Abstract
The inversion of Riesz potentials for Dunkl transform when is given by using the generalized wavelet transforms. It is also proved that the Riesz potentials are automorphisms on the Semyanistyi-Lizorkin spaces.
1. Introduction
Dunkl transform is a generalization of the Fourier transform associated with a family of weight functions, , invariant under a finite reflection group. Many papers devote to study the Dunkl transform; see [1–6] and the references therein.
In [7], the Riesz potentials for Dunkl transform were defined by the generalized translation operators, . The explicit expression and boundedness of are known only in some special cases such as when and the case when the kernel is a suitable radial function. The boundedness of was given only in the two cases mentioned above. Gorbachev et al. [8] studied the weighted ()-boundedness properties of Riesz potentials for Dunkl transform represented by the Stein-Weiss inequality. In this paper, we will study the inversion of in the case when . The paper is organized as follows. In Section 2, some necessary facts in Dunkl’s theory are reviewed. Section 3 is devoted to introduce the Semyanistyi-Lizorkin spaces associated with the reflection-invariant measure . In the final section, the inversion of the Riesz potentials will be given by the generalized wavelet transforms defined by the generalized translation operators.
2. Preliminaries
2.1. Dunkl Operator and Dunkl Transform
Let be a finite reflection group on with a fixed positive root system , normalized so that for all , where denotes the usual Euclidean inner product. Let be a nonnegative multiplicity function defined on with the property that whenever is conjugate to in ; then is a -invariant function. The weight function is positive homogeneous of degree defined by Note that is invariant under the reflection group .
Let be Dunkl’s differential-difference operators defined in [1] as where are the standard unit vectors of and denotes the reflection with respect to the hyperplane perpendicular to , , . The operators , , map to , where denotes the space of homogeneous polynomials of degree in variables, and they mutually commute; that is, , . For example, when , the Dunkl operator is
The intertwining operator is a linear operator determined uniquely by
Let , , where the superscript means that is applied to the variable. For , the Dunkl transform is defined by where is the constant defined by .
Define , and, for the sake of simplicity, set whenever the integral exists and denote .
The Dunkl transform shares many of the important properties with the usual Fourier transform, part of which are listed as follows ([2, 3]).
Proposition 1. (i) If , then and .
(ii) If , then .
(iii) The Dunkl transform is a topological automorphism on .
(iv) For , then and , where , .
(v) For all , we have .
(vi) There exists a unique extension of the Dunkl transform to with .
2.2. Generalized Translation Operator and Generalized Convolution
Let be given. The generalized translation operator is defined on by , .
For , the generalized convolution operator is defined bywhere . The main properties of the generalized translation operator and the generalized convolution are collected below [6, 9, 10].
Proposition 2. (i) For and being bounded, then (ii) For , When ,
(iii) for , ,
(iv) for , we have and
(v) let and . For , , .
2.3. Dunkl Transform of Distributions
References [5, 11, 12] study the actions of the Dunkl operators and Dunkl transform on the space . Reference [4] gives the definition of the Dunkl transform for the local integrable functions under the measure .
Let ; the generalized function associated with is defined by The Dunkl transform of is defined asThen, for , the Dunkl transform of is For , the dilation transform is defined as Let be the Dirac distribution associated with the measure ; that is, , .
Lemma 3. Let satisfy . For any , define ; then is a -sequence; that is, Then by (9) and (10), we obtain the Dunkl transform of as .
Define the action of the Dunkl operators , on the space asDenote and . Combining (9), (14), and Proposition 1 (iv), we have
2.4. Dunkl Riesz Potentials
For simplicity, we call the Riesz potential for Dunkl transform as the Dunkl Riesz potential , which is defined on in [7] aswhere and . The Dunkl transform and the Hardy-Littlewood-Sobolev theorem of are given in [7].
Proposition 4. Let . The identityholds in the sense that whenever .
Proposition 5. Let and . Let and satisfy (1)For , , .(2)For , the mapping is of weak type .
3. Semyanistyi-Lizorkin Spaces
The following spaces were introduced by Semyanistyi and generalized by Lizorkin and Samko; see [13–15].
Let be the class of functions in vanishing at the origin 0 with all their derivatives; that is,The space is a closed linear subspace of . It can be regarded as a linear topological space with the induced topology generated by the sequence of norms We claim that for , since implies for .
Let be the image of under the Dunkl transform; that is, . Since the Dunkl transform is an automorphism of , the space is a closed linear subspace of . We equip with the induced topology of the ambient space . Then becomes a linear topology space which is isomorphic to under the action of the Dunkl transform. According to the definition of , we conclude that the space consists of all functions which are orthogonal to all polynomials as for the measure ; that is, In fact, if , then , and for any multi-index , by Proposition 1 (iv), we have
Denote that and are the spaces of all semilinear functionals on and , respectively. Some properties of and are given in the following proposition.
Proposition 6. (i) The spaces and are not empty.
(ii) The space does not contain compactly supported infinitely differentiable functions, rather than 0.
(iii) The space is invariant under the generalized translations.
(iv) The space is dense in , .
(v) -distributions that coincide in the -sense differ from each other by a polynomial.
(vi) Let and , . If in the -sense, then almost everywhere.
The proof of the this proposition is similar to the ones in [13–15] except with the reflection-invariant measure . Now we sketch it below.
Proof. (i) Choose satisfying that . Then .
(ii) Suppose ; then we have If , , and , so for all . Then .
(iii) This conclusion can be obtained by Proposition 2 (iii).
(iv) It suffices to approximate a function by functions in the norm. We introduce the functions where such that for , for and . We define . Since and as , we have and then . In order to show that approximate the function , we represent them as where is the inverse Dunkl transform of the function . Then by Lemma 7, as .
(v) Suppose and in the sense of ; that is, for all , . Then, for all , This means that , which implies that is a finite linear combination of the derivatives of the delta function. Hence, by (15), is a polynomial.
(vi) For , denote and as the distributions of and , respectively. Then Then is finite since . The same argument gives that is finite as well. We claim that almost everywhere. In fact, by (v), where is a polynomial. Then, for all ,Thus a.e. So we have a.e. consequently.
Lemma 7. Let and , . Define Then, as .
Proof. According to the definition, If , Then as by Lebesgue’s dominated theorem.
When , by Proposition 2 (v),It suffices to verify for . Let be any number such that . By Hölder’s inequality, we have where . Then, by (33) tends to 0 as .
Theorem 8. Let . The operators are the automorphisms on the space .
Proof. According to the last equation in the proof for Proposition 4.1 in [7], for , Since belongs to and Dunkl transform maps isomorphically onto , it follows that the map is continuous from to itself. Owing to (36), is a linear continuous operator from to . Conversely, we claim that is surjective. In fact, for , let . Then and . Furthermore, the map is continuous in the topology of the space . This completes the proof.
4. The Inversion of
Now we can give the main result of this paper, the inversion of the Dunkl Riesz potentials when the group . The method follows the idea in [16]. Rubin [17] gave simpler proofs to some elementary approximate and explicit inversion formulae for the classical Riesz potentials.
Theorem 9. Let , , . Suppose that is a bounded radial function in and the integral is finite. Then where
Proof. DenoteWe claim that the operator is bounded on for any . Indeed, according to Proposition 2 (iv), we have Then the generalized Minkowski’s inequality gives thatIf we can show thatwhere denotes the inverse Dunkl transform of , then we have tending to 0 as , by Lebesgue’s theorem on dominated convergence.
Recall that . Let first , . Then Proposition 4 gives that when . And therefore, by replacing with in (42).
On the other hand, . According to Proposition 6 (vii), it suffices to prove (43) in the -sense; that is,After changing the order of the integration, the left-hand side of (45) equals Since the Dunkl transform of is , then Since is radial, then Combining the above, we obtain (45) as desired.
When , , the argument is the same as the first case, except with .
For the case , and . By interpolation, for all . Choosing in the interval and in the Hardy-Littlewood-Sobolev theorem, we can get the result by repeating the argument when . Thus we finish the proof.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by National Natural Science Foundation of China (no. 11275240).