are in fact isometric to each other, that is, the inclusionoperator G : B w → A w , G ( f ) = F is a Banach isometry. Now we can now state our main theorem: 9 ain Theorem.
Let J j = [ a j − h j , a j + h j ] , J = d (cid:89) j =1 J j . Let w be a weight defined on J.Let A w be the space of analytic functions defined above. Then we have the following:(a) B w ⊆ A w if and only if w ( r e i ξ ) ≡ w ( ξ ) is a product weight and w ∈ B .(b) B w is Banach equivalent to A w if and only if w ( r e i ξ ) ≡ w (1 − r )1 − r is a productweight and w ∈ D ∩ B . Remark 3.1.
The Main theorem essentially states that B w and A w are isomorphic as Banach spacesin the sense that1. B w and A w are both Banach spaces,2. f ∈ B w if and only if its analytic extension F ∈ A w ,3. F ∈ A w if and only if lim r → Re F ( r e i ξ ) ∈ B w ,4. (cid:107) f (cid:107) B w ≡ (cid:107) F (cid:107) A w . Remark 3.2.
1. In the first part of the Main Theorem, the weight function depends only on theargument ξ of z = r e i ξ , whereas in the second part, it depends only on the radius r .The condition that w ∈ B in the first part is weaker than the condition w ∈ D ∩ B in the second part since per Lemma 2.6, w ∈ D ∩ B ⊆ D = (cid:83) p> B p implies theexistence of p > such that w ∈ B p . There is no guarantee that this p will be 2as in the first part. However, what the two parts have in common is the necessarycondition that if B w is contained in A w , then the weight w ∈ D .2. In the Main Theorem, the weight w is a product weight, however general weights w defined on I d are not addressed in this manuscript and it would be a worthwhilefuture endeavor to have a holistic understanding of the role of the weight w . The proof of the Main Theorem relies on some crucial lemmas that will be statedbelow. The first lemma shows that partial derivatives of analytic extensions of specialatoms in higher dimensions are bounded. Henceforth, the constants C will be genericand when necessary, their dependence on an interval J will be specified accordingly. Lemma 3.3.
Let J j = [ a j − h j , a j + h j ] , J = d (cid:89) j =1 J j and F ( z ) = 1(2 π ) d (cid:90) J P ( z , ξ ) b w ( ξ ) d ξ .Then for any j = 1 , · · · , d ,
1) there exists a constant C ( J ) such that f j ( z j ) = C ( J ) K ( a j , h j , z j ) d (cid:89) l =1 l (cid:54) = j K ( a l , h l , z l ) , (3.1) where K ( a j , h j , z j ) = 1 i (cid:20) z j − e i ( a j − h j ) + 1 z j − e i ( a j + h j ) + 2 e ia j − z j (cid:21) ,K ( a l , h l , z l ) = 2 i (cid:2) ln (cid:0) e i ( a l − h l ) − z l (cid:1) + ln (cid:0) e i ( a l + h l ) − z l (cid:1) − (cid:0) e ia l − z l (cid:1)(cid:3) . (2) Moreover for i, j = 1 , · · · , k , there are absolute constants C and C such that | K ( a j , h j , z j ) | ≤ C , | K ( a l , h l , z l ) | ≤ C . Lemma 3.4.
Let real numbers a and h > , and z ∈ D . Let J = [ a − h, a + h ] .(a) If w ∈ B such that w ( re iξ ) ≡ w ( ξ ) , then there exists a constant C such that (cid:90) (cid:90) D | K ( a, h, z ) | w ( ξ ) dξdr ≤ C ( J ) < ∞ . (b) If w ∈ D ∩ B such that w ( re iξ ) ≡ w (1 − r )1 − r , then there exists a constant C suchthat (cid:90) (cid:90) D | K ( a, h, z ) | w (1 − r )1 − r dξdr ≤ C ( J ) < ∞ . Lemma 3.5.
Let ≤ j ≤ d and J j = [ a j − h j , a j + h j ] for real numbers a j and h j > .Consider z j = r j e iξ j ∈ D such that h j < | e ia j − z j | . Consider a product weight w suchthat w j ( t ) t ∈ L (0 , for all ≤ j ≤ d . Then there exists a constant C ( J j ) > such that h j w j ( J j ) (cid:90) ξ j / ∈ J ω ( ξ j ) ξ j dξ j ≤ C ( J j ) (cid:90) (cid:90) D | f j ( z ) | dξ j dr j . Lemma 3.6.
Let j = [ a − h, a + h ] for real numbers a and h > and w ( t ) /t and in L ( J ) . Fix ≤ j ≤ d .(a) Consider D = (cid:8) z = re iξ ∈ D : h < | e ia − z | (cid:9) . There exists an absolute constant C such that (cid:90) (cid:90) D | f j ( z ) | w (1 − r )1 − r dξdr ≥ C j (cid:90) h w ( u ) u du (b) Consider the subset D = (cid:8) z ∈ D : | e ia − z | ≤ h (cid:9) . There exists an absolute constant C j such that (cid:90) (cid:90) D | f j ( z ) | w j (1 − r )1 − r dξdr ≥ C j (cid:90) h √ w ( u ) u du . roof of the Main Theorem. Part (a): Let w be a product weight such that w ∈ B . To show that B w ⊆ A w , it will beenough to show that analytic extensions of weighted special atoms b w ( ξ ) are containedin A w , that is, we will show that for a special atom b w ( ξ ) ∈ B w , we have F ∈ A w where F ( z ) = 1(2 π ) d (cid:90) I P ( z , ξ ) b w ( ξ ) d ξ . Fix 1 ≤ j ≤ d . Let d ξ − j d r − j = dξ dr · · · dξ j − dr j − dξ j +1 dr j +1 · · · dξ d dr d . From Lemma3.3 above, we have | f j ( z ) | = C ( J ) | K ( a j , h j , z j ) | d (cid:89) l =1 l (cid:54) = j | K ( a l , h l , z l ) |≤ M ( J ) | K ( a j , h j , z j ) | where M ( J ) = C ( J ) C d − . Also from Lemma 3.3 above, K ( a j , h j , z j ) is bounded, for all 1 ≤ j ≤ d . Thereforesup ≤ j ≤ d | K ( a j , h j , z j ) | exists. Moreover using the definition of | F (cid:48) ( z ) | in equation (2.3), weobtain | F (cid:48) ( z ) | ≤ d / M ( J ) sup ≤ j ≤ d | K ( a j , h j , z j ) | . (3.2)We then have that (cid:90) (cid:90) D d | F (cid:48) ( z ) | w ( ξ ) d ξ d r = (cid:90) (cid:90) D d | F (cid:48) ( z ) | (cid:32) d (cid:89) j =1 w j ( ξ j ) (cid:33) d ξ d r ≤ d / M ( J ) (cid:18) sup ≤ j ≤ d (cid:90) (cid:90) D | K ( a j , h j , z j ) | w j ( ξ j ) dξ j dr j (cid:19) × (cid:90) (cid:90) D d − d (cid:89) l =1 l (cid:54) = j w l ( ξ l ) d ξ − j d r − j ≤ C sup ≤ j ≤ d (cid:18)(cid:90) (cid:90) D | K ( a j , h j , z j ) | w j ( ξ j ) dξ j dr j (cid:19) < ∞ . This proves that F ∈ A w .Conversely, suppose that F ∈ A w . We will show that in this case, w ∈ B . As above,it suffices to consider analytic extensions F of weighted special atoms. Let C > (cid:107) F (cid:107) A w < C . 12ix 1 ≤ j ≤ d . Then we know that (cid:107) F (cid:48) ( z ) (cid:107) ≥ (cid:107) F (cid:48) ( z ) (cid:107) ∞ ≥ | f j ( z ) | . Therefore C > (cid:90) (cid:90) D d | F (cid:48) ( z ) | w ( ξ ) d ξ d r = (cid:90) (cid:90) D d (cid:32) d (cid:88) l =1 | f l ( z ) | (cid:33) / w ( ξ ) d ξ d r ≥ (cid:90) (cid:90) D d − d (cid:89) l =1 l (cid:54) = j w l ( ξ l ) d ξ − j d r − j × (cid:18)(cid:90) (cid:90) D | f j ( z ) | w j ( ξ j ) dξ j dr j (cid:19) ≥ d (cid:89) l =1 l (cid:54) = j (cid:107) w l (cid:107) L × (cid:32)(cid:90) ξ j / ∈ J j w j ( ξ j ) ξ j dξ j (cid:33) , by Lemma 3.5Since j is arbitrary, it follows that | J j | p w j ( J j ) (cid:90) (cid:90) ξ j / ∈ J j w j ( ξ j ) ξ j dξ j dr j < C, ∀ j = 1 , · · · , d. Therefore w ∈ B and this concludes the proof of part (a) of the theorem.Part (b). Now assume w is a product weight such that w ∈ D ∩ B . Let F ∈ B w . Asabove, we will proceed by showing that analytic extensions F of special atoms are in A w .Thus consider F ( z ) = 1(2 π ) d (cid:90) J P ( z , ξ ) b w ( ξ ) d ξ . We know from above equation (3.2) that (cid:90) (cid:90) D d | F (cid:48) ( z ) | w (1 − r )1 − r d ξ d r ≤ d / M ( J ) (cid:18) sup ≤ j ≤ d (cid:90) (cid:90) D | K ( a j , h j , z j ) | w j (1 − r j )1 − r j dξ j dr j (cid:19) × (cid:90) (cid:90) D d − d (cid:89) l =1 l (cid:54) = j w l (1 − r l )1 − r l d ξ − j d r − j By Lemma 3.4, we havesup ≤ j ≤ d (cid:90) (cid:90) D | K ( a j , h j , z j ) | w j (1 − r j )1 − r j dξ j dr j ≤ sup ≤ j ≤ d C ( J j ) < ∞ . By the Dini condition, we have that C = (cid:90) (cid:90) D d − d (cid:89) l =1 l (cid:54) = j w l (1 − r l )1 − r l d ξ − j d r − j = (2 π ) d − d (cid:89) l =1 l (cid:54) = j (cid:18)(cid:90) w l ( u l ) u l du l (cid:19) < ∞ . Therefore, we can infer that F ∈ A w .Conversely, suppose that F ∈ A w . We will show that w ∈ D ∩ B . Let C > F (cid:107) A w ≤ C . Fix 1 ≤ j ≤ d . Then as above, C > (cid:90) (cid:90) D d | F (cid:48) ( z ) | w (1 − r )1 − r d ξ d r ≥ (cid:90) (cid:90) D d − d (cid:89) l =1 l (cid:54) = j w l (1 − r l )1 − r l d ξ − j d r − j × (cid:18)(cid:90) (cid:90) D | f j ( z ) | w j (1 − r j )1 − r j dξ j dr j (cid:19) = C × (cid:90) (cid:90) D | f j ( z ) | w j (1 − r j )1 − r j dξ j dr j . (3.3)We can first combine the latter with equation (3.3) and Lemma 3.6 (a) to obtain that (cid:90) h j w j ( u j ) u j dr j ≤ C j , that is, w j ∈ B . Since j is arbitrary, it follows that w ∈ B . Wecan also combine the latter with (3.3), Lemma 2.6, and Lemma 3.6 (b) to conclude that w j ∈ D . Since j is arbitrary, it follows that w ∈ D .It remains to show that B w and A w are norm-equivalent. Since B w ⊆ A w , there exists aconstant M > (cid:107) f (cid:107) A w ≤ M (cid:107) f (cid:107) B w . To obtain the reverse inequality, it sufficesto use a simple extension of the one dimension case to obtain that the dual B ∗ w of B w iscontinuously contained in the dual A ∗ w of A w . Hence by virtue of the inclusion B w ⊆ A w ,we have A ∗ w ⊆ B ∗ w , so that A ∗ w = B ∗ w . We then have the following situation:(a): B w ⊆ A w implies that the inclusion map G : B w → A w is an open map.(b): (cid:107) f (cid:107) A w ≤ M (cid:107) f (cid:107) B w implies that G is a bounded linear map.Thus by the Open Mapping Theorem, the range of G ( B w ) = B w is dense in A w .(c) Since A ∗ w = B ∗ w , it follows that B w and A w are norm-equivalent, see for example [9]page 160. Proof of Theorem 2.2.
In the proof that (cid:107)·(cid:107) B w is a norm, only the triangle inequalityrequires special care. Using the definition of the infimum, let (cid:15) > { α n } n ∈ N , { β n } n ∈ N such that f ( ξ ) = (cid:88) n ∈ N α n b w,n ( ξ ) and g ( ξ ) = (cid:88) n ∈ N β n b w,n ( ξ ) and (cid:88) n ∈ N | α n | < (cid:107) f (cid:107) B w + (cid:15)/ , (cid:88) n ∈ N | β n | < (cid:107) g (cid:107) B w + (cid:15)/
2. Hence ( f + g )( ξ ) = (cid:88) n ∈ N ( α n + β n ) b w,n ( ξ ) with (cid:88) n ∈ N | α n + β n | ≤ (cid:88) n ∈ N | α n | + (cid:88) n ∈ N | β n | < ∞ . Therefore, (cid:107) f + g (cid:107) B w ≤ (cid:88) n ∈ N | α n + β n | ≤ (cid:88) n ∈ N | α n | + (cid:88) n ∈ N | β n | < (cid:107) f (cid:107) B w + (cid:107) g (cid:107) B w + (cid:15) . Since (cid:15) is arbitrary, it follows that (cid:107) f + g (cid:107) B w ≤ (cid:107) f (cid:107) B w + (cid:107) g (cid:107) B w .Now, let us prove that B w is a Banach space. It will be sufficient to show that everyabsolutely convergent sequence is convergent. In short, it will be enough to show thatgiven a sequence { f n } n ∈ N , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) n ∈ N f n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B w ≤ (cid:88) n ∈ N (cid:107) f n (cid:107) B w .Let (cid:15) >
0. Given n ∈ N , there is a sequence α n k of real numbers such that f n ( ξ ) =14 k ∈ N α n k b w,n k ( ξ ) with (cid:88) k ∈ N | α n k | < (cid:107) f n (cid:107) B w + (cid:15) n . Therefore (cid:88) n ∈ N (cid:88) k ∈ N | α n k | < (cid:88) n ∈ N (cid:107) f n (cid:107) B w + (cid:88) n ∈ N (cid:15) n = (cid:88) n ∈ N (cid:107) f n (cid:107) B w + (cid:15) . Since (cid:15) is arbitrary, it follows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) n ∈ N f n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:88) n ∈ N (cid:107) f n (cid:107) B w . Proof of Lemma 3.3.
Let P (cid:48) j ( z j , ξ j ) = ∂P j ( z j , ξ j ) ∂z j = e iξ j ( e iξ j − z j ) , for j = 1 , , . . . , d .Let us start with d = 2. Let J = [ a − h , a + h ] × [ a − h , a + h ].Then F ( z , z ) = 1(2 π ) (cid:90) J P ( z , ξ ) P ( z , ξ ) b w ( ξ , ξ ) dξ dξ , and f ( z , z ) = 2(2 π ) (cid:90) J P (cid:48) ( z , ξ ) P ( z , ξ ) b w ( ξ , ξ ) dξ dξ = 2 w ( J )(2 π ) [ I + I − I − I ] , where I = (cid:90) a a − h (cid:90) a + h a P (cid:48) ( z , ξ ) P ( z , ξ ) dξ dξ , I = (cid:90) a + h a (cid:90) a a − h P (cid:48) ( z , ξ ) P ( z , ξ ) dξ dξ I = (cid:90) a a − h (cid:90) a a − h P (cid:48) ( z , ξ ) P ( z , ξ ) dξ dξ , I = (cid:90) a + h a (cid:90) a + h a P (cid:48) ( z , ξ ) P ( z , ξ ) dξ dξ . Therefore I − I = (cid:90) a a − h (cid:90) a + h a P (cid:48) ( z , ξ ) P ( z , ξ ) dξ dξ − (cid:90) a + h a (cid:90) a + h a P (cid:48) ( z , ξ ) P ( z , ξ ) dξ dξ = − K ( a , h , z ) M ( a , h , z ) , where K ( a , h , z ) = − (cid:90) a a − h e iξ ( e iξ − z ) dξ + (cid:90) a + h a e iξ ( e iξ − z ) dξ = 1 i (cid:20) z − e i ( a − h ) + 1 z − e i ( a + h ) + 2 e ia − z (cid:21) . Also, M ( a , h , z ) = (cid:90) a + h a P ( z , ξ ) dξ
15 1 i (cid:2) − ih + 2 ln( e ia − z ) − e i ( a − h ) − z ) (cid:3) . Likewise, we have I − I = K ( a , h , z ) M (cid:48) ( a , h , z ) with M (cid:48) ( a , h , z ) = 1 i (cid:2) − ih − e ia − z ) + 2 ln( e i ( a − h ) − z ) (cid:3) . It follows that I − I + I − I = K ( a , h , z ) [ M (cid:48) ( a , h , z ) − M ( a , h , z )] = K ( a , h , z ) K a , h , z ) , where K ( a , h , z ) = 2 i (cid:2) ln( e i ( a − h ) − z ) + ln( e i ( a + h ) − z ) − e ia − z ) (cid:3) . Hence f ( z , z ) = C ( J ) K ( a , h , z ) K ( a , h , z ) , where C ( J ) = 2 w ( J )(2 π ) . Similarly, we obtain f ( z , z ) = C ( J ) K ( a , h , z ) K ( a , h , z ) . For d ≥
2, we observe that the constant C ( J ) will remain the same, regardless of thevariable of differentiation. Moreover, the function K takes as arguments a j , h j , z j if weare differentiating with respect to z j and K takes as arguments a l , h l , h l , for all l (cid:54) = j .The product comes from the fact that the integrand is made of functions with separablevariables. Hence, we conclude that f j ( z ) = C ( J ) K ( a j , h j , z j ) d (cid:89) l =1 l (cid:54) = k K ( a l , h l , z l ) . Moreover, | K ( a l , h l , z l ) | ≤ (cid:2)(cid:12)(cid:12) ln( e i ( a l − h l ) − z l ) (cid:12)(cid:12) + (cid:12)(cid:12) ln( e i ( a l + h l ) − z l ) (cid:12)(cid:12) + 2 (cid:12)(cid:12) ln( e i ( a l ) − z l ) (cid:12)(cid:12)(cid:3) . Put Z l = e i ( a l − h l ) − z l . We know that | z l | <
1, thus ln( | Z il | ) ≤ ln(2). | ln( Z l ) | = (cid:112) (ln( | Z il | )) + arg( Z l ) . Consequently, | ln( Z l ) | ≤ (cid:113) (ln(2)) + π . Applying a similar argument to (cid:12)(cid:12) ln( e i ( a l − h l ) − z l ) (cid:12)(cid:12) and (cid:12)(cid:12) ln( e i ( a l − h l ) − z l ) (cid:12)(cid:12) , we obtain that | K ( a l , h l , z l ) | ≤ (cid:112) + π . We will use a similar argument to [3] to deal with K ( a , h , z ). However, this approachis much general than theirs in that they assumed that a = 0 which is not assumed here.We observe that iK ( a , h , z ) = 1 z − e i ( a − h ) + 1 z − e i ( a + h ) + 1 e ia − z
16 2 e ia ( z + e ia )(1 − cos h )( z − e i ( a − h ) )( z − e i ( a + h ) )( e ia − z ) . We have that( z − e i ( a − h ) )( z − e i ( a + h ) ) = (( e ia − z ) + 2 e ia z (1 − cos h ) , so the modulus of the denominator of iK ( a , h , z ) is (cid:12)(cid:12) ( z − e i ( a − h ) )( z − e i ( a + h ) )( e ia − z ) (cid:12)(cid:12) = (cid:12)(cid:12) ( e ia − z ) + 2 e ia z (1 − cos h ) (cid:12)(cid:12) (cid:12)(cid:12) e ia − z (cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:0) e ia − z (cid:1) − h (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) e ia − z (cid:12)(cid:12) . The last inequality is obtained by noticing that | z | < , | e ia | = 1, and 1 − cos h ≤ h .Now consider D = { z ∈ D : | e ia − z | > h } . For z ∈ D , the last inequality impliesthat (cid:12)(cid:12) ( z − e i ( a − h ) )( z − e i ( a + h ) )( e ia − z ) (cid:12)(cid:12) ≥ (cid:12)(cid:12) e ia − z (cid:12)(cid:12) ≥ h . On the other hand, the modulus of the numerator of iK ( a , h , z ) is bounded by 2 h on D , so that on D , one has | K ( a , h , z ) | ≤ h | e ia − z | ≤ h . (3.4)Now let z ∈ D c . Then we have that (cid:12)(cid:12) z − e i ( a − h ) (cid:12)(cid:12) , (cid:12)(cid:12) z − e i ( a + h ) (cid:12)(cid:12) , (cid:12)(cid:12) e ia − z (cid:12)(cid:12) ≤ h . PutΦ ∗ n = (cid:8) z ∈ D : 2 − n h < (cid:12)(cid:12) e ia − z (cid:12)(cid:12) ≤ − n h (cid:9) = (cid:26) z ∈ D : 2 n − h ≤ | e ia − z | < n − h (cid:27) . We note that (0 ,
1] = ∞ (cid:83) n =0 (cid:0) n +1 , n (cid:3) . It follows that D c ⊆ (cid:8) z ∈ D : (cid:12)(cid:12) e ia − z (cid:12)(cid:12) ≤ h (cid:9) ⊆ ∞ (cid:91) n =0 (cid:26) z ∈ D : 4 h n +1 < (cid:12)(cid:12) e ia − z (cid:12)(cid:12) ≤ h n (cid:27) = ∞ (cid:91) n =0 Φ ∗ n . (3.5)Thus, there exists an integer n such that | K ( a , h , z ) | ≤ | z − e i ( a − h ) | + 1 | z − e i ( a + h ) | + 1 | e ia − z | ≤ n − h . We conclude by taking C = max { n − h , h } and C = 3 (cid:112) + π .17 roof of Lemma 3.4. Part (a): Put J = [ a − h, a + h ] for some real numbers a and h >
0. Let N bethe smallest integer such that 2 N h ≥
1. Then for all n ≤ N , and z ∈ D , we have2 n h < | e ia − z | < < n +1 h .Also we observe that (cid:8) z = re iξ ∈ D : | e ia − z | ≤ ν (cid:9) ⊆ (cid:8) z = re iξ ∈ D : r ≥ − ν, | ξ − a | ≤ ν (cid:9) .Put Φ n = { z ∈ D : 2 n h ≤ | e ia − z | ≤ n +1 h } and U = (cid:90) (cid:90) D K ( a, h, z ) w ( ξ ) dξdr, U = (cid:90) (cid:90) D c K ( a, h, z ) w ( ξ ) dξdr . It follows that U ≤ h N (cid:88) n =0 (cid:90) (cid:90) Φ n w ( ξ ) | e ia − z | dξdr ≤ h N (cid:88) n =0 n h ) (cid:90) Q n +1 h ( a ) w ( ξ ) dξ ≤ C N (cid:88) n =0 n h ) (cid:90) Q nh ( a ) w ( ξ ) dξ, since w is doubling ≤ C N (cid:88) n =0 (cid:90) n h ≤| ξ − a |≤ n +1 h w ( ξ )( ξ − a ) dξ = C (cid:90) N +1 hh w ( ξ )( ξ − a ) dξ ≤ C w ( J ) | I | (cid:32) | J | w ( J ) (cid:90) ξ / ∈ J w ( ξ )( ξ − a ) dξ (cid:33) < Cw ( J ) , since w ∈ B For z ∈ D c , we have (cid:90) (cid:90) D c w ( ξ ) | e ia − z | dξdr ≤ ∞ (cid:88) n =0 (cid:90) (cid:90) Φ ∗ n w ( ξ ) | e ia − z | dξdr using equation (3.5) ≤ C ∞ (cid:88) n =0 n h (cid:90) − − n h (cid:90) Q − nh ( a ) w ( ξ ) dξdr using again equation (3.5) ≤ C ∞ (cid:88) n =0 (cid:90) Q − nh ( a ) w ( ξ ) dξ, since w is a doubling ≤ C (cid:90) Q h ( a ) w ( ξ ) dξ ≤ C (cid:90) Q h ( a ) w ( ξ ) dξ = Cw ( J ) . It follows that for z ∈ D c , U ≤ Cw ( J ) . Part (b): Suppose w ∈ D ∩ B such that w ( ξ, r ) ≡ w (1 − r )1 − r . Put V = (cid:90) (cid:90) D K ( a, h, z ) w (1 − r )1 − r drdξ, V = (cid:90) (cid:90) D c K ( a, h, z ) w (1 − r )1 − r drdξ . V ≤ h (cid:90) (cid:90) D | e ia − z | w (1 − r )1 − r drdξ ≤ h N (cid:88) n =0 n h ) (cid:90) | a | +2 n +1 h −| a |− n +1 h (cid:90) − n +1 h w (1 − r )1 − r drdξ ≤ h N (cid:88) n =0 (cid:18) | a | (2 n h ) + 1(2 n h ) (cid:19) (cid:90) n +1 h w ( u ) u du, by change of variable u = 1 − r ≤ h C N (cid:88) n =0 (cid:18) | a | (2 n h ) + 1(2 n h ) (cid:19) w (2 n h ) , since w ∈ D = C ( S + S )On one hand S = N (cid:88) n =0 n h ) w (2 n h )= N (cid:88) n =0 (cid:90) a +2 n +1 ha − n +1 h w (2 n h )(2 n h ) du ≤ C N (cid:88) n =0 (cid:90) a +2 n +1 ha − n +1 h w ( u )( u ) du, since w increasing and u < n h ) ≤ C (cid:90) a + ha − h w ( u ) u du ≤ C (cid:90) a − h w ( u ) u du ≤ since J ⊆ [0 , ≤ C w ( a − h )( a − h ) = Chw ( a − h ) , since w ∈ B ≤ Cw ( J ) , since w is increasing . On the other hand, we know that D ∩ B ⊆ D = (cid:83) p> B p . Let p > w ∈ B p . S = N (cid:88) n =0 | a | (2 n h ) w (2 n h ) ≤ | a | ( N + 1) (cid:32) N (cid:88) n =0 n h ) (cid:33) w (2 N h ) , since w is increasing ≤ C w (2 N h )(2 N h ) p − where C = | a | ( N + 1)(2 N h ) p − (cid:32) N (cid:88) n =0 n h ) (cid:33) ≤ C (cid:90) N +1 h N h w (2 N h )(2 N h ) p du ≤ p C (cid:90) N +1 h N h w ( u ) u p du since w is increasing ≤ p C w ( J ) | J | p (cid:18) | J | p w ( J ) (cid:90) u/ ∈ J w ( u ) u p du (cid:19) < Cw ( J ) , since 2 N h ≥ w ∈ B p .19ow V ≤ ∞ (cid:88) n =0 (cid:90) (cid:90) Φ ∗ n | e ia − z | w (1 − r )1 − r dξdr ≤ ∞ (cid:88) n =0 (cid:90) − − n h (cid:90) | a | +2 − n h −| a |− − n h | e ia − z | w (1 − r )1 − r dξdr ≤ ∞ (cid:88) n =0 n − h (cid:90) − − n h (cid:90) | a | +2 − n h −| a |− − n h w (1 − r )1 − r dξdr, since | e ia − z | < n − h on Φ ∗ n ≤ ∞ (cid:88) n =0 n h ( | a | + 2 − n h ) (cid:90) − n h w ( u ) u du, after the change of variable u = 1 − r = S + S . On one hand, S = ∞ (cid:88) n =0 n h − n h (cid:90) − n h w ( u ) u du = 4 ∞ (cid:88) n =0 (cid:90) − n h w ( u ) u du = 4 (cid:90) h w ( u ) u du ≤ Cw (4 h ) ≤ Cw ( h ) ≤ Cw ( J ) , since w ∈ D ∩ B ⊆ D . On the other hand S = | a | ∞ (cid:88) n =0 n h (cid:90) − n h w ( u ) u du ≤ C ∞ (cid:88) n =0 n h w (4 · − n h ) , since w ∈ D ≤ C ∞ (cid:88) n =0 n h w (2 − n h ) , since w ∈ D ≤ C ∞ (cid:88) n =0 (cid:90) − n h − n h w ( u ) u du, since − n h ≤ u ≤ − n h ≤ C (cid:90) h w ( u ) u du ≤ Cw (2 h ) ≤ Cw ( J ) , since w ∈ D . Proof of Lemma 3.5.
We will start again with the case d = 2. We know from above that iK ( a , h , z ) = 2 e ia ( z + e ia )(1 − cos h )( e ia − z )(( e ia − z ) + 2 e ia z (1 − cos h )) . h and a so that h < | e ia − z | . Then | z + e ia | ≥ − h . Since1 − h ≤ cos h ≤ − h + h , we obtain | K ( a , h , z ) | ≥ (2 − h ) (cid:16) h − h (cid:17) | ( e ia − z ) | | ( e ia − z ) + 2 e ia z (1 − cos h ) |≥ (2 − h ) (cid:16) h − h (cid:17) | e ia − z | = Ch | e ia − z | . (3.6)Now let us choose a and h such that 1 < h < | e ia − z | . We also know from abovethat K ( a , h , z ) = 2 i (cid:2) ln( e i ( a − h ) − z ) + ln( e i ( a + h ) − z ) − e ia − z ) (cid:3) . We know that | ln( e ia − z ) | ≥ ln( | ( e ia − z ) | ) ≥ ln( h ) > . Therefore | K ( a , h , z ) | ≥ h ) . (3.7)Combining both equations (3.6) and (3.7), it follows that for h < | e ia − z | and 1 2, we can generalize it so that given 1 ≤ j ≤ d and l (cid:54) = j one has | f j ( z ) | ≥ C ( h j ) h j | e ia j − z j | , for h j < (cid:12)(cid:12) e ia j − z j (cid:12)(cid:12) , < h l < (cid:12)(cid:12) e ia l − z l (cid:12)(cid:12) . (3.8)Assume that z j = r i e iξ j with r j < 1. Then (cid:12)(cid:12) e ia j − z j (cid:12)(cid:12) = 1 − r j − r j cos( ξ j − a j ) ≤ (1 − r j ) + ( ξ j − a j ) . Let D j = { z j ∈ D : h j < | e ia j − z j |} and D ∗ j = { z j ∈ D : h j < ξ j < a j + 1 } . Note thatthen D j ∩ D ∗ j (cid:54) = ∅ . We have that (cid:90) (cid:90) D | f j ( z ) | w j ( ξ ) dξ j dr j ≥ h j (cid:90) (cid:90) D j w j ( ξ i ) | e ia j − z j | dξ j dr j ≥ h j (cid:90) (cid:90) ξ>h j w j ( ξ i )((1 + r j ) + ( ξ j − a j ) ) / dξ j dr j ≥ h j (cid:90) ξ>h j w j ( ξ i )( ξ j − a j ) dξ j since x − > x − on (0,1) ≥ h j (cid:90) ξ>h j w j ( ξ i ) ξ j dξ j since 0 < h j < ξ j < a j + 1 . roof of Lemma 3.6. Firstly, let D j = (cid:8) z j ∈ D : h j < (cid:12)(cid:12) e ia j − z j (cid:12)(cid:12)(cid:9) . Therefore, if 1 − r j < | ξ j − a j | , we have h j ≤ | e ia j − z j | ≤ √ ξ j − a j ). There are twopossibilities: either 1 − r j < h j or 1 − r j ≥ h j . So let us consider the subset D ∗ j = { z j ∈ D : h j ≤ − r j < | ξ j − a j |} of D j . It follows from equation (3.8) that (cid:90) (cid:90) D | f j ( z ) | w j (1 − r j )1 − r j dξ j dr j ≥ (cid:90) (cid:90) D j | f j ( z ) | w j (1 − r j )1 − r j dξ j dr j ≥ C ( h j ) (cid:90) (cid:90) D j h j | e ia j − z j | w j (1 − r j )1 − r j dξ j dr j ≥ h j C ( h j )4 √ (cid:90) (cid:90) D j ξ j − a j ) w j (1 − r j )1 − r j dξ j dr j = h j C ( h j )4 √ (cid:90) − h j w j (1 − r j )1 − r j (cid:32)(cid:90) π − r j + a j ξ j − a j ) dξ j (cid:33) dr j ≥ C j (cid:90) h j w j ( u j ) u j du j . The latter inequality is obtained after the changes of variable ξ − a = θ and 1 − r = u respectively, with C j = h j C ( h j ) √ D j = (cid:26) z j ∈ D : (cid:12)(cid:12) e ia j − z j (cid:12)(cid:12) ≤ h j (cid:27) . As above, if | ξ j − a j | < − r j , we have | e ia j − z j | ≤ √ − r j ). So either √ − r j ) < h j or √ − r j ) ≥ h j . Thus, consider the subset D ∗ j = (cid:26) z j ∈ D : | ξ j − a j | < − r j < h j √ (cid:27) of D j . Therefore, we have (cid:90) (cid:90) D | f j ( z ) | w j (1 − r j )1 − r j dξ j dr j ≥ (cid:90) (cid:90) D j | f j ( z ) | w j (1 − r j )1 − r j dξ j dr j ≥ C ( h j ) (cid:90) (cid:90) D j h j | e ia j − z j | w j (1 − r j )1 − r j dξ j dr j = C ( h j )16 √ (cid:90) − hj √ w j (1 − r j )(1 − r j ) (cid:32)(cid:90) a j +1 − r j a j + r j − dξ j (cid:33) dr j = C j (cid:90) hj √ w j ( u j ) u j du j , after the change of variable u = 1 − r with C j = C ( h j )16 √ . Conclusion We have proposed in this paper a space that acts as the analytic extension of the so-calledspecial atom spaces in higher dimensions. What we find interesting and remarkable inthese spaces is their apparent simplicity. Certainly one could think of atoms defined onintervals that are different from the characteristic functions on these intervals, say forinstance polynomial type of atoms. However, the properties like orthonormality wouldbe difficult to prove, especially in higher dimensions. 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