Multipliers of Laplace transform type in certain Dunkl and Laguerre settings
aa r X i v : . [ m a t h . C A ] J a n MULTIPLIERS OF LAPLACE TRANSFORM TYPE IN CERTAIN DUNKLAND LAGUERRE SETTINGS
TOMASZ SZAREK
Abstract.
We investigate Laplace type and Laplace-Stieltjes type multipliers in the d -dimen-sional setting of the Dunkl harmonic oscillator with the associated group of reflections isomorphicto Z d and in the related context of Laguerre function expansions of convolution type. We useCalder´on-Zygmund theory to prove that these multiplier operators are bounded on weighted L p , 1 < p < ∞ , and from L to weak L . Introduction
In [20] the author defined and investigated square functions related to the Dunkl harmonicoscillator L α and the related group of reflections isomorphic to Z d . This paper continues thestudy of L p mapping properties of fundamental harmonic analysis operators associated with L α .We consider Laplace type and Laplace-Stieltjes type multiplier operators, see Section 2 for thedefinitions. The most typical examples of such operators are the imaginary powers L − iγα , γ ∈ R ,and the fractional integral operators L − δα , δ >
0, as pointed out in [21, Section 2]. Thus our mainresult, Theorem 2.2 below, may be regarded as a continuation of the investigations of Nowak andStempak [10, 11], where these operators were studied in the context of L α . A trivial choice ofthe multiplicity function reduces the Dunkl setting to the situation of classical Hermite functionexpansions. Thus, in particular, our considerations provide results in the Hermite setting, wherethe Laplace type multiplier operators were implicitly analyzed by Stempak and Torrea, see thecomment in [17, p. 46]. Moreover, our Dunkl situation reduces to the setting of Laguerre functionexpansions of convolution type after restricting to reflection invariant functions. Consequently,we obtain also results in the Laguerre context.Multipliers related to numerous classic kinds of orthogonal expansions were widely inves-tigated. In particular, Stempak and Trebels [18] studied multipliers of non-Laplace type ina one-dimensional Laguerre setting, which is deeply connected with our Dunkl situation (seeRemark 2.9 below). Some earlier results concerning multiplier operators of Laplace type fordiscrete and continuous orthogonal expansions can be found in [1, 2, 4, 5, 13, 21], among others.A general treatment of Laplace type multipliers in a context of symmetric diffusion semigroupscan be found in Stein’s monograph [15].We refer the reader to the survey article by R¨osler [12] for basic facts concerning Dunkl’stheory. A precise description of the Dunkl framework for the particular group of reflections G isomorphic to Z d can be found for instance in [9, Section 3]. Here we invoke only the most Mathematics Subject Classification.
Key words and phrases. multiplier, Dunkl harmonic oscillator, generalized Hermite expansions, Laguerre ex-pansions of convolution type, Calder´on-Zygmund operator, A p weight. relevant facts. We shall work on the 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 ) ∈ [ − / , ∞ ) d representsthe multiplicity function. Consider the reflection group G ≃ Z d generated by σ j , j = 1 , . . . , d , σ j ( x , . . . , x j , . . . , x d ) = ( x , . . . , − x j , . . . , x d ) . Clearly, the reflection σ j is in the hyperplane orthogonal to e j , the j th coordinate vector. Noticethat the 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. 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 our investigations a similar role to that of the Euclidean Laplacianin the classical harmonic analysis. Note that for α = ( − / , . . . , − / L α becomes the classicharmonic oscillator − ∆+ | x | . We shall consider a self-adjoint extension L α of L α , whose spectraldecomposition is discrete and given by the generalized Hermite functions h αn , see Section 2 fordetails.The main objects of our study are spectral multipliers associated with L α . More precisely, weinvestigate two kinds of such operators, see Definition 2.1 below. The first one is a multiplierof Laplace type, which originates in Stein’s monograph [15]. The second one is a multiplier ofLaplace-Stieltjes type, which was considered recently by Wr´obel [21] in the context of Laguerrefunction expansions of Hermite type, and its definition has roots in the work of De N´apoli,Drelichman and Dur´an [4]. Our main result, Theorem 2.2, says that the multiplier operatorsin question are bounded on weighted L p ( dw α ), 1 < p < ∞ , and satisfy weighted weak type(1,1) inequality, for a large class of weights. We note that the unweighted L p -boundedness,1 < p < ∞ , of the Laplace type multiplier operators follows also from the refinement of Stein’sgeneral Littlewood-Paley theory for semigroups (see [15, Corollary 3, p. 121]) due to Coifman,Rochberg and Weiss [3], see also [6, Theorem 2].In the proof of Theorem 2.2 we exploit the arguments from [10] that allow us to reducethe analysis to suitably defined Laguerre-type operators related to the smaller measure space( R d + , dw + α ); here R d + = (0 , ∞ ) d and w + α is the restriction of w α to R d + . Then we apply the theoryof Calder´on-Zygmund operators with the underlying space of homogeneous type ( R d + , dw + α , | · | ).An essential technical difficulty connected with this approach is to show the relevant kernel esti-mates. Here we employ a convenient technique having roots in Sasso’s paper [14] and developedby Nowak and Stempak [8, 9, 10] and the author [19, 20]. It is remarkable that similar methodshave been established recently in the contexts of Jacobi expansions [7] and Bessel operators [1]. ULTIPLIERS IN DUNKL AND LAGUERRE SETTINGS 3
The paper is organized as follows. Section 2 contains the setup, definitions of the investigatedmultipliers and statements of the main results. Also, suitable Laguerre-type operators, relatedto the restricted space ( R d + , dw + α ), are defined and the proof of the main theorem is reducedto showing that these auxiliary operators are Calder´on-Zygmund operators. Furthermore, weverify that the Laguerre-type operators are associated, in the Calder´on-Zygmund theory sense,with suitable integral kernels. The section is concluded by comments on the relation betweenthe Leguerre-type operators and the Laguerre setting studied in [8]. Finally, Section 3 is devotedto the proofs of the relevant kernel estimates.Throughout the paper we use a standard notation with essentially all symbols referring tothe spaces ( R d , dw α , | · | ) or ( R d + , dw + α , | · | ). Thus ∆ and ∇ denote the Euclidean Laplacianand 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 wemean R R d f ( x ) g ( x ) dw α ( x ) whenever the integral makes sense. In a similar way we define L p ( R d + , U 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 a positive constant C independent of significant quantities. We will write X ≃ Y whenboth X . Y and Y . X . Acknowledgments.
The author would like to thank Professor Krzysztof Stempak for sug-gesting the topic, Dr. Adam Nowak for valuable comments during preparation of the paper, andB la˙zej Wr´obel for sharing the manuscript [21] and discussions concerning the topic.2.
Preliminaries and main results
Let k = ( k , . . . , k d ) ∈ N d , N = { , , . . . } , and α = ( α , . . . , α d ) ∈ [ − / , ∞ ) d be multi-indices. The generalized Hermite functions in R d are defined as tensor products h αk ( x ) = h α k ( x ) · . . . · h α d k d ( x d ) , x = ( x , . . . , x d ) ∈ R d , where h α i k i are the one-dimensional generalized Hermite functions h α i k i ( x i ) = d k i ,α i e − x i / L α i k i ( x i ) ,h α i k i +1 ( x i ) = d k i +1 ,α i e − x i / x i L α i +1 k i ( x i );here L α i k i is the Laguerre polynomial of degree k i and order α i , and d n,α i , n ∈ N , are suitablenormalizing constants, see [9, p. 544] or [10, p. 4]. The system { h αk : k ∈ N d } is an orthonormalbasis in L ( R d , dw α ) consisting of eigenfunctions of L α , L α h αk = λ α | k | h αk , λ αn = 2 n + 2 | α | + 2 d, n ∈ N ;here | k | = k + . . . + k d is the length of k ; similarly | α | = α + . . . + α d . The operator L α f = ∞ X n =0 λ αn X | k | = n h f, h αk i dw α h αk , defined on the domain Dom( L α ) consisting of all functions f ∈ L ( R d , dw α ) for which thedefining series converges in L ( R d , dw α ), is a self-adjoint extension of L α considered on C ∞ c ( R d ).The heat semigroup associated with L α , defined by the spectral theorem as T αt f = exp( − t L α ) f = ∞ X n =0 e − tλ αn X | k | = n h f, h αk i dw α h αk , f ∈ L ( R d , dw α ) , MULTIPLIERS IN DUNKL AND LAGUERRE SETTINGS is a strongly continuous semigroup of contractions on L ( R d , dw α ). We have the integral repre-sentation 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 | k | = n h αk ( x ) h αk ( y ) . This oscillating series can be summed, see for instance [9, p. 544] or [10, p. 5], and the resultingformula is 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 considerthe functions z z ν and the Bessel function as analytic functions on C cut along the half-axis { ix : x ≤ } , see the references given above. Note that each G α,εt ( x, y ), ε ∈ Z d , is also expressedby the series (2.1), but with the summation in k restricted to the set N ε = (cid:8) k ∈ N d : k i is even if ε i = 0 , k i is odd if ε i = 1 , i = 1 , . . . , d (cid:9) . Definition 2.1.
Following E. M. Stein [15, p. 58, p. 121] we say that m is a multiplier of Laplace(transform) type associated with L α if m has the form m ( z ) = m η ( z ) = z Z ∞ e − tz η ( t ) dt, z ≥ λ α , (2.2) where η is a bounded measurable function on (0 , ∞ ) . We also consider multipliers of Laplace-Stieltjes type associated with L α (see [21, Section 2] and comments therein) having the form m ( z ) = m µ ( z ) = Z ∞ e − tz dµ ( t ) , z ≥ λ α , (2.3) where µ is a signed or complex Borel measure on (0 , ∞ ) with total variation | µ | satisfying Z ∞ e − tλ α d | µ | ( t ) < ∞ . (2.4)We are now prepared to define the main objects of our study. Given a Laplace type orLaplace-Stieltjes type multiplier m , we consider the multiplier operators m ( L α ) f = ∞ X n =0 m ( λ αn ) X | k | = n h f, h αk i dw α h αk , f ∈ L ( R d , dw α ) . (2.5)It is not hard to see that m defined either by (2.2) or by (2.3) is a bounded function on theinterval [ λ α , ∞ ). Since spec( L α ) ⊂ [ λ α , ∞ ), the operator m ( L α ) is a well-defined boundedoperator on L ( R d , dw α ).To state and prove our main result it is convenient to introduce 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 iseither even or odd with respect to the j th coordinate according to whether ε j = 0 or ε j = 1, ULTIPLIERS IN DUNKL AND LAGUERRE SETTINGS 5 respectively. If f is (0 , . . . , f is symmetric. Further, ifthere exists ε ∈ Z d such that 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 .The main result of the paper reads as follows. Theorem 2.2.
Assume that α ∈ [ − / , ∞ ) d , W is a weight on R d invariant under the reflections σ , . . . , σ d , and m is as in (2.2) or (2.3) . Then the multiplier operator m ( L α ) , defined initiallyon L ( R d , dw α ) by (2.5) , extends uniquely to a bounded linear operator on L p ( R d , W dw α ) , W + ∈ A α, + p , < p < ∞ , and to a bounded linear operator from L ( R d , W dw α ) to weak L ( R d , W dw α ) , W + ∈ A α, +1 . The proof we give relies on reducing the problem to showing similar mapping properties forcertain operators emerging from m ( L α ) and related to the restricted space ( R d + , dw + α ). Thedetails are as follows. Let 1 ≤ p < ∞ be fixed and let W be a symmetric weight on R d suchthat W + ∈ A α, + p . We decompose m ( L α ) into a finite sum of L -bounded operators m ( L α ) = X ε ∈ Z d m ε ( L α ) , where m ε ( L α ) = ∞ X n =0 m ( λ αn ) X | k | = nk ∈N ε h f, h αk i dw α h αk , f ∈ L ( R d , dw α ) . Then we proceed as in [10, Section 3], see also [20, Section 2]. We split a function f ∈ L ( R d , dw α )into a finite sum of ε -symmetric functions f ε , f = X ε ∈ Z d f ε . We also introduce the Laguerre-type operators acting on L ( R d + , dw + α ), m ε, + ( L α ) f = ∞ X n =0 m ( λ αn ) X | k | = nk ∈N ε h f, h αk i dw + α h αk . (2.6)Since h αk is ε -symmetric if and only if k ∈ N ε , we see that m ( L α ) f = X ε ∈ Z d m ε ( L α ) f ε , f ∈ L ( R d , dw α ) . Taking into account the fact that m ε ( L α ) f ε is ε -symmetric, and the relation h f ε , h αk i dw α =2 d h f + ε , h αk i dw + α for k ∈ N ε , we obtain k m ( L α ) f k L p ( R d ,W dw α ) ≤ d/p X ε ∈ Z d k m ε ( L α ) f ε k L p ( R d + ,W + dw + α ) = 2 d + d/p X ε ∈ Z d k m ε, + ( L α ) f + ε k L p ( R d + ,W + dw + α ) , for f ∈ L ( R d , dw α ) ∩ L p ( R d , W dw α ). 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 ) , MULTIPLIERS IN DUNKL AND LAGUERRE SETTINGS the above estimates together with the bounds k m ε, + ( L α ) 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 m ( L α ) f k L p ( R d ,W dw α ) . k f k L p ( R d ,W dw α ) . An analogous implication involving weighted weak type (1 ,
1) inequalities is also valid. Thus wereduced proving Theorem 2.2 to showing the following.
Theorem 2.3.
Assume that α ∈ [ − / , ∞ ) d , ε ∈ Z d and m is of the form (2.2) or (2.3) .Then the Laguerre-type operators m ε, + ( L α ) , defined initially on L ( R d + , dw + α ) by (2.6) , extenduniquely to bounded linear operators on L p ( R d + , U dw + α ) , U ∈ A α, + p , < p < ∞ , and to boundedlinear operators from L ( R d + , U dw + α ) to weak L ( R d + , U dw + α ) , U ∈ A α, +1 . The proof of Theorem 2.3 will be obtained by means of the general Calder´on-Zygmund theory.More precisely, we will show that each of the operators m ε, + ( L α ), ε ∈ Z d , is a Calder´on-Zygmundoperator in the sense of the space of homogeneous type ( R d + , dw + α , | · | ). It is well known that theclassical Calder´on-Zygmund theory works, with appropriate adjustments, when the underlyingspace is of homogeneous type; see for instance the comments and references in [8, p. 649].The following result combined with the Calder´on-Zygmund theory implies Theorem 2.3, andthus also Theorem 2.2. The corresponding proof splits naturally into proving Proposition 2.5and Theorem 2.6 below. Theorem 2.4.
Assume that α ∈ [ − / , ∞ ) d and ε ∈ Z d . Then the Laguerre-type operatorsfrom Theorem 2.3 are Calder´on-Zygmund operators in the sense of the space of homogeneoustype ( R d + , dw + α , | · | ) . Formal computations, similar to those from [21, Section 2], suggest that m ε, + ( L α ), ε ∈ Z d ,where m is either as in (2.2) or as in (2.3), are associated with the kernels K α,ε ( x, y ) = K α,εη ( x, y ) = − Z ∞ ∂ t G α,εt ( x, y ) η ( t ) dt, ε ∈ Z d , (2.7) K α,ε ( x, y ) = K α,εµ ( x, y ) = Z ∞ G α,εt ( x, y ) dµ ( t ) , ε ∈ Z d , (2.8)respectively. The next result shows that this is indeed the case, at least in the Calder´on-Zygmundtheory sense. Proposition 2.5.
Let α ∈ [ − / , ∞ ) d and ε ∈ Z d . (a) If m = m η is a Laplace type multiplier, then m ε, + ( L α ) is associated with the kernel K α,ε ( x, y ) in the sense that for any f, g ∈ C ∞ c ( R d + ) with disjoint supports h m ε, + ( L α ) f, g i dw + α = Z R d + Z R d + K α,ε ( x, y ) f ( y ) dw + α ( y ) g ( x ) dw + α ( x ) . (2.9)(b) If m = m µ is a Laplace-Stieltjes type multiplier, then m ε, + ( L α ) is associated with thekernel K α,ε ( x, y ) in the sense that for any f, g ∈ C ∞ c ( R d + ) with disjoint supports h m ε, + ( L α ) f, g i dw + α = Z R d + Z R d + K α,ε ( x, y ) f ( y ) dw + α ( y ) g ( x ) dw + α ( x ) . (2.10) ULTIPLIERS IN DUNKL AND LAGUERRE SETTINGS 7
Proof.
Taking into account that for each ε ∈ Z d the system { d/ h αk : k ∈ N ε } is an orthonormalbasis in L ( R d + , dw + α ), by (2.6) and Parseval’s identity we see that h m ε, + ( L α ) f, g i dw + α = ∞ X n =0 m ( λ αn ) X | k | = nk ∈N ε h f, h αk i dw + α h h αk , g i dw + α . (2.11)To finish the proof it suffices to show that the right-hand side of (2.11) coincides with the right-hand side of (2.9) or (2.10), according to whether m is as in (2.2) or (2.3), respectively. Thistask reduces to justifying the possibility of changing the order of integration, summation anddifferentiation in the relevant expressions. Then the arguments are similar as for other operators,see for instance [8, Proposition 3.3]. The crucial facts are the estimate (cf. [20, (2.3)]), | h αk ( x ) | . ( | k | + 1) c d,α , k ∈ N d , x ∈ R d + , and the bounds Z ∞ (cid:12)(cid:12) ∂ t G α,εt ( x, y ) η ( t ) (cid:12)(cid:12) dt . w + α ( B ( x, | y − x | )) , x, y ∈ R d + , x = y, Z ∞ G α,εt ( x, y ) d | µ | ( t ) . w + α ( B ( x, | y − x | )) , x, y ∈ R d + , x = y, which are verified in the proof of Theorem 2.6. We leave further details to the reader. (cid:3) Let B ( x, r ) denote the ball centered at x and of radius r , restricted to R d + . Theorem 2.6.
Assume that α ∈ [ − / , ∞ ) d and ε ∈ Z d . (a) The kernel K α,ε ( x, y ) satisfies the growth estimate | K α,ε ( x, y ) | . w + α ( B ( x, | y − x | )) , x, y ∈ R d + , x = y, and the gradient condition |∇ x,y K α,ε ( x, y ) | . | x − y | w + α ( B ( x, | y − x | )) , x, y ∈ R d + , x = y. (b) Analogous estimates hold for K α,ε ( x, y ) . The proof of Theorem 2.6 is the most technical part of the paper and is located in Section 3.We conclude this section with various comments and remarks related to the main result. First,note that our methods allow also to treat Laplace and Laplace-Stieltjes type multipliers relatedto the square root of L α . Indeed, consider the multiplier operator m ( p L α ) f = ∞ X n =0 m ( p λ αn ) X | k | = n h f, h αk i dw α h αk , f ∈ L ( R d , dw α ) , (2.12)and for ε ∈ Z d the Poisson-Laguerre-type operators m ε, + ( p L α ) f = ∞ X n =0 m ( p λ αn ) X | k | = nk ∈N ε h f, h αk i dw + α h αk , f ∈ L ( R d + , dw + α ) , (2.13)where m is as in Definition 2.1 with λ α replaced by p λ α . Then the operators m ( √L α ) and m ε, + ( √L α ) are well defined and bounded on L ( R d , dw α ) and L ( R d + , dw + α ), respectively. The MULTIPLIERS IN DUNKL AND LAGUERRE SETTINGS integral kernels associated with m ε, + ( √L α ) have analogous forms to (2.7) and (2.8), where G α,εt ( x, y ) should be replaced by the subordinated kernel P α,εt ( x, y ) = Z ∞ G α,εt / (4 u ) ( x, y ) e − u du √ πu . Proceeding similarly as in the case of m ( L α ), with only slightly more effort we obtain thefollowing result. Theorem 2.7.
Assume that α ∈ [ − / , ∞ ) d and ε ∈ Z d . Then the Poisson-Laguerre-typeoperators m ε, + ( √L α ) , ε ∈ Z d , defined by (2.13) , are Calder´on-Zygmund operators in the senseof the space of homogeneous type ( R d + , dw + α , | · | ) . Consequently, each of these operators, definedinitially on L ( R d + , dw + α ) , extends uniquely to a bounded linear operator on L p ( R d + , U dw + α ) , U ∈ A α, + p , < p < ∞ , and to a bounded linear operator from L ( R d + , U dw + α ) to weak L ( R d + , U dw + α ) , U ∈ A α, +1 . Corollary 2.8.
Assume that α ∈ [ − / , ∞ ) d and W is a weight on R d invariant under thereflections σ , . . . , σ d . Then the multiplier operator m ( √L α ) , defined initially on L ( R d , dw α ) by (2.12) , extends uniquely to a bounded linear operator on L p ( R d , W dw α ) , W + ∈ A α, + p , < p < ∞ ,and to a bounded linear operator from L ( R d , W dw α ) to weak L ( R d , W dw α ) , W + ∈ A α, +1 . Next, we comment on the relation between the auxiliary operators m ε, + ( L α ), ε ∈ Z d , andthe Laguerre setting from [8]. Note that for the particular ε = (0 , . . . ,
0) the Laguerre-typeoperator m ε , + ( L α ) coincides, up to the factor 2 − d , with the multiplier operator m ( L ℓα ) relatedto the Laguerre Laplacian L ℓα considered in [8] and [19]. More precisely, m ( L ℓα ) f = ∞ X n =0 m ( λ α n ) X | k | = n h f, ℓ αk i dµ α ℓ αk , f ∈ L ( R d + , dµ α ) , (2.14)where ℓ αk are the Laguerre functions of convolution type and µ α ≡ w + α . Therefore the results ofthis section deliver also analogous results in the Laguerre setting of convolution type; see [20,Section 2] for further explanations concerning the connection between the Dunkl and Laguerresettings. Theorem 2.9.
Assume that α ∈ [ − / , ∞ ) d and m is as in (2.2) or (2.3) . Then the Laguerremultiplier operator m ( L ℓα ) , defined initially on L ( R d + , dµ α ) by (2.14) , is a Calder´on-Zygmundoperator in the sense of the space of homogeneous type ( R d + , dµ α , | · | ) . Consequently, m ( L ℓα ) extends uniquely to a bounded linear operator on L p ( R d + , U dµ α ) , U ∈ A α, + p , < p < ∞ , and toa bounded linear operator from L ( R d + , U dµ α ) to weak L ( R d + , U dµ α ) , U ∈ A α, +1 . A similar result holds for Laplace and Laplace-Stieltjes type multipliers related to the squareroot of L ℓα , see Theorem 2.7 and Corollary 2.8 above.3. Kernel estimates
This section delivers proofs of the relevant kernel estimates. We use the technique developedby Nowak and Stempak [8, 9, 10], which is based on Schl¨afli’s integral representation for themodified Bessel function I ν involved in the Dunkl heat kernel. This method was refined by theauthor in [19] and [20] to obtain standard estimates for various kernels related to L ℓα and L α .Below, we will frequently invoke estimates obtained in the latter paper. Recall that we alwaysassume that α ∈ [ − / , ∞ ) d . ULTIPLIERS IN DUNKL AND LAGUERRE SETTINGS 9
Given ε ∈ Z d , the ε -component of the Dunkl heat kernel is given by, see [9, Section 5], 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 ) , (3.1)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(3.2) t = t ( ζ ) = 12 log 1 + ζ − ζ . The measure Π β appearing in (3.1) is a product of one-dimensional measures, Π β = N di =1 Π β i ,where Π β i is given by the densityΠ 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 Π − / = (cid:0) δ − + δ (cid:1) / √ π , with δ − and δ denoting the point masses at − G α,εt ( x, y ) we will need several auxiliary results. In par-ticular, the following modification of [16, Lemma 1.1] will be useful. Lemma 3.1.
Given a > , we have Z (1 − ζ ) − / ζ − a exp( − T ζ − ) dζ . T − a +1 , T > . Proof.
We split the region of integration onto (0 , /
2) and (1 / , ζ ∈ (0 , /
2) we have(1 − ζ ) − / ≃
1, so the bound for the integral over (0 , /
2) is a straightforward consequence of[16, Lemma 1.1]. It remains to estimate the integral over (1 / , ζ ≃ u ≥ u a − e − u < ∞ , we see that ζ − a exp( − T ζ − ) . T − a +1 ( T ζ − ) a − exp( − T ζ − ) . T − a +1 , ζ ∈ (1 / , , T > . Now the conclusion follows because R / (1 − ζ ) − / dζ < ∞ . (cid:3) The lemma below establishes an important connection between the estimates emerging fromthe representation (3.1) and the standard estimates related to the space ( R d + , dw + α , | · | ). Lemma 3.2. ([9, Lemma 5.3] , [10, Lemma 4]) Assume that α ∈ [ − / , ∞ ) d and let δ, κ ∈ [0 , ∞ ) d be fixed. 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 | )) . To state the next lemma and to perform the relevant kernel estimates we will use the sameabbreviation as in [20], E xp( ζ, q ± ) = exp (cid:16) − ζ q + ( x, y, s ) − ζ q − ( x, y, s ) (cid:17) . Also, we will often neglect the set of integration [ − , d in integrals against Π α and will fre-quently write shortly q + and q − omitting the arguments. Lemma 3.3.
Assume that α ∈ [ − / , ∞ ) d and ε, ξ, ρ ∈ Z d are fixed and such that ξ ≤ ε , ρ ≤ ε .Given C > and u ≥ , define the function p u acting on R d + × R d + × (0 , 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 ) . Then p u satisfies the integral 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 (3.2) .Proof. Changing the variable as in (3.2) and then using sequently the Fubini-Tonelli theorem,Lemma 3.1 (specified to a = d + | α | + | ε |−| ξ | / −| ρ | / u/ /
2) and the inequality | x − y | ≤ q + ,we obtain k p u ( x, y, ζ ( t )) k L ( dt ) = x ε − ξ y ε − ρ Z Z (1 − ζ ) − / ζ − d −| α |−| ε | + | ξ | / | ρ | / − u/ − / (cid:0) E xp( ζ, q ± ) (cid:1) C dζ Π α + ε ( ds ) . x ε − ξ y ε − ρ Z ( q + ) − d −| α |−| ε | + | ξ | / | ρ | / − u/ / Π α + ε ( ds ) ≤ ( x + y ) ε − ξ − ρ | x − y | u − Z ( q + ) − d −| α |−| ε − ξ/ − ρ/ | Π α + ε ( ds ) . This, in view of Lemma 3.2 (taken with δ = ε − ξ/ − ρ/ κ = ξ/ ρ/ (cid:3) Lemma 3.4. ([19, Lemma 4.2])
Given b ≥ and c > , we have (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) , A > , uniformly in q ± .Proof of Theorem 2.6 (a). The growth condition is a consequence of the estimate (cid:12)(cid:12) ∂ t G α,εt ( x, y ) (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 ) , which is stated explicitly in [20, (4.6)], the fact that η is bounded and Lemma 3.3 (applied with u = 1, ξ = ρ = 0).To prove the gradient condition, by symmetry reasons, it suffices to show that (cid:12)(cid:12) ∂ x j K α,ε ( x, y ) (cid:12)(cid:12) . | x − y | w + α ( B ( x, | y − x | )) , x = y, j = 1 , . . . , d. ULTIPLIERS IN DUNKL AND LAGUERRE SETTINGS 11
Differentiating (2.7) in x j (passing with ∂ x j under the integral sign can be easily justified, see[19, Section 4.1]) we get ∂ x j K α,ε ( x, y ) = − Z ∞ ∂ x j ∂ t G α,εt ( x, y ) η ( t ) dt, x = y. Applying now the first inequality in [20, (4.7)], (cid:12)(cid:12) ∂ x j ∂ t G α,εt ( x, y ) (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 )+ χ { ε j =1 } p − ζ ζ − d −| α |−| ε |− x ε − e j y ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds ) , the fact that η is bounded and Lemma 3.3 twice (specified to u = 2 and either ξ = ρ = 0 or ξ = e j , ρ = 0) leads directly to the required bound.The proof of (a) in Theorem 2.6 is finished. (cid:3) Proof of Theorem 2.6 (b).
We first verify the growth condition. Using Lemma 3.4 (taken with b = d + | α | + | ε | , c = 1 / A = ζ − ) we get e t (2 d +2 | α | ) G α,εt ( x, y ) ≤ (cid:16) ζ − ζ (cid:17) d + | α | (cid:16) − ζ ζ (cid:17) d + | α | + | ε | ( xy ) ε Z E xp( ζ, q ± ) Π α + ε ( ds ) . ζ − d −| α |−| ε | ( xy ) ε Z E xp( ζ, q ± ) Π α + ε ( ds ) . ( x + y ) ε Z ( q + ) − d −| α |−| ε | Π α + ε ( ds ) . Now the conclusion follows with the aid of Lemma 3.2 (specified to δ = ε and κ = 0) and theassumption (2.4) concerning the measure µ .Next, our task is to show the gradient condition. By symmetry reasons, it suffices to provethat (cid:12)(cid:12) ∂ x j K α,ε ( x, y ) (cid:12)(cid:12) . | x − y | w + α ( B ( x, | y − x | )) , x = y, j = 1 , . . . , d. Differentiating (2.8) in x j (exchanging ∂ x j with the integral sign is legitimate, which can bejustified by a slightly modified version of [20, Lemma 4.7], see for instance [19, Section 4.3]) weget ∂ x j K α,ε ( x, y ) = Z ∞ ∂ x j G α,εt ( x, y ) dµ ( t ) , x = y. Using the estimate (cid:12)(cid:12) ∂ x j G α,εt ( x, y ) (cid:12)(cid:12) . (cid:16) − ζ ζ (cid:17) d + | α | + | ε | ζ − / ( xy ) ε Z (cid:0) E xp( ζ, q ± ) (cid:1) / Π α + ε ( ds )+ χ { ε j =1 } (cid:16) − ζ ζ (cid:17) d + | α | + | ε | x ε − e j y ε Z E xp( ζ, q ± ) Π α + ε ( ds ) , which is implicitly contained in [20, Section 4.2], and then proceeding in a similar way as in thecase of the growth condition, applying this time Lemma 3.4 twice (with b = d + | α | + | ε | + 1 / c = 1 / A = ζ − and b = d + | α | + | ε | , c = 1 / A = ζ − ), we obtain e t (2 d +2 | α | ) (cid:12)(cid:12) ∂ x j G α,εt ( x, y ) (cid:12)(cid:12) . ( x + y ) ε Z ( q + ) − d −| α |−| ε |− / Π α + ε ( ds )+ χ { ε j =1 } ( x + y ) ε − e j Z ( q + ) − d −| α |−| ε − e j / |− / Π α + ε ( ds ) . Taking into account (2.4), this estimate together with Lemma 3.2 (specified to δ = ε , κ = 0 and δ = ε − e j / κ = e j /
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