On Lusin's area integrals and g-functions in certain Dunkl and Laguerre settings
aa r X i v : . [ m a t h . C A ] N ov ON LUSIN’S AREA INTEGRALS AND G-FUNCTIONS IN CERTAIN DUNKL ANDLAGUERRE SETTINGS
TOMASZ SZAREK
Abstract.
We investigate g -functions and Lusin’s area type integrals related to certain multi-dimen-sional Dunkl and Laguerre settings. We prove that the considered square functions are bounded onweighted L p , 1 < p < ∞ , and from L into weak L . Introduction
This paper embraces a completion and extension of the research initiated by the author in [15] thatconcerned square functions related to the so-called Laguerre expansions of convolution type. Here wegeneralize the results of [15] by studying square functions in the context of the Dunkl harmonic oscillatorand the related group of reflections isomorphic to Z d . This Dunkl setting reduces to that of [15] afterrestricting to reflection invariant functions. Consequently, the results delivered by the present paperimplicitly contain, in particular, those of [15]. Moreover, a trivial choice of the multiplicity functionreduces the Dunkl setting to the situation of classical Hermite function expansions. Thus our results mayalso be seen as a continuation and extension of the investigations of Thangavelu [16], Harboure, de Rosa,Segovia and Torrea [4] and Stempak and Torrea [14], concerning g -functions in the context of the classicharmonic oscillator.An essential novelty in comparison with the previous study is the investigation of Lusin’s area typeintegrals. These objects have more complex structure than the vertical and horizontal g -functions andhence their treatment requires additional arguments and effort. The results obtained in the Dunkl settingimply similar results in the Hermite setting and in the Laguerre situation of [15], where Lusin’s area typeintegrals were not considered.It is commonly known that square functions play an important role in harmonic analysis (see [15,Section 1] for brief comments and references), being valuable tools with several significant applications.Also the results we prove have some interesting potential applications, which remain to be investigated;this concerns, in particular, multiplier theorems and characterizations of Hardy spaces. Similarly to [15],the present work contributes to the development of Littlewood-Paley theory for discrete and continuousorthogonal expansions, which receives a considerable attention in recent years, see [15, Section 1] forreferences. In particular, Lusin’s area type integrals in the context of another, one-dimensional, Laguerresetting, and also in the one-dimensional Hermite context, were studied very recently by Betancor, Molinaand Rodr´ıguez-Mesa [2].We refer the reader to the survey article by R¨osler [11] for basic facts concerning Dunkl’s theory. Aprecise description of the Dunkl framework for the particular group of reflections G isomorphic to Z d canbe found for instance in [9, Section 3]. Here we only invoke the most relevant facts. We shall work onthe space R d , d ≥
1, equipped with the measure dw α ( x ) = d Y j =1 | x j | α j +1 dx, x = ( x , . . . , x d ) ∈ R d , and with the Euclidean norm | · | . The multi-index α = ( α , . . . , α d ) will always be assumed to belong to[ − / , ∞ ) d . Consider the reflection group G generated by σ j , j = 1 , . . . , d , σ j ( x , . . . , x j , . . . , x d ) = ( x , . . . , − x j , . . . , x d ) . Mathematics Subject Classification.
Key words and phrases. square function, g -function, Lusin’s area integral, Dunkl’s harmonic oscillator, generalizedHermite expansions, Laguerre semigroup, Laguerre expansions of convolution type, Calder´on-Zygmund operator, A p weight. Clearly, the reflection σ j is in the hyperplane orthogonal to e j , the j th coordinate vector. Notice thatthe measure w α is G -invariant. The Dunkl differential-difference operators T αj , j = 1 , . . . , d , are given by T αj f ( x ) = ∂ x j f ( x ) + ( α j + 1 / f ( x ) − f ( σ j x ) x j , f ∈ C ( R d ) , j = 1 , . . . , d, and form a commuting system. The Dunkl Laplacian,∆ α f ( x ) = d X j =1 (cid:0) T αj (cid:1) f ( x ) = d X j =1 (cid:18) ∂ f∂x j ( x ) + 2 α j + 1 x j ∂f∂x j ( x ) − ( α j + 1 / f ( x ) − f ( σ j x ) x j (cid:19) , is formally self-adjoint in L ( R d , dw α ). The Dunkl harmonic oscillator is defined as L α = − ∆ α + | x | . This operator will play in the present paper a similar role to that of the Euclidean Laplacian in theclassical harmonic analysis. Note that for α = ( − / , . . . , − / L α becomes the classic harmonicoscillator − ∆ + | x | . We shall consider a self-adjoint extension L α of L α , whose spectral decompositionis discrete and given by the generalized Hermite functions h αn , see Section 2 for details. Natural partialderivatives related to L α are obtained from the symmetric decomposition L α = 12 d X j =1 ( δ ∗ j δ j + δ j δ ∗ j ) , where δ j = T αj + x j , δ ∗ j = − T αj + x j , j = 1 , . . . , d ;here δ ∗ j is the formal adjoint of δ j in L ( R d , dw α ).The main objects of our study are vertical and horizontal g -functions and Lusin’s type area integralsbased on the semigroup generated by L α . Our main result, Theorem 2.1 below, says that each of thesquare functions is bounded on weighted L p ( dw α ), 1 < p < ∞ , and satisfies weighted weak type (1,1)inequality for a large class of weights. To prove this, we exploit the arguments from [10] that allow toreduce the analysis to the context of the smaller measure space ( R d + , dw + α ) and suitably defined Laguerre-type square functions, where R d + = (0 , ∞ ) d and w + α is the restriction of w α to R d + . Then we apply thegeneral theory of vector-valued Calder´on-Zygmund operators with the underlying space of homogeneoustype ( R d + , dw + α , | · | ). The main technical difficulty connected with this approach is to show the relevantkernel estimates. Here, similarly as in [15], we use a convenient technique having roots in Sasso’s work [12]and developed later by Nowak and Stempak in [8]. For our purposes we derive some further generalizationsof this interesting method, which may be of independent interest. It is remarkable that essentially thesame procedure applies as well to higher order square functions in the investigated setting. The relatedanalysis, however, is because of its length beyond the scope of this article.The paper is organized as follows. Section 2 contains the setup, definitions of the investigated squarefunctions, statements of the main results and the accompanying comments and remarks. Also, suitableLaguerre-type square functions, related to the restricted space ( R d + , dw + α ), are defined and the proof ofthe main theorem is reduced to showing that these auxiliary square functions can be viewed as vector-valued Calder´on-Zygmund operators. In Section 3 the Laguerre-type square functions are proved to be L -bounded and associated, in the Calder´on-Zygmund theory sense, with the relevant kernels. Finally,Section 4 is devoted to the proofs of all necessary kernel estimates. This is the largest and most technicalpart of the work.Throughout the paper we use a standard notation with essentially all symbols referring to the spaces( R d , dw α , |·| ) or ( R d + , dw + α , |·| ). Thus ∆ and ∇ denote the Euclidean Laplacian and gradient, respectively.Further, L p ( R d , W dw α ) stands for the weighted L p ( R d , dw α ) space, W being a nonnegative weight on R d ; we write simply L p ( dw α ) if W ≡
1. By h f, g i dw α we mean R R d f ( x ) g ( x ) dw α ( x ) whenever the integralmakes sense. In a similar way we define L p ( R d + , W dw + α ) and h f, g i dw + α . For 1 ≤ p < ∞ we denote by A α, + p the Muckenhoupt class of A p weights associated to the space ( R d + , dw + α , | · | ).While writing estimates we will frequently use the notation X . Y to indicate that X ≤ CY with apositive constant C independent of significant quantities. We will write X ≃ Y when X . Y and Y . X . QUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS 3
Acknowledgments.
The author would like to thank Dr. Adam Nowak for suggesting the topic andfor many discussions related to this paper.2.
Preliminaries and statement of results
Let m = ( m , . . . , m d ) ∈ N d , N = { , , . . . } , and α = ( α , . . . , α d ) ∈ [ − / , ∞ ) d be multi-indices. Thegeneralized Hermite functions in R d are defined as the tensor products h αm ( x ) = h α m ( x ) · . . . · h α d m d ( x d ) , x = ( x , . . . , x d ) ∈ R d , where h α i m i are the one-dimensional generalized Hermite functions h α i m i ( x i ) = d m i ,α i e − x i / L α i m i ( x i ) ,h α i m i +1 ( x i ) = d m i +1 ,α i e − x i / x i L α i +1 m i ( x i );here L α i m i is the Laguerre polynomial of degree m i and order α i , and d k,α i , k ∈ N , are proper normalizingconstants, see [9, p. 544] or [10, p. 4]. The system { h αm : m ∈ N d } is an orthonormal basis in L ( R d , dw α )consisting of eigenfunctions of L α , L α h αm = λ α | m | h αm , λ αn = 2 n + 2 | α | + 2 d, n ∈ N ;here | m | = m + . . . + m d is the length of m . The operator L α f = ∞ X n =0 λ αn X | m | = n h f, h αm i dw α h αm , defined on the domainDom( L α ) = n f ∈ L ( R d , dw α ) : X m ∈ N d (cid:12)(cid:12) λ α | m | h f, h αm i dw α (cid:12)(cid:12) < ∞ o , is a self-adjoint extension of L α considered on C ∞ c ( R d ) as the natural domain (the inclusion C ∞ c ( R d ) ⊂ Dom( L α ) may be easily verified).The heat semigroup T αt = exp( − t L α ), t ≥
0, generated by L α is a strongly continuous semigroup ofcontractions on L ( R d , dw α ). By the spectral theorem, T αt f = ∞ X n =0 e − tλ αn X | m | = n h f, h αm i dw α h αm , f ∈ L ( R d , dw α ) . We have the integral representation T αt f ( x ) = Z R d G αt ( x, y ) f ( y ) dw α ( y ) , x ∈ R d , t > , where the Dunkl heat kernel is given by(2.1) G αt ( x, y ) = ∞ X n =0 e − tλ αn X | m | = n h αm ( x ) h αm ( y ) . This oscillating series can be summed, see for instance [9, p. 544] or [10, p. 5], and the resulting formulais G αt ( x, y ) = X ε ∈ Z d G α,εt ( x, y ) , with the component kernels G α,εt ( x, y ) = (2 sinh 2 t ) − d exp (cid:16) −
12 coth(2 t ) (cid:0) | x | + | y | (cid:1)(cid:17) d Y i =1 ( x i y i ) ε i I α i + ε i (cid:0) x i y i sinh 2 t (cid:1) ( x i y i ) α i + ε i , where I ν denotes the modified Bessel function of the first kind and order ν . Here we consider the functions z z ν and the Bessel function as analytic functions on C cut along the half axis { ix : x ≤ } , see thereferences given above. Note that G α,εt ( x, y ) is also expressed by the series (2.1), but with the summationin m restricted to the set N ε = (cid:8) m ∈ N d : m i is even if ε i = 0 , m i is odd if ε i = 1 , i = 1 , . . . , d (cid:9) . SQUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS
The operators determined by integration against G α,εt ( x, y ) dw α ( y ), ε ∈ Z d , will be denoted by T α,εt .Clearly, we have the decomposition T αt = X ε ∈ Z d T α,εt . (2.2)We consider the following vertical and horizontal square functions based on the Dunkl heat semigroup: g V ( f )( x ) = (cid:13)(cid:13) ∂ t T αt f ( x ) (cid:13)(cid:13) L ( tdt ) ,g jH ( f )( x ) = (cid:13)(cid:13) δ j T αt f ( x ) (cid:13)(cid:13) L ( dt ) , j = 1 , . . . , d,g jH, ∗ ( f )( x ) = (cid:13)(cid:13) δ ∗ j T αt f ( x ) (cid:13)(cid:13) L ( dt ) , j = 1 , . . . , d,S V ( f )( x ) = (cid:16) R A ( x ) t (cid:12)(cid:12) ∂ t T αt f ( z ) (cid:12)(cid:12) dw α ( z ) V α √ t ( x ) dt (cid:17) / ,S jH ( f )( x ) = (cid:16) R A ( x ) (cid:12)(cid:12) δ j T αt f ( z ) (cid:12)(cid:12) dw α ( z ) V α √ t ( x ) dt (cid:17) / , j = 1 , . . . , d,S jH, ∗ ( f )( x ) = (cid:16) R A ( x ) (cid:12)(cid:12) δ ∗ j T αt f ( z ) (cid:12)(cid:12) dw α ( z ) V α √ t ( x ) dt (cid:17) / , j = 1 , . . . , d, where A ( x ) is the parabolic cone with vertex at x , A ( x ) = ( x,
0) +
A, A = n ( z, t ) ∈ R d × (0 , ∞ ) : | z | < √ t o , and V αt ( x ) is the w α measure of the cube centered at x and with side lengths 2 t . More precisely, V αt ( x ) = d Y j =1 V α j t ( x j ) , V α j t ( x j ) = w α j (cid:0) ( x j − t, x j + t ) (cid:1) , x ∈ R d , t > . The above definitions of S V , S jH , S jH, ∗ fit into a general concept of Lusin’s area integrals in a context ofspaces of homogeneous type; see for instance [5, (2.10)] or [2, Section 1]. It is not hard to see that thearea type integrals just defined can be written as S V ( f )( x ) = (cid:13)(cid:13)(cid:13) ∂ t T αt f ( x + z ) r w α ( x + z ) V α √ t ( x ) (cid:13)(cid:13)(cid:13) L ( A,tdtdz ) ,S jH ( f )( x ) = (cid:13)(cid:13)(cid:13) δ j T αt f ( x + z ) r w α ( x + z ) V α √ t ( x ) (cid:13)(cid:13)(cid:13) L ( A,dtdz ) , j = 1 , . . . , d,S jH, ∗ ( f )( x ) = (cid:13)(cid:13)(cid:13) δ ∗ j T αt f ( x + z ) r w α ( x + z ) V α √ t ( x ) (cid:13)(cid:13)(cid:13) L ( A,dtdz ) , j = 1 , . . . , d. Our main result concerns mapping properties of the square functions under consideration.
Theorem 2.1.
Assume that α ∈ [ − / , ∞ ) d and W is a weight on R d invariant under the reflections σ , . . . , σ d . Then each of the square functions g V , g jH , g jH, ∗ , S V , S jH , S jH, ∗ , j = 1 , . . . , d, is bounded on L p ( R d , W dw α ) , W + ∈ A α, + p , < p < ∞ , and from L ( R d , W dw α ) to weak L ( R d , W dw α ) , W + ∈ A α, +1 . Proving Theorem 2.1 can be reduced to showing similar mapping properties for certain square functionsemerging from those defined above and related to the restricted space ( R d + , dw + α ); recall that w + α is therestriction of w α to R d + . The details are as follows. For ε ∈ Z d , we consider the operators acting on L ( R d + , dw + α ) and defined by T α,ε, + t f = ∞ X n =0 e − tλ αn X | m | = nm ∈N ε h f, h αm i dw + α h αm , f ∈ L ( R d + , dw + α ) . The integral representation of T α,ε, + t is T α,ε, + t f ( x ) = Z R d + G α,εt ( x, y ) f ( y ) dw + α ( y ) , x ∈ R d + , t > . QUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS 5
The estimates | h αm ( x ) | . ( | m | + 1) c d,α , m ∈ N d , x ∈ R d + , (2.3) |h f, h αm i dw + α | . (cid:0) | m | + 1 (cid:1) c d,α,p k f k L p ( R d + ,Udw + α ) , m ∈ N d , (2.4)which hold for general f ∈ L p ( R d + , U dw + α ), U ∈ A α, + p , 1 ≤ p < ∞ , allow to check that for each ε ∈ Z d the series defining T α,ε, + t converges pointwise for such f and produces a smooth function of ( t, x ) ∈ (0 , ∞ ) × R d + . An analogous claim is true for the integral representation. The bound (2.3) is a consequenceof Muckenhoupt’s generalization [6] of the classical estimates for the standard Laguerre functions due toAskey and Wainger [1]. Actually, those estimates imply a sharper version of (2.3) that involves someexponential decay in x , which together with the arguments from the proof of [7, Lemma 4.2] justifies(2.4).Next, we define the Laguerre-type square functions g ε, + V ( f )( x ) = (cid:13)(cid:13) ∂ t T α,ε, + t f ( x ) (cid:13)(cid:13) L ( tdt ) ,g j,ε, + H ( f )( x ) = (cid:13)(cid:13) δ j T α,ε, + t f ( x ) (cid:13)(cid:13) L ( dt ) , j = 1 , . . . , d,g j,ε, + H, ∗ ( f )( x ) = (cid:13)(cid:13) δ ∗ j T α,ε, + t f ( x ) (cid:13)(cid:13) L ( dt ) , j = 1 , . . . , d,S ε, + V ( f )( x ) = (cid:16) R A ( x ) t (cid:12)(cid:12) ∂ t T α,ε, + t f ( z ) (cid:12)(cid:12) χ { z ∈ R d + } dw + α ( z ) V α, + √ t ( x ) dt (cid:17) / ,S j,ε, + H ( f )( x ) = (cid:16) R A ( x ) (cid:12)(cid:12) δ j T α,ε, + t f ( z ) (cid:12)(cid:12) χ { z ∈ R d + } dw + α ( z ) V α, + √ t ( x ) dt (cid:17) / , j = 1 , . . . , d,S j,ε, + H, ∗ ( f )( x ) = (cid:16) R A ( x ) (cid:12)(cid:12) δ ∗ j T α,ε, + t f ( z ) (cid:12)(cid:12) χ { z ∈ R d + } dw + α ( z ) V α, + √ t ( x ) dt (cid:17) / , j = 1 , . . . , d. Here V α, + t ( x ) denotes the w + α measure of the cube centered at x and with side lengths 2 t , restricted to R d + . More precisely, V α, + t ( x ) = d Y j =1 V α j , + t ( x j ) , x ∈ R d + , t > , (2.5)and for j = 1 , . . . , d , V α j , + t ( x j ) = w + α j (cid:0) ( x j − t, x j + t ) ∩ R + (cid:1) = ( x j + t ) αj +2 α j +2 , x j < t ( x j + t ) αj +2 − ( x j − t ) αj +2 α j +2 , x j ≥ t . Notice that V α, + t ( x ) ≃ t d d Y j =1 ( x j + t ) α j +1 , x ∈ R d + , t > . (2.6)Observe also that the Laguerre-type Lusin’s area integrals can be written as S ε, + V ( f )( x ) = (cid:13)(cid:13) ∂ t T α,ε, + t f ( x + z ) p ϕ α ( x, z, t ) χ { x + z ∈ R d + } (cid:13)(cid:13) L ( A,tdtdz ) ,S j,ε, + H ( f )( x ) = (cid:13)(cid:13) δ j T α,ε, + t f ( x + z ) p ϕ α ( x, z, t ) χ { x + z ∈ R d + } (cid:13)(cid:13) L ( A,dtdz ) , j = 1 , . . . , d,S j,ε, + H, ∗ ( f )( x ) = (cid:13)(cid:13) δ ∗ j T α,ε, + t f ( x + z ) p ϕ α ( x, z, t ) χ { x + z ∈ R d + } (cid:13)(cid:13) L ( A,dtdz ) , j = 1 , . . . , d, where the function ϕ α is given by(2.7) ϕ α ( x, z, t ) = d Y j =1 ( x j + z j ) α j +1 V α j , + √ t ( x j ) , x ∈ R d + , z ∈ R d , x + z ∈ R d + . We are now in a position to reduce the proof of Theorem 2.1 to showing the following.
Theorem 2.2.
Assume that α ∈ [ − / , ∞ ) d and ε ∈ Z d . Then each of the Laguerre-type square functions g ε, + V , g j,ε, + H , g j,ε, + H, ∗ , S ε, + V , S j,ε, + H , S j,ε, + H, ∗ , j = 1 , . . . , d, is bounded on L p ( R d + , U dw + α ) , U ∈ A α, + p , < p < ∞ , and from L ( R d + , U dw + α ) to weak L ( R d + , U dw + α ) , U ∈ A α, +1 . SQUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS
For the sake of brevity, we give a detailed description of the reduction only in the case of S jH , adaptingsuitably arguments from the proof of [10, Theorem 1]. The remaining cases are treated in a similar wayand the cases of g V , g jH and g jH, ∗ are even simpler. In what follows, we shall use the following terminology.Given ε ∈ Z d , we say that a function f : R d → C is ε -symmetric if for each j = 1 , . . . , d , f is either evenor odd with respect to the j th coordinate according to whether ε j = 0 or ε j = 1, respectively. If f is(0 , . . . , f is symmetric. Furthermore, if there exists ε ∈ Z d suchthat f is ε -symmetric, then we denote by f + the restriction of f to R d + . This convention pertains also to ε -symmetric weights defined on R d .Let j ∈ { , . . . , d } and 1 ≤ p < ∞ be fixed, and let W be a symmetric weight on R d such that W + ∈ A α, + p . According to (2.2), we decompose δ j T αt into a finite sum, δ j T αt f = X ε ∈ Z d δ j T α,εt f. Next, we invoke the differentiation rule (see [9, (4.4)]) δ j h αm = Φ( m j , α j ) h αm − e j , where Φ( m j , α j ) = (cid:26) p m j if m j is even p m j + 4 α j + 2 if m j is odd ;here and elsewhere we use the convention that h αm = 0 if m / ∈ N d . Then, in view of the estimates similarto (2.3) and (2.4), but adjusted to the space ( R d , dw α , | · | ), we may write δ j T α,εt f = ∞ X n =0 e − tλ αn X | m | = nm ∈N ε h f, h αm i dw α Φ( m j , α j ) h αm − e j , f ∈ L p ( R d , W dw α ) . Proceeding as in [10, Section 3], we split a function f ∈ L p ( R d , W dw α ) into a sum of ε -symmetricfunctions f ε , f = X ε ∈ Z d f ε , f ε ( x ) = 12 d X η ∈{− , } d η ε f ( ηx ) , where η ε = η ε · . . . · η ε d d and ηx = ( η x , . . . , η d x d ). Since h αm is ε -symmetric if and only if m ∈ N ε , wesee that δ j T αt f = X ε ∈ Z d δ j T α,εt f = X ε ∈ Z d δ j T α,εt f ε , (2.8)and the function δ j T α,εt f ε is ( ε ± e j )-symmetric, depending on whether ε j = 0 or ε j = 1.Consider the auxiliary square functions S j,εH , ε ∈ Z d , acting on functions on R d and defined by S j,εH h ( x ) = (cid:18) Z A (cid:12)(cid:12) δ j T α,εt h ( x + z ) (cid:12)(cid:12) w α ( x + z ) V α √ t ( x ) dz dt (cid:19) / . Since | δ j T α,εt f ε | and w α are symmetric, and A is a symmetric set, it follows that S j,εH f ε is also symmetric.Moreover, by (2.8) we see that S jH ( f )( x ) ≤ X ε ∈ Z d S j,εH f ε ( x ) . Now, by the inclusions (cid:8) z ∈ R d : | z − x | < √ t (cid:9) ⊂ (cid:16) [ η ∈ Z d (cid:8) z ∈ R η : | z − σ η ( x ) | < √ t (cid:9)(cid:17) ∪ M, x ∈ R d + , t > , where M = (cid:8) z ∈ R d : there exists i ∈ { , . . . , d } such that z i = 0 (cid:9) , R η = (cid:8) z ∈ R d : z i > η i = 0 , z i < η i = 1 , i = 1 , . . . , d (cid:9) , QUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS 7 and σ η = σ η ◦ . . . ◦ σ η d d , we get for any x ∈ R d + , (cid:0) S j,εH f ε ( x ) (cid:1) = Z | z − x | < √ t (cid:12)(cid:12) δ j T α,εt f ε ( z ) (cid:12)(cid:12) w α ( z ) V α √ t ( x ) dz dt ≤ X η ∈ Z d Z | z − σ η ( x ) | < √ t (cid:12)(cid:12) δ j T α,εt f ε ( z ) (cid:12)(cid:12) w α ( z ) V α √ t ( x ) χ { z ∈ R η } dz dt, since M has the Lebesgue measure 0. Then the change of variable z σ η ( z ) reveals that (cid:0) S j,εH f ε ( x ) (cid:1) ≤ d Z | z − x | < √ t (cid:12)(cid:12) δ j T α,εt f ε ( z ) (cid:12)(cid:12) w α ( z ) V α √ t ( x ) χ { z ∈ R d + } dz dt. Thus, in view of the above estimates, the inequality V α, + √ t ( x ) ≤ V α √ t ( x ) and the fact that for each m ∈ N ε we have h f ε , h αm i dw α = 2 d h f + ε , h αm i dw + α and consequently δ j T α,εt f ε = 2 d δ j T α,ε, + t ( f + ε ) on R d + , we get S j,εH f ε ( x ) ≤ d/ S j,ε, + H ( f + ε )( x ) , x ∈ R d + . Taking into account the symmetry of S j,εH f ε and W dw α , we obtain k S jH ( f ) k L p ( R d ,W dw α ) ≤ d/p X ε ∈ Z d kS j,εH f ε k L p ( R d + ,W + dw + α ) . X ε ∈ Z d k S j,ε, + H ( f + ε ) k L p ( R d + ,W + dw + α ) and similarly Z { x ∈ R d : S jH ( f )( x ) >λ } W ( y ) dw α ( y ) ≤ d X ε ∈ Z d Z { x ∈ R d + : S j,ε, + H ( f + ε )( x ) > − d/ λ } W + ( y ) dw + α ( y ) , λ > . Since we have (see [10, p. 6] for the unweighted case) k f k L p ( R d ,W dw α ) ≃ X ε ∈ Z d k f + ε k L p ( R d + ,W + dw + α ) , this shows that the estimates k S j,ε, + H ( f + ε ) k L p ( R d + ,W + dw + α ) . k f + ε k L p ( R d + ,W + dw + α ) , ε ∈ Z d , imply the estimate k S jH ( f ) k L p ( R d ,W dw α ) . k f k L p ( R d ,W dw α ) . and an analogous implication is true for the weighted weak type (1 ,
1) inequalities.Thus we reduced proving Theorem 2.1 to showing Theorem 2.2. The proof of the latter result isbased on the general Calder´on-Zygmund theory. Clearly, the square functions are not linear, but in thewell-known way they can be viewed as vector-valued linear operators, see [15, Section 2]. In fact, wewill show that each of the square functions from Theorem 2.2, viewed as a vector-valued operator, isa Calder´on-Zygmund operator in the sense of the space of homogeneous type ( R d + , dw + α , | · | ). We shallneed a slightly more general version of the Calder´on-Zygmund theory than the one used in [15]. Moreprecisely, here we allow weaker smoothness estimates as indicated below.Let B be a Banach space and K ( x, y ) be a kernel defined on R d + × R d + \{ ( x, y ) : x = y } and takingvalues in B . We say that K ( x, y ) is a standard kernel in the sense of the space of homogeneous type( R d + , dw + α , | · | ) if it satisfies the growth estimate k K ( x, y ) k B . w + α ( B ( x, | y − x | ))(2.9)and the smoothness estimates k K ( x, y ) − K ( x ′ , y ) k B . (cid:18) | x − x ′ || x − y | (cid:19) δ w + α ( B ( x, | y − x | )) , | x − y | > | x − x ′ | , (2.10) k K ( x, y ) − K ( x, y ′ ) k B . (cid:18) | y − y ′ || x − y | (cid:19) δ w + α ( B ( x, | y − x | )) , | x − y | > | y − y ′ | , (2.11) SQUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS for some fixed δ >
0; here B ( x, r ) denotes the ball centered at x and with radius r , restricted to R d + .Notice that the bounds (2.10) and (2.11) imply analogous estimates with any 0 < δ ′ < δ replacing δ > T assigning to each f ∈ L ( R d + , dw + α ) a measurable B -valued function T f on R d + isa (vector-valued) Calder´on-Zygmund operator in the sense of the space ( R d + , dw + α , | · | ) if(i) T is bounded from L ( R d + , dw + α ) to L B ( R d + , dw + α ),(ii) there exists a standard B -valued kernel K ( x, y ) such that T f ( x ) = Z R d + K ( x, y ) f ( y ) dw + α ( y ) , a.e. x / ∈ supp f, for every f ∈ L ( R d + , dw + α ) vanishing outside a compact set contained in R d + (we write shortly T ∼ K ( x, y ) for this kind of association).Here integration of B -valued functions is understood in Bochner’s sense, and L B is the Bochner-Lebesguespace of all B -valued dw + α -square integrable functions on R d + . It is well known that a large part of theclassical theory of Calder´on-Zygmund operators remains valid, with appropriate adjustments, when theunderlying space is of homogeneous type and the associated kernels are vector-valued, see the commentsin [8, p. 649] and references given there.The following result, combined with the general theory of Calder´on-Zygmund operators and argumentssimilar to those from the proof of [15, Corollary 2.5], implies Theorem 2.2, and thus also Theorem 2.1 bythe reduction reasoning described above. Theorem 2.3.
Assume that α ∈ [ − / , ∞ ) d and ε ∈ Z d . Then each of the square functions g ε, + V , g j,ε, + H , g j,ε, + H, ∗ , S ε, + V , S j,ε, + H , S j,ε, + H, ∗ , j = 1 , . . . , d, viewed as a vector-valued operator related to either B = L ( tdt ) (the case of g ε, + V ), or B = L ( dt ) (thecases of g j,ε, + H and g j,ε, + H, ∗ ), or B = L ( A, tdtdz ) (the case of S ε, + V ), or B = L ( A, dtdz ) (the cases of S j,ε, + H and S j,ε, + H, ∗ ), is a Calder´on-Zygmund operator in the sense of the space of homogeneous type ( R d + , dw + α , |·| ) . The proof of Theorem 2.3 splits naturally into proving the following three results. Showing them willcomplete the whole reasoning justifying Theorem 2.1.
Proposition 2.4.
Let α ∈ [ − / , ∞ ) d and ε ∈ Z d . Then the square functions from Theorem 2.3 arebounded on L ( R d + , dw + α ) . Consequently, each of them, viewed as a vector-valued operator, is boundedfrom L ( R d + , dw + α ) to L B ( R d + , dw + α ) , where B is as in Theorem 2.3. Formal computations suggest that S ε, + V , S j,ε, + H , S j,ε, + H, ∗ are associated with the kernels K α,ε,Vz,t ( x, y ) = ∂ t (cid:0) G α,εt ( x + z, y ) (cid:1)p ϕ α ( x, z, t ) χ { x + z ∈ R d + } , (2.12) K α,ε,H,jz,t ( x, y ) = δ j,x (cid:0) G α,εt ( x + z, y ) (cid:1)p ϕ α ( x, z, t ) χ { x + z ∈ R d + } , j = 1 , . . . , d,K α,ε,H, ∗ ,jz,t ( x, y ) = δ ∗ j,x (cid:0) G α,εt ( x + z, y ) (cid:1)p ϕ α ( x, z, t ) χ { x + z ∈ R d + } , j = 1 , . . . , d, respectively. A part of the next result shows that this is indeed true, at least in the Calder´on-Zygmundtheory sense. Proposition 2.5.
Let α ∈ [ − / , ∞ ) d and ε ∈ Z d . Then the square functions from Theorem 2.3, viewedas vector-valued linear operators related to B as in Theorem 2.3, are associated with the following kernels: g ε, + V ∼ (cid:8) ∂ t G α,εt ( x, y ) (cid:9) t> , S ε, + V ∼ (cid:8) K α,ε,Vz,t ( x, y ) (cid:9) ( z,t ) ∈ A ,g j,ε, + H ∼ (cid:8) δ j,x G α,εt ( x, y ) (cid:9) t> , S j,ε, + H ∼ (cid:8) K α,ε,H,jz,t ( x, y ) (cid:9) ( z,t ) ∈ A , j = 1 , . . . , d,g j,ε, + H, ∗ ∼ (cid:8) δ ∗ j,x G α,εt ( x, y ) (cid:9) t> , S j,ε, + H, ∗ ∼ (cid:8) K α,ε,H, ∗ ,jz,t ( x, y ) (cid:9) ( z,t ) ∈ A , j = 1 , . . . , d. Theorem 2.6.
Assume that α ∈ [ − / , ∞ ) d and ε ∈ Z d . Let K ( x, y ) be any of the vector-valued kernelslisted in Proposition 2.5. Then K ( x, y ) satisfies the standard estimates (2.9) , (2.10) and (2.11) with therelevant space B and either δ = 1 in the cases of g -functions, or δ = 1 / in the cases of area integrals. The proofs of Propositions 2.4 and 2.5 are given in Section 3 (in fact we show somewhat stronger resultthan Proposition 2.4). The proof of Theorem 2.6 is the most technical and tricky part of the paper andis located in Section 4.
QUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS 9
We conclude this section with various comments and remarks related to the main result. First, we notethat our results imply analogous results for g -functions emerging from the Poisson semigroup related tothe Dunkl harmonic oscillator. To be more precise, consider the semigroup { P αt } t> generated by √L α , P αt f = e − t √L α f = ∞ X n =0 e − t √ λ αn X | m | = n h f, h αm i dw α h αm , and the auxiliary operators P α,ε, + t f = ∞ X n =0 e − t √ λ αn X | m | = nm ∈N ε h f, h αm i dw + α h αm , ε ∈ Z d . Clearly, by the subordination principle, P αt f ( x ) = Z ∞ T αt / (4 u ) f ( x ) e − u du √ πu , P α,ε, + t f ( x ) = Z ∞ T α,ε, + t / (4 u ) f ( x ) e − u du √ πu . (2.13)We consider the following g -functions: g V,P ( f )( x ) = (cid:13)(cid:13) ∂ t P αt f ( x ) (cid:13)(cid:13) L ( tdt ) , g ε, + V,P ( f )( x ) = (cid:13)(cid:13) ∂ t P α,ε, + t f ( x ) (cid:13)(cid:13) L ( tdt ) ,g jH,P ( f )( x ) = (cid:13)(cid:13) δ j P αt f ( x ) (cid:13)(cid:13) L ( tdt ) , g j,ε, + H,P ( f )( x ) = (cid:13)(cid:13) δ j P α,ε, + t f ( x ) (cid:13)(cid:13) L ( tdt ) , j = 1 , . . . , d,g jH, ∗ ,P ( f )( x ) = (cid:13)(cid:13) δ ∗ j P αt f ( x ) (cid:13)(cid:13) L ( tdt ) , g j,ε, + H, ∗ ,P ( f )( x ) = (cid:13)(cid:13) δ ∗ j P α,ε, + t f ( x ) (cid:13)(cid:13) L ( tdt ) , j = 1 , . . . , d. The result below is a consequence of (2.13) and Theorems 2.1 and 2.2.
Theorem 2.7.
Assume that α ∈ [ − / , ∞ ) d and W is a weight on R d invariant under the reflections σ , . . . , σ d . Then each of the g -functions g V,P , g jH,P , g jH, ∗ ,P , j = 1 , . . . , d, is bounded on L p ( R d , W dw α ) , W + ∈ A α, + p , < p < ∞ , and from L ( R d , W dw α ) to weak L ( R d , W dw α ) , W + ∈ A α, +1 . Furthermore, the Laguerre-type square functions g ε, + V,P , g j,ε, + H,P , g j,ε, + H, ∗ ,P , j = 1 , . . . , d, ε ∈ Z d , are bounded on L p ( R d + , U dw + α ) , U ∈ A α, + p , < p < ∞ , and from L ( R d + , U dw + α ) to weak L ( R d + , U dw + α ) , U ∈ A α, +1 . Treatment of Lusin’s area integrals associated to the Poisson semigroup is more subtle. In particular,one cannot apply the arguments from [2, Section 2] since in the present situation the function V αt ( x )depends not only on t , but also on x .Next, we note that for the particular α = ( − / , . . . , − /
2) the generalized Hermite functions becomethe classic Hermite functions and L α is the Euclidean harmonic oscillator. Thus Theorem 2.1 provides,in particular, results in the Hermite setting for which certain square functions were studied earlier. Tobe more precise, the vertical g -function g V was considered by Thangavelu [16, Chapter 4] to prove theMarcinkiewicz multiplier theorem for Hermite function expansions. The Poisson semigroup based g -functions g V,P , g jH,P , g jH, ∗ ,P , j = 1 , . . . , d , were studied by Harboure, de Rosa, Segovia and Torrea [4],in connection with Riesz transforms associated to the Hermite setting. All the abovementioned squarefunctions were reinvestigated later by Stempak and Torrea [14]. Lusin’s area integrals for Hermite functionexpansions were studied recently, in the one-dimensional case, by Betancor, Molina and Rodr´ıguez-Mesa[2]. The area integral g W there coincides, up to a multiplicative constant, with our area integral S V withslightly modified aperture of the parabolic cone A (see also Remark 2.10 below).We now focus on the relation between the Laguerre-type square functions studied in this paper andthe Laguerre setting from [15]. We note that for the particular ε = (0 , . . . , T α,ε , + t , t >
0, coincides, up to the factor 2 − d , with the Laguerre semigroup T αt considered in [15]. Moreover, for ε = e j , j = 1 , . . . , d , the operators T α,e j , + t are related to the modified Laguerre semigroups e T α,jt (see [15,Section 2] for the definition) by e T α,jt = 2 d e − t T α,e j , + t . (2.14) Therefore many results of [15] can be seen as special cases of Theorem 2.3. More precisely, these obser-vations, or rather analogous observations concerning the integral kernels of the semigroups in question,combined with Theorem 2.3 show that the g -functions g V,T , g iH,T , g j,iH, e T , i, j = 1 , . . . , d , investigated in[15] can be viewed as vector-valued Calder´on-Zygmund operators. The fact that g jV, e T , j = 1 , . . . , d , from[15] may be interpreted as vector-valued Calder´on-Zygmund operators can be, in principle, also recoveredfrom the results and reasonings of this paper; this, however, is less explicit because of the factor e − t in(2.14), which does not affect the horizontal g -functions.Further, we define Lusin’s area type integrals in the Laguerre function setting of convolution type;such operators were not considered in [15]. We adopt the notation from [15], but to avoid a confusion,here we denote the Laguerre heat semigroup by T αt . Let S V, T ( f )( x ) = (cid:16) R A ( x ) t (cid:12)(cid:12) ∂ t T αt f ( z ) (cid:12)(cid:12) χ { z ∈ R d + } dµ α ( z ) V α, + √ t ( x ) dt (cid:17) / ,S jH, T ( f )( x ) = (cid:16) R A ( x ) (cid:12)(cid:12) δ j T αt f ( z ) (cid:12)(cid:12) χ { z ∈ R d + } dµ α ( z ) V α, + √ t ( x ) dt (cid:17) / , j = 1 , . . . , d,S j,iH, e T ( f )( x ) = (cid:16) R A ( x ) (cid:12)(cid:12) δ i e T α,jt f ( z ) (cid:12)(cid:12) χ { z ∈ R d + } dµ α ( z ) V α, + √ t ( x ) dt (cid:17) / , i, j = 1 , . . . , d, i = j,S j,jH, e T ( f )( x ) = (cid:16) R A ( x ) (cid:12)(cid:12) δ ∗ j e T α,jt f ( z ) (cid:12)(cid:12) χ { z ∈ R d + } dµ α ( z ) V α, + √ t ( x ) dt (cid:17) / , j = 1 , . . . , d, where V α, + t ( x ) is defined by (2.5), because dµ α ≡ dw + α . Thus Theorems 2.2 and 2.3 provide, in particular,the following result for the Laguerre area integrals. Theorem 2.8.
Assume that α ∈ [ − / , ∞ ) d . Then each of the Lusin’s area type integrals S V, T , S jH, T , S j,iH, e T , i, j = 1 , . . . , d, viewed as a vector-valued operator related to either B = L ( A, tdtdz ) (the case of S V, T ), or B = L ( A, dtdz ) (the cases of S jH, T and S j,iH, e T ), is a Calder´on-Zygmund operator in the sense of the space of homogeneoustype ( R d + , dµ α , | · | ) . Consequently, these square functions are bounded on L p ( R d + , U dµ α ) , U ∈ A α, + p , < p < ∞ , and from L ( R d + , U dµ α ) to weak L ( R d + , U dµ α ) , U ∈ A α, +1 . Remark 2.9.
Theorem 2.1, Theorem 2.2, the first identity of Proposition 3.1 and the analogous equalitiesfor g V , g ε, + V,P and g V,P , together with standard arguments, see [15, Remark 2.6] , allow to show also lowerweighted L p estimates for the vertical g -functions under consideration. With the assumption α ∈ [ − / , ∞ ) d , for ε ∈ Z d and U ∈ A α, + p , < p < ∞ , we have k f k L p ( R d + ,Udw + α ) . k g ε, + V ( f ) k L p ( R d + ,Udw + α ) , f ∈ L p ( R d + , U dw + α ) , k f k L p ( R d + ,Udw + α ) . k g ε, + V,P ( f ) k L p ( R d + ,Udw + α ) , f ∈ L p ( R d + , U dw + α ) . Consequently, if W is a symmetric weight on R d , W + ∈ A α, + p , < p < ∞ , we also have k f k L p ( R d ,W dw α ) . k g V ( f ) k L p ( R d ,W dw α ) , f ∈ L p ( R d , W dw α ) , k f k L p ( R d ,W dw α ) . k g V,P ( f ) k L p ( R d ,W dw α ) , f ∈ L p ( R d , W dw α ) . Remark 2.10.
The exact aperture of the parabolic cone A is not essential for our developments. Indeed,if we fix β > and write A β = n ( z, t ) ∈ R d × (0 , ∞ ) : | z | < β √ t o instead of A in the definitions ofLusin’s area type integrals, then the results of this paper, and in particular Theorem 2.1, remain valid. L -Boundedness and Kernel associations In this section we check that the Laguerre-type square functions under consideration are boundedon the Hilbert space L ( R d + , dw + α ). We also show that these square functions, viewed as vector-valuedoperators, are associated with the relevant kernels.The following result is essentially a slight generalization of [15, Proposition 3.1] and [15, Proposition3.2]. The proof is nearly identical and thus is omitted. A crucial fact needed in the proof is that for each ε ∈ Z d the system { d/ h αm : m ∈ N ε } is an orthonormal basis in L ( R d + , dw + α ). QUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS 11
Proposition 3.1.
Assume that α ∈ [ − / , ∞ ) d and ε ∈ Z d . Then k g ε, + V ( f ) k L ( R d + ,dw + α ) = 2 − d − k f k L ( R d + ,dw + α ) , f ∈ L ( R d + , dw + α ) , (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:0) g ,ε, + H ( f ) , . . . , g d,ε, + H ( f ) (cid:1)(cid:12)(cid:12) ℓ (cid:13)(cid:13)(cid:13) L ( R d + ,dw + α ) . k f k L ( R d + ,dw + α ) , f ∈ L ( R d + , dw + α ) , (3.1) (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:0) g ,ε, + H, ∗ ( f ) , . . . , g d,ε, + H, ∗ ( f ) (cid:1)(cid:12)(cid:12) ℓ (cid:13)(cid:13)(cid:13) L ( R d + ,dw + α ) ≃ k f k L ( R d + ,dw + α ) , f ∈ L ( R d + , dw + α ) . Moreover, if ε = (0 , . . . , , then the relation ” . ” in (3.1) can be replaced by ” ≃ ”. The same is true for ε = (0 , . . . , provided that f is taken from the subspace { h α (0 ,..., } ⊥ ⊂ L ( R d + , dw + α ) . Proposition 3.2.
Assume that α ∈ [ − / , ∞ ) d and ε ∈ Z d . Then k S ε, + V ( f ) k L ( R d + ,dw + α ) ≃ k f k L ( R d + ,dw + α ) , f ∈ L ( R d + , dw + α ) , (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:0) S ,ε, + H ( f ) , . . . , S d,ε, + H ( f ) (cid:1)(cid:12)(cid:12) ℓ (cid:13)(cid:13)(cid:13) L ( R d + ,dw + α ) . k f k L ( R d + ,dw + α ) , f ∈ L ( R d + , dw + α ) , (3.2) (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:0) S ,ε, + H, ∗ ( f ) , . . . , S d,ε, + H, ∗ ( f ) (cid:1)(cid:12)(cid:12) ℓ (cid:13)(cid:13)(cid:13) L ( R d + ,dw + α ) ≃ k f k L ( R d + ,dw + α ) , f ∈ L ( R d + , dw + α ) . Moreover, if ε = (0 , . . . , , then the relation ” . ” in (3.2) can be replaced by ” ≃ ”. The same is true for ε = (0 , . . . , provided that f is taken from the subspace { h α (0 ,..., } ⊥ ⊂ L ( R d + , dw + α ) .Proof. We give a justification only for the first relation. The remaining cases, being similar, are left tothe reader. Using the Fubini-Tonelli theorem, the estimate (2.6) of V α, + √ t ( x ) and then the inequalities Z ∞ χ {| x j − z j | < √ t } x α j +1 j √ t ( x j + √ t ) α j +1 dx j ≤ , which is legitimate since the integrand is dominated by t − / , and Z ∞ χ {| x j − z j | < √ t } x α j +1 j √ t ( x j + √ t ) α j +1 dx j ≥ t − / Z z j + √ tz j + √ t/ (cid:18) x j x j + √ t (cid:19) α j +1 dx j ≥
12 3 − α j − , which holds because the function x j (cid:16) x j x j + √ t (cid:17) α j +1 is increasing for x j >
0, we obtain k S ε, + V ( f ) k L ( R d + ,dw + α ) ≃ Z R d + Z ∞ t | ∂ t T α,ε, + t f ( z ) | dt dw + α ( z ) = k g ε, + V ( f ) k L ( R d + ,dw + α ) . Now the conclusion follows from the first identity of Proposition 3.1. (cid:3)
Proposition 3.1 together with Proposition 3.2 imply Proposition 2.4.Next we prove that each of the Laguerre-type square functions under consideration, viewed as a vector-valued linear operator, is indeed associated with the relevant kernel in the sense of the Calder´on-Zygmundtheory. We adapt essentially the reasoning given in the proof of [15, Proposition 2.3], see also commentsand references given there.
Proof of Proposition 2.5.
A careful repetition of the arguments given in the proof of [15, Proposition2.3], see also [14, Section 2], leads to the desired conclusions for the g -functions g ε, + V , g j,ε, + H and g j,ε, + H, ∗ ,since we have a suitable estimate for the generalized Hermite functions, see (2.3), and the relevant kernelestimates, see Theorem 2.6.Treatment of the area integrals S ε, + V , S j,ε, + H and S j,ε, + H, ∗ , is slightly different, but relies on similararguments. Hence we give the details only in the case of S ε, + V , leaving the remaining cases to the reader.Let B = L ( A, tdtdz ). Proceeding as in the proof of [15, Proposition 2.3] one reduces the task to checkingthat Dn ∂ t T α,ε, + t f ( x + z ) χ { x + z ∈ R d + } p ϕ α ( x, z, t ) o ( z,t ) ∈ A , r E L B ( R d + ,dw + α ) = (cid:28) Z R d + (cid:8) K α,ε,Vz,t ( x, y ) (cid:9) ( z,t ) ∈ A f ( y ) dw + α ( y ) , r (cid:29) L B ( R d + ,dw + α ) (3.3) for every f ∈ C ∞ c ( R d + ) and r ( x, z, t ) = r ( x ) r ( z, t ), where r ∈ C ∞ c ( R d + ), r ∈ C ∞ c ( A ) and supp f ∩ supp r = ∅ . We first deal with the left-hand side of (3.3), Dn ∂ t T α,ε, + t f ( x + z ) χ { x + z ∈ R d + } p ϕ α ( x, z, t ) o ( z,t ) ∈ A , r E L B ( R d + ,dw + α ) = Z A tr ( z, t ) Z R d + ∂ t T α,ε, + t f ( x + z ) χ { x + z ∈ R d + } p ϕ α ( x, z, t ) r ( x ) dw + α ( x ) dt dz = − Z A tr ( z, t ) Z R d + (cid:18) ∞ X n =0 λ αn e − tλ αn X | m | = nm ∈N ε h f, h αm i dw + α h αm ( x + z ) (cid:19) χ { x + z ∈ R d + } × p ϕ α ( x, z, t ) r ( x ) dw + α ( x ) dt dz. The first identity above follows by Fubini’s theorem; the possibility of its application can be justified withthe aid of the boundedness of S ε, + V on L ( R d + , dw + α ). The second equality is obtained by exchanging theorder of ∂ t and P , which is legitimate in view of (2.3).Now we focus on the right-hand side of (3.3). Changing the order of integrals, which is justified by thegrowth condition for the kernel (cid:8) K α,ε,Vz,t ( x, y ) (cid:9) , see Theorem 2.6, and using the fact that the supports of f and r are disjoint and compact, we see that the expression in question is equal Z A tr ( z, t ) Z R d + Z R d + K α,ε,Vz,t ( x, y ) f ( y ) r ( x ) dw + α ( y ) dw + α ( x ) dt dz. Then expressing K α,ε,Vz,t by means of the series and then using Fubini’s theorem, whose application islegitimate in view of (2.3), we get Z R d + K α,ε,Vz,t ( x, y ) f ( y ) dw + α ( y )= − Z R d + (cid:18) ∞ X n =0 λ αn e − tλ αn X | m | = nm ∈N ε h αm ( x + z ) h αm ( y ) (cid:19) χ { x + z ∈ R d + } p ϕ α ( x, z, t ) f ( y ) dw + α ( y )= − ∞ X n =0 λ αn e − tλ αn X | m | = nm ∈N ε h f, h αm i dw + α h αm ( x + z ) χ { x + z ∈ R d + } p ϕ α ( x, z, t ) . Integrating the last identity against tr ( x ) r ( z, t ) dw + α ( x ) dt dz , we see that both sides of (3.3) coincide. (cid:3) Kernel estimates
This section is devoted to the proofs of the relevant kernel estimates for all the considered squarefunctions. We generalize the arguments developed in [8, 9], which are based on Schl¨afli’s integral repre-sentation for the modified Bessel function I ν involved in the Dunkl heat kernel. This method was usedalso by the author in [15] to obtain the standard estimates for the kernel G α,εt ( x, y ) in the extreme casewhen ε = (0 , . . . , α ∈ [ − / , ∞ ) d .Given ε ∈ Z d , the ε -component of the Dunkl heat kernel is given by, see [9, Section 5],(4.1) G α,εt ( x, y ) = 12 d (cid:16) − ζ ζ (cid:17) d + | α | + | ε | ( xy ) ε Z [ − , d exp (cid:16) − ζ q + ( x, y, s ) − ζ q − ( x, y, s ) (cid:17) Π α + ε ( ds ) , where ( xy ) ε = ( x y ) ε · . . . · ( x d y d ) ε d , q ± ( x, y, s ) = | x | + | y | ± d X i =1 x i y i s i , and t > ζ ∈ (0 ,
1) are related by ζ = tanh t ; equivalently(4.2) t = t ( ζ ) = 12 log 1 + ζ − ζ . QUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS 13
The measure Π β appearing in (4.1) is a product of one-dimensional measures, Π β = N di =1 Π β i , whereΠ β i is given by the density Π β i ( ds i ) = (1 − s i ) β i − / ds i √ π β i Γ( β i + 1 / , β i > − / , and in the limiting case of β i = − /
2, Π − / = (cid:0) η − + η (cid:1) / √ π , with η − and η denoting the pointmasses at − G α,εt ( x, y ) we will use several technical lemmas which are gatheredbelow. Some of them were obtained elsewhere, but we state them anyway for the sake of completenessand reader’s convenience.To begin with, notice that we have the asymptotics(4.3) log 1 + ζ − ζ ∼ ζ, ζ → + and log 1 + ζ − ζ ∼ − log(1 − ζ ) , ζ → − . The following result is a compilation of [15, Lemma 4.1, Lemma 4.2, Lemma 4.4].
Lemma 4.1.
Let b ≥ and c > be fixed. Then for any j = 1 , . . . , d, we have (a) | x j ± y j s j | ≤ p q ± ( x, y, s ) and | y j ± x j s j | ≤ p q ± ( x, y, s ) , (b) (cid:0) q ± ( x, y, s ) (cid:1) b exp (cid:0) − cAq ± ( x, y, s ) (cid:1) . A − b exp (cid:16) − cA q ± ( x, y, s ) (cid:17) , (c) Z ζ − log 1 + ζ − ζ exp (cid:16) − cζ q + ( x, y, s ) (cid:17) dζ . q + ( x, y, s ) , uniformly in x, y ∈ R d + , s ∈ [ − , d , and also in A > if (b) is considered. Lemma 4.2. ([13, Lemma 1.1])
Given a > , we have Z ζ − a exp( − T ζ − ) dζ . T − a +1 , T > . The following result is a slight extension of [15, Lemma 4.5], the proof being nearly identical.
Lemma 4.3. If x, y, z ∈ R d + are such that | x − y | > | x − z | , then q ± ( x, y, s ) ≤ q ± ( z, y, s ) ≤ q ± ( x, y, s ) , s ∈ [ − , d . The same holds after exchanging the roles of x and y . Lemma 4.4. ([9, Lemma 5.3] , [10, Lemma 4]) Assume that α ∈ [ − / , ∞ ) d and let δ, κ ∈ [0 , ∞ ) d befixed. Then for x, y ∈ R d + , x = y , ( x + y ) δ Z [ − , d (cid:0) q + ( x, y, s ) (cid:1) − d −| α |−| δ | Π α + δ + κ ( ds ) . w + α ( B ( x, | y − x | )) and ( x + y ) δ Z [ − , d (cid:0) q + ( x, y, s ) (cid:1) − d −| α |−| δ |− / Π α + δ + κ ( ds ) . | x − y | w + α ( B ( x, | y − x | )) . Lemma 4.5.
Let γ > be fixed. On the set { ( x, y, z ) ∈ R d + × R d + × R d + : | x − y | > | x − z |} we have (cid:18) | z − y | (cid:19) γ w + α ( B ( z, | z − y | )) ≃ (cid:18) | x − y | (cid:19) γ w + α ( B ( x, | y − x | )) . Proof.
Observe that12 | y − x | ≤ | y − x | − | x − z | ≤ | y − z | ≤ | y − x | + | x − z | ≤ | y − x | . Now the conclusion is an easy consequence of the doubling property of the measure w + α . (cid:3) To state the next lemma, and also to perform the relevant kernel estimates, we will use the sameabbreviations as in [15], L og( ζ ) = log 1 + ζ − ζ , E xp( ζ, q ± ) = exp (cid:16) − ζ q + ( x, y, s ) − ζ q − ( x, y, s ) (cid:17) . Furthermore, we will often neglect the set of integration [ − , d in integrals against Π α and write shortly q ± omitting the arguments. Lemma 4.6.
Assume that α ∈ [ − / , ∞ ) d and ξ, ρ, ε ∈ Z d are fixed and such that ξ ≤ ε , ρ ≤ ε . Given C > and u ∈ R , consider the function acting on R d + × R d + × (0 , and defined by p u ( x, y, ζ ) = p − ζ ζ − d −| α |−| ε | + | ξ | / | ρ | / − u/ x ε − ξ y ε − ρ Z [ − , d (cid:0) E xp ( ζ, q ± ) (cid:1) C Π α + ε ( ds ) . (a) If u ≥ , then we have the estimate k p u ( x, y, ζ ( t )) k L ( dt ) . | x − y | u − w + α ( B ( x, | y − x | )) , x = y, where t and ζ are related as in (4.2) . (b) If u ≥ , then we also have k p u ( x, y, ζ ( t )) k L ( tdt ) . | x − y | u − w + α ( B ( x, | y − x | )) , x = y. Proof.
We start with proving the first estimate. Changing the variable according to (4.2) and then usingsequently the Minkowski integral inequality, Lemma 4.1 (b) (applied with b = 2 d + 2 | α | + 2 | ε | − | ξ | − | ρ | + u − c = C/ A = ζ − ), Lemma 4.2 (with a = 2 and T = Cq + ) and the inequality | x − y | ≤ q + , weobtain k p u ( x, y, ζ ( t )) k L ( dt ) = x ε − ξ y ε − ρ (cid:18) Z (cid:16) ζ (cid:17) d +2 | α | +2 | ε |−| ξ |−| ρ | + u (cid:18) Z (cid:0) E xp( ζ, q ± ) (cid:1) C Π α + ε ( ds ) (cid:19) dζ (cid:19) / ≤ x ε − ξ y ε − ρ Z (cid:18) Z (cid:16) ζ (cid:17) d +2 | α | +2 | ε |−| ξ |−| ρ | + u (cid:0) E xp( ζ, q ± ) (cid:1) C dζ (cid:19) / Π α + ε ( ds ) . x ε − ξ y ε − ρ Z ( q + ) − d −| α |−| ε | + | ξ | / | ρ | / − u/ (cid:18) Z (cid:16) ζ (cid:17) (cid:0) E xp( ζ, q ± ) (cid:1) C dζ (cid:19) / Π α + ε ( ds ) . x ε − ξ y ε − ρ Z ( q + ) − d −| α |−| ε | + | ξ | / | ρ | / / − u/ Π α + ε ( ds ) . | x − y | u − ( x + y ) ε − ξ − ρ Z ( q + ) − d −| α |−| ε − ξ/ − ρ/ | Π α + ε ( ds ) . Now an application of Lemma 4.4 (with δ = ε − ξ/ − ρ/ κ = ξ/ ρ/
2) leads to the desiredconclusion.Similar arguments (using this time Lemma 4.1 (c) instead of Lemma 4.2) justify the second estimate. (cid:3)
Lemma 4.7. ([15, Lemma 4.7])
Let F : (0 , ∞ ) × [ − , d → R be a function such that F ( · , s ) is contin-uously differentiable for each fixed s , and F ( z, · ) ∈ L (Π α ( ds )) for any z > . Further, assume that foreach v > there exists a < v < b and a function f a,b ∈ L (Π α ( ds )) such that | ∂ z F ( z, s ) | ≤ f a,b ( s ) for all z ∈ [ a, b ] and s ∈ [ − , d . Then ∂ z Z [ − , d F ( z, s ) Π α ( ds ) = Z [ − , d ∂ z F ( z, s ) Π α ( ds ) , z > . In what follows it is convenient to use the following notation. Given x, y ∈ R d + , we write x ≤ y if x j ≤ y j for each j = 1 , . . . , d . We denote by max { x, y } the point in R d + having the coordinatesmax { x j , y j } , j = 1 , . . . , d , and similarly for min { x, y } . QUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS 15
Vertical g -function based on { T α,ε, + t } . Proof of Theorem 2.6; the case of g ε, + V . We first deal with the growth condition. Differentiating (4.1) in t (passing with ∂ t under the integral sign can be easily justified by Lemma 4.7, see [15, Section 4]) we get ∂ t G α,εt ( x, y ) = − d ( xy ) ε (cid:16) − ζ ζ (cid:17) d + | α | + | ε | h ( x, y, ζ ) , (4.4)where the auxiliary function h is given by h ( x, y, ζ ) =( d + | α | + | ε | ) 1 + ζ ζ Z E xp( ζ, q ± ) Π α + ε ( ds )+ 1 − ζ ζ Z E xp( ζ, q ± ) h ζ q − − ζ q + i Π α + ε ( ds ) . Notice that the function h depends on α and ε , but to shorten the notation we do not indicate thatexplicitly (a similar convention will concern other auxiliary functions appearing in the sequel).Using Lemma 4.1 (b) (first with b = 1, c = 1 / A = ζ and then with b = 1, c = 1 / A = ζ − ) weobtain | h ( x, y, ζ ) | . ζ − Z E xp( ζ, q ± ) Π α + ε + ζ − Z E xp( ζ, q ± ) h ζq − + q + ζ i Π α + ε ( ds ) . ζ − Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) . (4.5)This, in view of (4.4), gives | ∂ t G α,εt ( x, y ) | . p − ζ ζ − d −| α |−| ε |− ( xy ) ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) . (4.6)Finally, Lemma 4.6 (b) (specified to u = 2, ξ = ρ = 0) leads directly to the desired bound.We pass to proving the smoothness estimates. By symmetry reasons, it suffices to show that (cid:13)(cid:13) ∂ t G α,εt ( x, y ) − ∂ t G α,εt ( x ′ , y ) (cid:13)(cid:13) L ( tdt ) . | x − x ′ || x − y | w + α ( B ( x, | y − x | )) , | x − y | > | x − x ′ | . By the mean value theorem (cid:12)(cid:12) ∂ t G α,εt ( x, y ) − ∂ t G α,εt ( x ′ , y ) (cid:12)(cid:12) ≤ | x − x ′ | (cid:12)(cid:12) ∇ x ∂ t G α,εt ( θ, y ) (cid:12)(cid:12) , where θ is a convex combination of x and x ′ that depends also on t . Thus our task reduces to provingthat k ∂ x i ∂ t G α,εt ( θ, y ) k L ( tdt ) . | x − y | w + α ( B ( x, | y − x | )) , | x − y | > | x − x ′ | , for each i = 1 , . . . , d . To proceed we first analyze the derivative ∂ x i ∂ t G α,εt ( x, y ) = − d ( xy ) ε (cid:16) − ζ ζ (cid:17) d + | α | + | ε | ∂ x i h ( x, y, ζ ) − χ { ε i =1 } d x ε − e i y ε (cid:16) − ζ ζ (cid:17) d + | α | + | ε | h ( x, y, ζ ) . An elementary computation shows that ∂ x i h ( x, y, ζ ) = − ( d + | α | + | ε | ) 1 + ζ ζ Z E xp( ζ, q ± ) h ζ ( x i + y i s i ) + ζ x i − y i s i ) i Π α + ε ( ds ) − − ζ ζ Z E xp( ζ, q ± ) h ζ ( x i + y i s i ) + ζ x i − y i s i ) ih ζ q − − ζ q + i Π α + ε ( ds )+ 1 − ζ ζ Z E xp( ζ, q ± ) h ζ x i − y i s i ) − ζ ( x i + y i s i ) i Π α + ε ( ds ) . Applying Lemma 4.1 (a) and then repeatedly Lemma 4.1 (b) (specified to b = 1 / b = 1) we get | ∂ x i h ( x, y, ζ ) | . ζ − Z E xp( ζ, q ± ) h √ q + ζ + ζ √ q − i Π α + ε ( ds ) + ζ − Z E xp( ζ, q ± ) h √ q + ζ + ζ √ q − ih ζq − + q + ζ i Π α + ε ( ds ) . ζ − / Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) . Denote x ∗ = max { x, x ′ } and observe that θ ≤ x ∗ and also | x − x ∗ | ≤ | x − x ′ | . Then using the last estimateof ∂ x i h ( x, y, ζ ), (4.5) and then Lemma 4.3 (first with z = θ and then with z = x ∗ ) produces | ∂ x i ∂ t G α,εt ( θ, y ) | (4.7) . p − ζ ζ − d −| α |−| ε |− / ( θy ) ε Z (cid:0) E xp( ζ, q ± ( θ, y, s )) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } p − ζ ζ − d −| α |−| ε |− θ ε − e i y ε Z (cid:0) E xp( ζ, q ± ( θ, y, s )) (cid:1) / Π α + ε ( ds ) . p − ζ ζ − d −| α |−| ε |− / ( x ∗ y ) ε Z (cid:0) E xp( ζ, q ± ( x ∗ , y, s )) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } p − ζ ζ − d −| α |−| ε |− ( x ∗ ) ε − e i y ε Z (cid:0) E xp( ζ, q ± ( x ∗ , y, s )) (cid:1) / Π α + ε ( ds ) , provided that | x − y | > | x − x ′ | . Now Lemma 4.6 (b) (taken with u = 3, ξ = ρ = 0 and ξ = e i , ρ = 0)combined with Lemma 4.5 (specified to z = x ∗ ) gives the desired smoothness condition.The proof of the case of g ε, + V in Theorem 2.6 is finished. (cid:3) Horizontal g -functions based on { T α,ε, + t } . Proof of Theorem 2.6; the case of g j,ε, + H . To compute δ j,x G α,εt ( x, y ), observe that δ j,x may be replacedeither by δ ej,x or δ oj,x (see [9, p. 548]), δ ej,x = ∂ x j + x j , δ oj,x = ∂ x j + x j + 2 α j + 1 x j , depending on whether ε j = 0 or ε j = 1, respectively. Then we see that δ j,x G α,εt ( x, y ) = 12 d (cid:16) − ζ ζ (cid:17) d + | α | + | ε | h j ( x, y, ζ ) , where the auxiliary functions h j are given by h j ( x, y, ζ ) = − ( xy ) ε Z E xp( ζ, q ± ) h ζ ( x j + y j s j ) + ζ x j − y j s j ) i Π α + ε ( ds )+ x j ( xy ) ε Z E xp( ζ, q ± ) Π α + ε ( ds )+ χ { ε j =1 } (2 α j + 2) x ε − e j y ε Z E xp( ζ, q ± ) Π α + ε ( ds ) . Using Lemma 4.1 (a), the fact that x j ≤ √ q + + √ q − and then Lemma 4.1 (b) (taken with b = 1 / A = ζ − and A = ζ ) we obtain | δ j,x G α,εt ( x, y ) | . p − ζ ζ − d −| α |−| ε | ( xy ) ε Z E xp( ζ, q ± ) h √ q + ζ + ζ √ q − i Π α + ε ( ds )+ p − ζ ζ − d −| α |−| ε | ( xy ) ε Z E xp( ζ, q ± )( √ q + + √ q − ) Π α + ε ( ds )+ χ { ε j =1 } p − ζ ζ − d −| α |−| ε | x ε − e j y ε Z E xp( ζ, q ± ) Π α + ε ( ds ) . p − ζ ζ − d −| α |−| ε |− / ( xy ) ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )(4.8) + χ { ε j =1 } p − ζ ζ − d −| α |−| ε | x ε − e j y ε Z E xp( ζ, q ± ) Π α + ε ( ds ) . Now an application of Lemma 4.6 (a) (specified to u = 1, ξ = ρ = 0 and ξ = e j , ρ = 0) leads to thegrowth condition for (cid:8) δ j,x G α,εt ( x, y ) (cid:9) t> . QUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS 17
To prove the smoothness estimates we first show that (cid:13)(cid:13) δ j,x G α,εt ( x, y ) − δ j,x G α,εt ( x ′ , y ) (cid:13)(cid:13) L ( dt ) . | x − x ′ || x − y | w + α ( B ( x, | y − x | )) , | x − y | > | x − x ′ | . Using the mean value theorem we get (cid:12)(cid:12) δ j,x G α,εt ( x, y ) − δ j,x G α,εt ( x ′ , y ) (cid:12)(cid:12) ≤ | x − x ′ | (cid:12)(cid:12) ∇ x δ j,x G α,εt ( θ, y ) (cid:12)(cid:12) , where θ is a convex combination of x and x ′ (notice that θ depends also on t ). Thus it suffices to showthat for any i, j = 1 , . . . , d, k ∂ x i δ j,x G α,εt ( θ, y ) k L ( dt ) . | x − y | w + α ( B ( x, | y − x | )) , | x − y | > | x − x ′ | . (4.9)We shall first estimate ∂ x i h j ( x, y, ζ ). It is convenient to distinguish two cases. Case 1: i = j . An elementary computation produces ∂ x i h j ( x, y, ζ )= ( xy ) ε Z E xp( ζ, q ± ) h ζ ( x i + y i s i ) + ζ x i − y i s i ) ih ζ ( x j + y j s j ) + ζ x j − y j s j ) i Π α + ε ( ds ) − x j ( xy ) ε Z E xp( ζ, q ± ) h ζ ( x i + y i s i ) + ζ x i − y i s i ) i Π α + ε ( ds ) − χ { ε j =1 } (2 α j + 2) x ε − e j y ε Z E xp( ζ, q ± ) h ζ ( x i + y i s i ) + ζ x i − y i s i ) i Π α + ε ( ds ) − χ { ε i =1 } x ε − e i y ε Z E xp( ζ, q ± ) h ζ ( x j + y j s j ) + ζ x j − y j s j ) i Π α + ε ( ds )+ χ { ε i =1 } x j x ε − e i y ε Z E xp( ζ, q ± ) Π α + ε ( ds )+ χ { ε i =1 } χ { ε j =1 } (2 α j + 2) x ε − e i − e j y ε Z E xp( ζ, q ± ) Π α + ε ( ds ) . Using sequently Lemma 4.1 (a), the fact that x j ≤ √ q + + √ q − and then Lemma 4.1 (b) (taken with b = 1 / A = ζ − and A = ζ , respectively) we get | ∂ x i h j ( x, y, ζ ) | . ( xy ) ε Z E xp( ζ, q ± ) h ζ √ q + + ζ √ q − i Π α + ε ( ds )+ ( xy ) ε Z E xp( ζ, q ± ) (cid:0) √ q + + √ q − (cid:1)h ζ √ q + + ζ √ q − i Π α + ε ( ds )+ χ { ε j =1 } x ε − e j y ε Z E xp( ζ, q ± ) h ζ √ q + + ζ √ q − i Π α + ε ( ds )+ χ { ε i =1 } x ε − e i y ε Z E xp( ζ, q ± ) h ζ √ q + + ζ √ q − i Π α + ε ( ds )+ χ { ε i =1 } x ε − e i y ε Z E xp( ζ, q ± ) (cid:0) √ q + + √ q − (cid:1) Π α + ε ( ds )+ χ { ε i =1 } χ { ε j =1 } x ε − e i − e j y ε Z E xp( ζ, q ± ) Π α + ε ( ds ) . ζ − ( xy ) ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )+ χ { ε j =1 } ζ − / x ε − e j y ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } ζ − / x ε − e i y ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } χ { ε j =1 } x ε − e i − e j y ε Z E xp( ζ, q ± ) Π α + ε ( ds ) . Case 2: i = j . We have ∂ x j h j ( x, y, ζ ) = ( xy ) ε Z E xp( ζ, q ± ) h ζ ( x j + y j s j ) + ζ x j − y j s j ) i Π α + ε ( ds ) − ( xy ) ε Z E xp( ζ, q ± ) h ζ + ζ i Π α + ε ( ds ) + (cid:0) χ { ε j =1 } (cid:1) ( xy ) ε Z E xp( ζ, q ± ) Π α + ε ( ds ) − x j ( xy ) ε Z E xp( ζ, q ± ) h ζ ( x j + y j s j ) + ζ x j − y j s j ) i Π α + ε ( ds ) − χ { ε j =1 } (2 α j + 3) x ε − e j y ε Z E xp( ζ, q ± ) h ζ ( x j + y j s j ) + ζ x j − y j s j ) i Π α + ε ( ds ) . Proceeding similarly as in Case 1 (and using the inequality ζ − ≥
1) we obtain | ∂ x j h j ( x, y, ζ ) | . ζ − ( xy ) ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )+ χ { ε j =1 } ζ − / x ε − e j y ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) . Now using the above estimates of ∂ x i h j ( x, y, ζ ), the fact that θ ≤ x ∗ and Lemma 4.3 twice (with z = θ and z = x ∗ ) we see that | ∂ x i δ j,x G α,εt ( θ, y ) | (4.10) . p − ζ ζ − d −| α |−| ε |− ( x ∗ y ) ε Z (cid:0) E xp( ζ, q ± ( x ∗ , y, s )) (cid:1) / Π α + ε ( ds )+ χ { ε j =1 } p − ζ ζ − d −| α |−| ε |− / ( x ∗ ) ε − e j y ε Z (cid:0) E xp( ζ, q ± ( x ∗ , y, s )) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } p − ζ ζ − d −| α |−| ε |− / ( x ∗ ) ε − e i y ε Z (cid:0) E xp( ζ, q ± ( x ∗ , y, s )) (cid:1) / Π α + ε ( ds )+ χ { i = j } χ { ε i =1 } χ { ε j =1 } p − ζ ζ − d −| α |−| ε | ( x ∗ ) ε − e i − e j y ε Z (cid:0) E xp( ζ, q ± ( x ∗ , y, s )) (cid:1) / Π α + ε ( ds ) , provided that | x − y | > | x − x ′ | . From here (4.9) follows with the aid of Lemma 4.6 (a) (specified toeither u = 2, ρ = 0 and ξ = 0 or ξ = e j , ξ = e i , or ξ = e i + e j ) and Lemma 4.5 (taken with z = x ∗ ).The proof will be finished once we show that (cid:13)(cid:13) δ j,x G α,εt ( x, y ) − δ j,x G α,εt ( x, y ′ ) (cid:13)(cid:13) L ( dt ) . | y − y ′ || x − y | w + α ( B ( x, | y − x | )) , | x − y | > | y − y ′ | . By the mean value theorem it is enough to verify that for any i, j = 1 , . . . , d, we have k ∂ y i δ j,x G α,εt ( x, θ ) k L ( dt ) . | x − y | w + α ( B ( x, | y − x | )) , | x − y | > | y − y ′ | , where θ is a convex combination of y and y ′ . When considering ∂ y i h j ( x, y, ζ ) again it is natural todistinguish two cases. Case 1: i = j . A simple computation gives ∂ y i h j ( x, y, ζ )= ( xy ) ε Z E xp( ζ, q ± ) h ζ ( x j + y j s j ) + ζ x j − y j s j ) ih ζ ( y i + x i s i ) + ζ y i − x i s i ) i Π α + ε ( ds ) − x j ( xy ) ε Z E xp( ζ, q ± ) h ζ ( y i + x i s i ) + ζ y i − x i s i ) i Π α + ε ( ds ) − χ { ε j =1 } (2 α j + 2) x ε − e j y ε Z E xp( ζ, q ± ) h ζ ( y i + x i s i ) + ζ y i − x i s i ) i Π α + ε ( ds ) − χ { ε i =1 } x ε y ε − e i Z E xp( ζ, q ± ) h ζ ( x j + y j s j ) + ζ x j − y j s j ) i Π α + ε ( ds )+ χ { ε i =1 } x j x ε y ε − e i Z E xp( ζ, q ± ) Π α + ε ( ds ) QUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS 19 + χ { ε i =1 } χ { ε j =1 } (2 α j + 2) x ε − e j y ε − e i Z E xp( ζ, q ± ) Π α + ε ( ds ) . Proceeding as before (see the estimate of ∂ x i h j ( x, y, ζ ) above) we obtain | ∂ y i h j ( x, y, ζ ) | . ζ − ( xy ) ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )+ χ { ε j =1 } ζ − / x ε − e j y ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } ζ − / x ε y ε − e i Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } χ { ε j =1 } x ε − e j y ε − e i Z E xp( ζ, q ± ) Π α + ε ( ds ) . Case 2: i = j . It is not hard to check that ∂ y j h j ( x, y, ζ )= ( xy ) ε Z E xp( ζ, q ± ) h ζ ( x j + y j s j ) + ζ x j − y j s j ) ih ζ ( y j + x j s j ) + ζ y j − x j s j ) i Π α + ε ( ds )+ ( xy ) ε Z E xp( ζ, q ± ) h − ζ s j + ζ s j i Π α + ε ( ds ) − x j ( xy ) ε Z E xp( ζ, q ± ) h ζ ( y j + x j s j ) + ζ y j − x j s j ) i Π α + ε ( ds ) − χ { ε j =1 } x ε y ε − e j Z E xp( ζ, q ± ) h ζ ( x j + y j s j ) + ζ x j − y j s j ) i Π α + ε ( ds )+ χ { ε j =1 } x j x ε y ε − e j Z E xp( ζ, q ± ) Π α + ε ( ds )+ χ { ε j =1 } (2 α j + 2)( xy ) ε − e j Z E xp( ζ, q ± ) Π α + ε ( ds ) − χ { ε j =1 } (2 α j + 2) x ε − e j y ε Z E xp( ζ, q ± ) h ζ ( y j + x j s j ) + ζ y j − x j s j ) i Π α + ε ( ds )and therefore (see Case 2 in the estimate of ∂ x i h j ( x, y, ζ ) above) | ∂ y j h j ( x, y, ζ ) | . ζ − ( xy ) ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )+ χ { ε j =1 } ζ − / x ε y ε − e j Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )+ χ { ε j =1 } ( xy ) ε − e j Z E xp( ζ, q ± ) Π α + ε ( ds )+ χ { ε j =1 } ζ − / x ε − e j y ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) . Using the above estimates of ∂ y i h j ( x, y, ζ ), the fact that θ ≤ y ∗ and Lemma 4.3 twice we get | ∂ y i δ j,x G α,εt ( x, θ ) | (4.11) . p − ζ ζ − d −| α |−| ε |− ( xy ∗ ) ε Z (cid:0) E xp( ζ, q ± ( x, y ∗ , s )) (cid:1) / Π α + ε ( ds )+ χ { ε j =1 } p − ζ ζ − d −| α |−| ε |− / x ε − e j ( y ∗ ) ε Z (cid:0) E xp( ζ, q ± ( x, y ∗ , s )) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } p − ζ ζ − d −| α |−| ε |− / x ε ( y ∗ ) ε − e i Z (cid:0) E xp( ζ, q ± ( x, y ∗ , s )) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } χ { ε j =1 } p − ζ ζ − d −| α |−| ε | x ε − e j ( y ∗ ) ε − e i Z (cid:0) E xp( ζ, q ± ( x, y ∗ , s )) (cid:1) / Π α + ε ( ds ) , provided that | x − y | > | y − y ′ | . Now Lemma 4.6 (a) (applied with u = 2 and: ξ = ρ = 0 or ξ = e j , ρ = 0, or ξ = 0, ρ = e i , or ξ = e j , ρ = e i ) together with Lemma 4.5 gives the desired bound. The proof of the case of g j,ε, + H in Theorem 2.6 is complete. (cid:3) Proof of Theorem 2.6; the case of g j,ε, + H, ∗ . We first show the growth condition. Since δ ∗ j,x = − δ j,x + 2 x j ,in view of Theorem 2.6 (the case of g j,ε, + H ) it suffices to show that k x j G α,εt ( x, y ) k L ( dt ) . w + α ( B ( x, | y − x | )) , x = y, j = 1 , . . . , d. Taking into account (4.1), the fact that x j ≤ √ q + + √ q − and Lemma 4.1 (b) (specified to b = 1 / A = ζ − and A = ζ ) we get x j G α,εt ( x, y ) . p − ζ ζ − d −| α |−| ε | ( xy ) ε Z E xp( ζ, q ± )( √ q + + √ q − ) Π α + ε ( ds ) . p − ζ ζ − d −| α |−| ε |− / ( xy ) ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) . (4.12)Now an application of Lemma 4.6 (a) (taken with u = 1 and ξ = ρ = 0) leads to the required bound.To prove the smoothness estimates, again in view of the relation δ ∗ j,x = − δ j,x + 2 x j and Theorem 2.6(the case of g j,ε, + H ) it suffices to verify that k x j G α,εt ( x, y ) − x ′ j G α,εt ( x ′ , y ) k . | x − x ′ || x − y | w + α ( B ( x, | y − x | )) , | x − y | > | x − x ′ | , k x j G α,εt ( x, y ) − x j G α,εt ( x, y ′ ) k . | y − y ′ || x − y | w + α ( B ( x, | y − x | )) , | x − y | > | y − y ′ | . Using the mean value theorem we obtain (cid:12)(cid:12) x j G α,εt ( x, y ) − x ′ j G α,εt ( x ′ , y ) (cid:12)(cid:12) ≤| x − x ′ | (cid:12)(cid:12) ∇ x (cid:0) x j G α,εt ( x, y ) (cid:1)(cid:12)(cid:12) x = θ (cid:12)(cid:12) , (cid:12)(cid:12) x j G α,εt ( x, y ) − x j G α,εt ( x, y ′ ) (cid:12)(cid:12) ≤| y − y ′ | (cid:12)(cid:12) ∇ y (cid:0) x j G α,εt ( x, y ) (cid:1)(cid:12)(cid:12) y = ψ (cid:12)(cid:12) , where θ , ψ are convex combinations of x , x ′ , and y , y ′ , respectively, that depend also on t . Thus it sufficesto show that for any i, j = 1 , . . . , d, (cid:13)(cid:13) ∂ x i (cid:0) x j G α,εt ( x, y ) (cid:1)(cid:12)(cid:12) x = θ (cid:13)(cid:13) L ( dt ) . | x − y | w + α ( B ( x, | y − x | )) , | x − y | > | x − x ′ | , (cid:13)(cid:13) ∂ y i (cid:0) x j G α,εt ( x, y ) (cid:1)(cid:12)(cid:12) y = ψ (cid:13)(cid:13) L ( dt ) . | x − y | w + α ( B ( x, | y − x | )) , | x − y | > | y − y ′ | . An elementary computation gives ∂ x i (cid:0) x j G α,εt ( x, y ) (cid:1) = − d (cid:16) − ζ ζ (cid:17) d + | α | + | ε | x j ( xy ) ε Z E xp( ζ, q ± ) h ζ ( x i + y i s i ) + ζ x i − y i s i ) i Π α + ε ( ds )+ (cid:0) χ { ε i =1 } + χ { i = j } (cid:1) d (cid:16) − ζ ζ (cid:17) d + | α | + | ε | x j x ε − e i y ε Z E xp( ζ, q ± ) Π α + ε ( ds ) ,∂ y i (cid:0) x j G α,εt ( x, y ) (cid:1) = − d (cid:16) − ζ ζ (cid:17) d + | α | + | ε | x j ( xy ) ε Z E xp( ζ, q ± ) h ζ ( y i + x i s i ) + ζ y i − x i s i ) i Π α + ε ( ds )+ χ { ε i =1 } d (cid:16) − ζ ζ (cid:17) d + | α | + | ε | x j x ε y ε − e i Z E xp( ζ, q ± ) Π α + ε ( ds ) . Applying the inequality x j ≤ √ q + + √ q − and Lemma 4.1 (a), (b) (with b = 1 /
2) we get (cid:12)(cid:12) ∂ x i (cid:0) x j G α,εt ( x, y ) (cid:1)(cid:12)(cid:12) . p − ζ ζ − d −| α |−| ε |− ( xy ) ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )(4.13) + χ { ε i =1 } p − ζ ζ − d −| α |−| ε |− / x ε − e i y ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) , (cid:12)(cid:12) ∂ y i (cid:0) x j G α,εt ( x, y ) (cid:1)(cid:12)(cid:12) . p − ζ ζ − d −| α |−| ε |− ( xy ) ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )(4.14) QUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS 21 + χ { ε i =1 } p − ζ ζ − d −| α |−| ε |− / x ε y ε − e i Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) . Now using the fact that θ ≤ x ∗ , ψ ≤ y ∗ , and Lemma 4.3, we obtain the estimates (cid:12)(cid:12) ∂ x i (cid:0) x j G α,εt ( x, y ) (cid:1)(cid:12)(cid:12) x = θ (cid:12)(cid:12) . p − ζ ζ − d −| α |−| ε |− ( x ∗ y ) ε Z (cid:0) E xp( ζ, q ± ( x ∗ , y, s )) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } p − ζ ζ − d −| α |−| ε |− / ( x ∗ ) ε − e i y ε Z (cid:0) E xp( ζ, q ± ( x ∗ , y, s )) (cid:1) / Π α + ε ( ds ) , (cid:12)(cid:12) ∂ y i (cid:0) x j G α,εt ( x, y ) (cid:1)(cid:12)(cid:12) y = ψ (cid:12)(cid:12) . p − ζ ζ − d −| α |−| ε |− ( xy ∗ ) ε Z (cid:0) E xp( ζ, q ± ( x, y ∗ , s )) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } p − ζ ζ − d −| α |−| ε |− / x ε ( y ∗ ) ε − e i Z (cid:0) E xp( ζ, q ± ( x, y ∗ , s )) (cid:1) / Π α + ε ( ds ) , provided that | x − y | > | x − x ′ | and | x − y | > | y − y ′ | , respectively. Finally, combining Lemma 4.6 (a)with Lemma 4.5 gives the smoothness conditions. (cid:3) Lusin’s area integrals based on { T α,ε, + t } . In this subsection we show the standard estimates for the kernels K α,ε,Vz,t ( x, y ) , K α,ε,H,jz,t ( x, y ) , K α,ε,H, ∗ ,jz,t ( x, y ) , j = 1 , . . . , d, valued in the Banach spaces L ( A, tdtdz ) (the case of K α,ε,Vz,t ( x, y )) or L ( A, dtdz ) (the remaining cases),where A = { ( z, t ) ∈ R d × (0 , ∞ ) : | z | < √ t } . To achieve this we shall need several additional technicallemmas. Lemma 4.8.
Let x, y ∈ R d + , z ∈ R d , s ∈ [ − , d . Then q ± ( x + z, y, s ) ≥ q ± ( x, y, s ) − | z | . Proof.
Since q − ( x, y, s ) = q + ( x, y, − s ) we may consider q + only. Moreover, by the structure of q + wemay restrict to the one-dimensional case. Then a simple computation shows that q + ( x + z, y, s ) − q + ( x, y, s ) + z = 12 ( x + ys + 2 z ) + 12 (1 − s ) y . Since | s | ≤
1, the conclusion follows. (cid:3)
Lemma 4.9.
Assume that α ∈ [ − / , ∞ ) d . Let x, x ′ ∈ R d + and z ∈ R d be such that x + z ∈ R d + and let ϕ α be the function given by (2.7) . If θ = θ ( x, x ′ , z, ζ ( t )) is a convex combination of x, x ′ , then Z | z | < √ L og ( ζ ) / (cid:12)(cid:12) ∇ x ϕ α ( x, z, t ( ζ )) (cid:12)(cid:12) x = θ (cid:12)(cid:12) χ { x + z ∈ R d + } χ { x ′ + z ∈ R d + } dz . ζ − / uniformly in x, x ′ , ζ , where ζ is related to t as in (4.2) .Proof. It suffices to show that for every j = 1 , . . . , d , we have Z | z | < √ L og( ζ ) / (cid:12)(cid:12) ∂ x j ϕ α ( x, z, t ( ζ )) (cid:12)(cid:12) x = θ (cid:12)(cid:12) χ { x + z ∈ R d + } χ { x ′ + z ∈ R d + } dz . (cid:0) L og( ζ ) (cid:1) − / , since ζ . L og( ζ ). An elementary computation gives ∂ x j ϕ α ( x, z, t ) (cid:12)(cid:12) x = θ = (2 α j + 1)( θ j + z j ) α j V α j , + √ t ( θ j ) − ( θ j + z j ) α j +1 ∂ x j V α j , + √ t ( x j ) (cid:12)(cid:12) x j = θ j (cid:0) V α j , + √ t ( θ j ) (cid:1) Y i = j ( θ i + z i ) α i +1 V α i , + √ t ( θ i ) . We estimate this derivative on the set of integration by using the inequality | z j | ≤ p L og( ζ ) / V α j , + √ t ( x j ) ≃ p L og( ζ ) (cid:0) x j + p L og( ζ ) / (cid:1) α j +1 , (cid:12)(cid:12) ∂ x j V α j , + √ t ( x j ) (cid:12)(cid:12) ≤ (cid:0) x j + p L og( ζ ) / (cid:1) α j +1 , (4.15) obtaining (cid:12)(cid:12) ∂ x j ϕ α ( x, z, t ( ζ )) (cid:12)(cid:12) x = θ (cid:12)(cid:12) . (cid:0) L og( ζ ) (cid:1) − d/ (cid:20) (2 α j + 1) ( θ j + z j ) α j (cid:0) θ j + p L og( ζ ) / (cid:1) α j +1 + (cid:0) L og( ζ ) (cid:1) − / (cid:21) ≡ I + I . Now it is not hard to see that the required bound holds for the integral involving I . To estimate theintegral related to I , we consider three cases. The case when α j = − / α j ∈ ( − / , s s + z j s + √ L og( ζ ) / is increasing for s ≥ θ j + z j ) α j (cid:0) θ j + p L og( ζ ) / (cid:1) α j +1 ≤ (( x ∗ ) j + z j ) α j (cid:0) ( x ∗ ) j + p L og( ζ ) / (cid:1) α j +1 , where x ∗ = min { x, x ′ } . Using this inequality and observing that z j > − ( x ∗ ) j if x j + z j > x ′ j + z j > Z | z | < √ L og( ζ ) / I χ { x + z ∈ R d + } χ { x ′ + z ∈ R d + } dz . (cid:0) L og( ζ ) (cid:1) − / Z √ L og( ζ ) / − ( x ∗ ) j (( x ∗ ) j + z j ) α j (cid:0) ( x ∗ ) j + p L og( ζ ) / (cid:1) α j +1 dz j . (cid:0) L og( ζ ) (cid:1) − / . Finally, if α j ≥ θ j + z j ) α j ≤ ( θ j + p L og( ζ ) / α j and I . (cid:0) L og( ζ ) (cid:1) − ( d +1) / , so the conclusion again follows. (cid:3) Lemma 4.10.
Assume that α ∈ [ − / , ∞ ) d and ξ, ρ, η, ε ∈ Z d are fixed and such that ξ + η ≤ ε and ρ ≤ ε . Given C > and u ∈ R , consider the function acting on R d + × R d + × (0 , and defined by p u ( x, y, ζ ) = p − ζ ζ − d −| α |−| ε | + | ξ | / | ρ | / − u/ (cid:0) L og ( ζ ) (cid:1) | η | / x ε − ξ − η y ε − ρ exp (cid:18) L og ( ζ )8 ζ (cid:19) × Z [ − , d (cid:0) E xp ( ζ, q ± ) (cid:1) C Π α + ε ( ds ) . (a) If u ≥ , then we have the estimate k p u ( x, y, ζ ( t )) k L ( dt ) . | x − y | u − w + α ( B ( x, | y − x | )) , x = y, where t and ζ are related as in (4.2) . (b) If u ≥ , then we also have k p u ( x, y, ζ ( t )) k L ( tdt ) . | x − y | u − w + α ( B ( x, | y − x | )) , x = y. Proof.
We will prove only the first inequality, leaving the remaining one to the reader. To show therequired estimate we change the variable according to (4.2) and split the region of integration in ζ onto(0 , /
2) and (1 / , I and I , respectively. Then the conclusionfor I is a straightforward consequence of Lemma 4.6 (a), see the asymptotics (4.3). We now focus on I . Since exp( − s ) . exp( − s ), when ζ ∈ (1 / ,
1) we have the estimates E xp( ζ, q ± ) . exp( − | x | − | y | ),exp (cid:0) L og( ζ )8 ζ (cid:1) . (1 − ζ ) − / and ζ − ≃
1. Thus for ζ ∈ (1 / ,
1) we obtain I . (cid:18) Z / (cid:0) L og( ζ ) (cid:1) | η | x ε − ξ − η y ε − ρ (1 − ζ ) − / (cid:18) Z exp (cid:16) − C | x | − C | y | (cid:17) Π α + ε ( ds ) (cid:19) dζ (cid:19) / . x ε − ξ − η y ε − ρ exp (cid:16) − C | x | − C | y | (cid:17) . ( | x | + | y | ) − d − | α | . w + α ( B ( x, | y − x | )) , as desired. (cid:3) QUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS 23
Lemma 4.11.
Assume that α ∈ [ − / , ∞ ) d and ξ, ρ, ε ∈ Z d are fixed and such that ξ + η ≤ ε and ρ ≤ ε .Given C > and u ∈ R , consider the function acting on R d + × R d + × (cid:8) ( z, ζ ) : | z | < p L og ( ζ ) / (cid:9) anddefined by p u ( x, y, z, ζ ) = p − ζ ζ − d −| α |−| ε | + | ξ | / | ρ | / − u/ (cid:0) L og ( ζ ) (cid:1) − d/ ( x + z ) ε − ξ y ε − ρ χ { x + z ∈ R d + } × exp (cid:18) L og ( ζ )8 ζ (cid:19) Z [ − , d (cid:0) E xp ( ζ, q ± ) (cid:1) C Π α + ε ( ds ) . (a) If u ≥ , then we have the estimate k p u ( x, y, z, ζ ( t )) k L ( A,dtdz ) . | x − y | u − w + α ( B ( x, | y − x | )) , x = y, where t and ζ are related as in (4.2) . (b) If u ≥ , then we also have k p u ( x, y, z, ζ ( t )) k L ( A,tdtdz ) . | x − y | u − w + α ( B ( x, | y − x | )) , x = y. Proof.
As in the proof of Lemma 4.10 we show only the first estimate. Since | z | < p L og( ζ ) / A , we get(4.16) ( x + z ) ε − ξ χ { x + z ∈ R d + } . X ≤ η ≤ ε − ξ x ε − ξ − η (cid:0) L og( ζ ) (cid:1) | η | . Thus we have Z | z | < √ L og( ζ ) / ( x + z ) ε − ξ χ { x + z ∈ R d + } dz . X ≤ η ≤ ε − ξ x ε − ξ − η (cid:0) L og( ζ ) (cid:1) | η | + d/ . Now changing the variable according to (4.2) and then applying the above estimate we obtain k p u ( x, y, z, ζ ( t )) k L ( A,dtdz ) = (cid:18) Z Z | z | < √ L og( ζ ) / (cid:16) ζ (cid:17) d +2 | α | +2 | ε |−| ξ |−| ρ | + u (cid:0) L og( ζ ) (cid:1) − d/ ( x + z ) ε − ξ y ε − ρ χ { x + z ∈ R d + } × exp (cid:18) L og( ζ )4 ζ (cid:19)(cid:18) Z (cid:0) E xp( ζ, q ± ) (cid:1) C Π α + ε ( ds ) (cid:19) dz dζ (cid:19) / . X ≤ η ≤ ε − ξ (cid:18) Z (cid:16) ζ (cid:17) d +2 | α | +2 | ε |−| ξ |−| ρ | + u (cid:0) L og( ζ ) (cid:1) | η | x ε − ξ − η y ε − ρ × exp (cid:18) L og( ζ )4 ζ (cid:19)(cid:18) Z (cid:0) E xp( ζ, q ± ) (cid:1) C Π α + ε ( ds ) (cid:19) dζ (cid:19) / . This, in view of Lemma 4.10 (a), gives the conclusion. (cid:3)
Proof of Theorem 2.6; the case of S ε, + V . Notice that on the set A ∩ { ( z, t ) : x + z ∈ R d + } we have, see(4.15),(4.17) ϕ α ( x, z, t ) . (cid:0) L og( ζ ) (cid:1) − d/ . Using this observation, the estimate (4.6) of ∂ t G α,εt ( x, y ), Lemma 4.8 and the fact that | z | < p L og( ζ ) / A , we obtain | K α,ε,Vz,t ( x, y ) | . p − ζ ζ − d −| α |−| ε |− (cid:0) L og( ζ ) (cid:1) − d/ ( x + z ) ε y ε χ { x + z ∈ R d + } × Z (cid:0) E xp( ζ, q ± ( x + z, y, s )) (cid:1) / Π α + ε ( ds ) . p − ζ ζ − d −| α |−| ε |− (cid:0) L og( ζ ) (cid:1) − d/ ( x + z ) ε y ε χ { x + z ∈ R d + } (4.18) × exp (cid:18) L og( ζ )8 ζ (cid:19) Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) . Now the growth estimate follows with the aid of Lemma 4.11 (b) (specified to u = 2, ξ = ρ = 0).Next, our task is to show that (cid:13)(cid:13) K α,ε,Vz,t ( x, y ) − K α,ε,Vz,t ( x ′ , y ) (cid:13)(cid:13) L ( A,tdtdz ) . s | x − x ′ || x − y | w + α ( B ( x, | y − x | )) , | x − y | > | x − x ′ | . It is convenient to split the region of integration A above onto four subsets depending on whether x + z, x ′ + z are in R d + or not. More precisely, let A = A ∩ { ( z, t ) : x + z ∈ R d + , x ′ + z ∈ R d + } ,A = A ∩ { ( z, t ) : x + z ∈ R d + , x ′ + z / ∈ R d + } ,A = A ∩ { ( z, t ) : x + z / ∈ R d + , x ′ + z ∈ R d + } ,A = A ∩ { ( z, t ) : x + z / ∈ R d + , x ′ + z / ∈ R d + } . We will estimate separately the L ( A i , tdtdz ) norms, i = 1 , . . . ,
4, of the relevant difference. The treatmentof the integral norm over A is trivial since the integrand vanishes. For the remaining norms we considerthree cases. Case 1: The norm in L ( A , tdtdz ) . Using the triangle inequality we get | K α,ε,Vz,t ( x, y ) − K α,ε,Vz,t ( x ′ , y ) | ≤ (cid:12)(cid:12) ∂ t G α,εt ( x + z, y ) − ∂ t G α,εt ( x ′ + z, y ) (cid:12)(cid:12)p ϕ α ( x ′ , z, t )+ (cid:12)(cid:12) ∂ t G α,εt ( x + z, y ) (cid:12)(cid:12)(cid:12)(cid:12)p ϕ α ( x, z, t ) − p ϕ α ( x ′ , z, t ) (cid:12)(cid:12) ≡ I ( x, x ′ , y, z, t ) + I ( x, x ′ , y, z, t ) . We will treat I and I separately. By the mean value theorem I ( x, x ′ , y, z, t ) ≤| x − x ′ | (cid:12)(cid:12) ∇ x ∂ t G α,εt ( x + z, y ) (cid:12)(cid:12) x = θ (cid:12)(cid:12)p ϕ α ( x ′ , z, t ) , where θ is a convex combination of x and x ′ that depends also on z and t . To show the desired boundfor the norm of I it suffices to check that for each i = 1 , . . . , d , we have (cid:13)(cid:13) ∂ x i (cid:0) ∂ t G α,εt ( x + z, y ) (cid:1)(cid:12)(cid:12) x = θ p ϕ α ( x ′ , z, t ) (cid:13)(cid:13) L ( A ,tdtdz ) . | x − y | w + α ( B ( x, | y − x | )) , for | x − y | > | x − x ′ | . Applying (4.7), (4.17), Lemma 4.8 and then Lemma 4.3 (with z = θ and then z = x ∗ ) we get (cid:12)(cid:12) ∂ x i (cid:0) ∂ t G α,εt ( x + z, y ) (cid:1)(cid:12)(cid:12) x = θ p ϕ α ( x ′ , z, t ) (cid:12)(cid:12) . p − ζ ζ − d −| α |−| ε |− / (cid:0) L og( ζ ) (cid:1) − d/ ( θ + z ) ε y ε Z (cid:0) E xp( ζ, q ± ( θ + z, y, s )) (cid:1) / Π α + ε ( ds )(4.19) + χ { ε i =1 } p − ζ ζ − d −| α |−| ε |− (cid:0) L og( ζ ) (cid:1) − d/ ( θ + z ) ε − e i y ε × Z (cid:0) E xp( ζ, q ± ( θ + z, y, s )) (cid:1) / Π α + ε ( ds ) . p − ζ ζ − d −| α |−| ε |− / (cid:0) L og( ζ ) (cid:1) − d/ ( x ∗ + z ) ε y ε × exp (cid:18) L og( ζ )16 ζ (cid:19) Z (cid:0) E xp( ζ, q ± ( x ∗ , y, s )) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } p − ζ ζ − d −| α |−| ε |− (cid:0) L og( ζ ) (cid:1) − d/ ( x ∗ + z ) ε − e i y ε × exp (cid:18) L og( ζ )8 ζ (cid:19) Z (cid:0) E xp( ζ, q ± ( x ∗ , y, s )) (cid:1) / Π α + ε ( ds ) , provided that | x − y | > | x − x ′ | . Now Lemma 4.11 (b) (taken with u = 3, ξ = ρ = 0 and ξ = e i , ρ = 0; the application is possible since on A we have x ∗ + z ∈ R d + ) together with Lemma 4.5 (taken with z = x ∗ ) leads to the required bound involving I .To show the norm estimate of I we use the inequality ( a − b ) ≤ | a − b | , which holds for any a, b ≥ (cid:12)(cid:12)p ϕ α ( x, z, t ) − p ϕ α ( x ′ , z, t ) (cid:12)(cid:12) ≤| ϕ α ( x, z, t ) − ϕ α ( x ′ , z, t ) | ≤ | x − x ′ | (cid:12)(cid:12) ∇ x ϕ α ( x, z, t ) (cid:12)(cid:12) x = θ (cid:12)(cid:12) , (4.20) QUARE FUNCTIONS IN CERTAIN DUNKL AND LAGUERRE SETTINGS 25 where θ is a convex combination of x and x ′ depending also on z and t = t ( ζ ). Changing the variableaccording to (4.2) and then applying sequently the above estimate, (4.6), Lemma 4.8, inequality (4.16)(with ξ = 0) and Lemma 4.9, we get k I ( x, x ′ , y, z, t ) k L ( A ,tdtdz ) . p | x − x ′ | (cid:18) Z Z | z | < √ L og( ζ ) / L og( ζ ) (cid:16) ζ (cid:17) d +2 | α | +2 | ε | +2 ( x + z ) ε y ε χ { x + z ∈ R d + } χ { x ′ + z ∈ R d + } × (cid:12)(cid:12) ∇ x ϕ α ( x, z, t ( ζ )) (cid:12)(cid:12) x = θ (cid:12)(cid:12) exp (cid:18) L og( ζ )4 ζ (cid:19)(cid:18) Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) (cid:19) dz dζ (cid:19) / . p | x − x ′ | X ≤ η ≤ ε (cid:18) Z (cid:0) L og( ζ ) (cid:1) | η | (cid:16) ζ (cid:17) d +2 | α | +2 | ε | +5 / x ε − η y ε exp (cid:18) L og( ζ )4 ζ (cid:19) × (cid:18) Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) (cid:19) dζ (cid:19) / , provided that | x − y | > | x − x ′ | . Finally, an application of Lemma 4.10 (b) (specified to u = 5 / ξ = ρ = 0) gives the desired estimate, so the conclusion related to A follows. Case 2: The norm in L ( A , tdtdz ) . For k = 1 , . . . , d , we define the sets A k = A ∩ { ( z, t ) : x + z ∈ R d + , z k ≤ − x ′ k } . Since these sets cover A and on each of them K α,ε,Vz,t ( x ′ , y ) = 0, our task reduces to showing that (cid:13)(cid:13) K α,ε,Vz,t ( x, y ) (cid:13)(cid:13) L ( A k ,tdtdz ) . s | x − x ′ || x − y | w + α ( B ( x, | y − x | )) , | x − y | > | x − x ′ | . Changing the variable according to (4.2), applying the estimate (4.18) and then the inequality (4.16)(with ξ = 0), we obtain (cid:13)(cid:13) K α,ε,Vz,t ( x, y ) (cid:13)(cid:13) L ( A k ,tdtdz ) . X ≤ η ≤ ε (cid:18) Z Z | z | < √ L og( ζ ) / χ {− x k
Case 1: The norm in L ( A , dtdz ) . Using the triangle inequality we get (cid:12)(cid:12)(cid:12) ( a j + z j ) G α,εt ( a + z, y ) p ϕ α ( a, z, t ) χ { a + z ∈ R d + } (cid:12)(cid:12)(cid:12) a = xa = x ′ (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) ( x j + z j ) G α,εt ( x + z, y ) − ( x ′ j + z j ) G α,εt ( x ′ + z, y ) (cid:12)(cid:12)p ϕ α ( x ′ , z, t )+ (cid:12)(cid:12) ( x j + z j ) G α,εt ( x + z, y ) (cid:12)(cid:12)(cid:12)(cid:12)p ϕ α ( x, z, t ) − p ϕ α ( x ′ , z, t ) (cid:12)(cid:12) ≡ L ( x, x ′ , y, z, t ) + L ( x, x ′ , y, z, t ) . First, we analyze L . By the mean value theorem and (4.17) it is enough to check that for any i, j =1 , . . . , d , we have(4.25) (cid:13)(cid:13) ∂ x i (cid:0) ( x j + z j ) G α,εt ( x + z, y ) (cid:1)(cid:12)(cid:12) x = θ (cid:0) L og( ζ ) (cid:1) − d/ (cid:13)(cid:13) L ( A ,dtdz ) . | x − y | w + α ( B ( x, | y − x | )) , for | x − y | > | x − x ′ | , where θ is a convex combination of x and x ′ . Using the inequality (4.13), Lemma4.8 and then Lemma 4.3, we obtain (cid:12)(cid:12) ∂ x i (cid:0) ( x j + z j ) G α,εt ( x + z, y ) (cid:1)(cid:12)(cid:12) x = θ (cid:12)(cid:12) . p − ζ ζ − d −| α |−| ε |− ( x ∗ + z ) ε y ε exp (cid:18) L og( ζ )16 ζ (cid:19) × Z (cid:0) E xp( ζ, q ± ( x ∗ , y, s )) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } p − ζ ζ − d −| α |−| ε |− / ( x ∗ + z ) ε − e i y ε × exp (cid:18) L og( ζ )8 ζ (cid:19) Z (cid:0) E xp( ζ, q ± ( x ∗ , y, s )) (cid:1) / Π α + ε ( ds ) , provided that | x − y | > | x − x ′ | . Then combining Lemma 4.11 (a) (notice that on A we have x ∗ + z ∈ R d + )with Lemma 4.5 gives (4.25), and hence also the required bound for the norm of L .We now focus on L . Changing the variable according to (4.2) and then applying sequently the estimate(4.12) of x j G α,εt ( x, y ), (4.20), Lemma 4.8, inequality (4.16) (with ξ = 0) and Lemma 4.9, we get k L ( x, x ′ , y, z, t ) k L ( A ,dtdz ) . p | x − x ′ | X ≤ η ≤ ε (cid:18) Z (cid:0) L og( ζ ) (cid:1) | η | (cid:16) ζ (cid:17) d +2 | α | +2 | ε | +3 / x ε − η y ε × exp (cid:18) L og( ζ )4 ζ (cid:19)(cid:18) Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) (cid:19) dζ (cid:19) / , provided that | x − y | > | x − x ′ | . From here the conclusion follows with the aid of Lemma 4.10 (a) (takenwith u = 3 / ξ = ρ = 0). Case 2: The norm in L ( A , dtdz ) . It suffices to verify that (cid:13)(cid:13) ( x j + z j ) G α,εt ( x + z, y ) p ϕ α ( x, z, t ) (cid:13)(cid:13) L ( A k ,dtdz ) . s | x − x ′ || x − y | w + α ( B ( x, | y − x | )) , for | x − y | > | x − x ′ | , where A k are the sets from the part of the proof of Theorem 2.6 concerning S ε, + V .Changing the variable as in (4.2), using the inequalities (4.24), (4.16) and then (4.21), we obtain (cid:13)(cid:13) ( x j + z j ) G α,εt ( x + z, y ) p ϕ α ( x, z, t ) (cid:13)(cid:13) L ( A k ,dtdz ) . p | x − x ′ | X ≤ η ≤ ε (cid:18) Z (cid:0) L og( ζ ) (cid:1) | η | (cid:16) ζ (cid:17) d +2 | α | +2 | ε | +3 / x ε − η y ε × exp (cid:18) L og( ζ )4 ζ (cid:19)(cid:18) Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) (cid:19) dζ (cid:19) / . Now an application of Lemma 4.10 (a) (specified to u = 3 / ξ = ρ = 0) leads to the desired bound. Case 3: The norm in L ( A , dtdz ) . Here the arguments are essentially the same as in Case 2 andthus are omitted.
The proof will be finished once we show the remaining smoothness condition. Again by the relation δ ∗ j,x = − δ j,x + 2 x j , the already justified case of S j,ε, + H in Theorem 2.6 and the mean value theorem, itsuffices to prove that (cid:13)(cid:13) ∂ y i (cid:0) ( x j + z j ) G α,εt ( x + z, y ) (cid:1)(cid:12)(cid:12) y = θ p ϕ α ( x, z, t ) χ { x + z ∈ R d + } (cid:13)(cid:13) L ( A,dtdz ) . | x − y | w + α ( B ( x, | y − x | )) , for | x − y | > | y − y ′ | , where θ is a convex combination of y and y ′ . Using the estimates (4.14), (4.17),Lemma 4.8 and Lemma 4.3, we obtain (cid:12)(cid:12) ∂ y i (cid:0) ( x j + z j ) G α,εt ( x + z, y ) (cid:1)(cid:12)(cid:12) y = θ (cid:12)(cid:12)p ϕ α ( x, z, t ) χ { x + z ∈ R d + } . p − ζ ζ − d −| α |−| ε |− (cid:0) L og( ζ ) (cid:1) − d/ ( x + z ) ε ( y ∗ ) ε χ { x + z ∈ R d + } × exp (cid:18) L og( ζ )16 ζ (cid:19) Z (cid:0) E xp( ζ, q ± ( x, y ∗ , s )) (cid:1) / Π α + ε ( ds )+ χ { ε i =1 } p − ζ ζ − d −| α |−| ε |− / (cid:0) L og( ζ ) (cid:1) − d/ ( x + z ) ε ( y ∗ ) ε − e i χ { x + z ∈ R d + } × exp (cid:18) L og( ζ )8 ζ (cid:19) Z (cid:0) E xp( ζ, q ± ( x, y ∗ , s )) (cid:1) / Π α + ε ( ds ) , provided that | x − y | > | y − y ′ | . Now the desired bound follows by applying Lemma 4.11 (a) and Lemma4.5.The proof of the case of S j,ε, + H, ∗ in Theorem 2.6 is complete. This finishes proving Theorem 2.6. (cid:3) References [1] R. Askey and S. Wainger,
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Tomasz Szarek,ul. W. Rutkiewicz 29/43, PL-50–571 Wroc law, Poland
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