The inversion of Riesz potentials for Dunkl transform when G=Z2d is given by using the generalized wavelet transforms. It is also proved that the Riesz potentials Iακ are automorphisms on the Semyanistyi-Lizorkin spaces.

National Natural Science Foundation of China112752401. Introduction

Dunkl transform is a generalization of the Fourier transform associated with a family of weight functions, hκ, 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 Iακ for Dunkl transform were defined by the generalized translation operators, τy. The explicit expression and boundedness of τy are known only in some special cases such as when G=Z2d and the case when the kernel is a suitable radial function. The boundedness of Iακ was given only in the two cases mentioned above. Gorbachev et al. [8] studied the weighted (Lp,Lq)-boundedness properties of Riesz potentials for Dunkl transform represented by the Stein-Weiss inequality. In this paper, we will study the inversion of Iακ in the case when G=Z2d. 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 hκ2(x)dx. In the final section, the inversion of the Riesz potentials Iακ will be given by the generalized wavelet transforms defined by the generalized translation operators.

2. Preliminaries2.1. Dunkl Operator and Dunkl Transform

Let G be a finite reflection group on Rd with a fixed positive root system R+, normalized so that 〈v,v〉=2 for all v∈R+, where 〈x,y〉 denotes the usual Euclidean inner product. Let κ be a nonnegative multiplicity function defined on R+ with the property that κu=κv whenever σu is conjugate to σv in G; then v⟼κv is a G-invariant function. The weight function is positive homogeneous of degree γκ:=∑v∈R+κv defined by (1)hκx=∏v∈R+x,vκv,x∈Rd.Note that hκ is invariant under the reflection group G.

Let Dj be Dunkl’s differential-difference operators defined in [1] as (2)Djfx=∂jfx+∑v∈R+κvfx-fxσvx,vv,εj,1≤j≤d,where ε1,ε2,…,εd are the standard unit vectors of Rd and σv denotes the reflection with respect to the hyperplane perpendicular to v, xσv:=x-2(〈x,v〉/v2)v, x∈Rd. The operators Dj, 1≤j≤d, map Pnd to Pn-1d, where Pnd denotes the space of homogeneous polynomials of degree n in d variables, and they mutually commute; that is, DiDj=DjDi, 1≤i,j≤d. For example, when d=1, the Dunkl operator is (3)Dfx=f′x+κ+12fx-f-xx.

The intertwining operator Vκ is a linear operator determined uniquely by (4)VκPnd⊂Pnd,Vκ1=1,DiVκ=Vκ∂i,1≤i≤d.

Let E(x,iy)=Vκ(x)[ei〈x,y〉], x,y∈Rd, where the superscript means that Vκ is applied to the x variable. For f∈L1(Rd,hκ2), the Dunkl transform is defined by (5)f^y=ch∫RdfxEx,-iyhκ2xdxwhere ch is the constant defined by ch-1=∫Rdhκ2(x)e-∥x∥2/2dx.

Define Aκ(Rd)={f∈L1(Rd,hκ2):f^∈L1(Rd,hκ2)}, and, for the sake of simplicity, set 〈f,g〉κ=∫Rdf(x)g(x)hκ2(x)dx whenever the integral exists and denote λκ=2γκ+d.

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 f∈L1(Rd,hκ2), then f^∈C0(Rd) and f^∞≤fκ,1.

(ii) If f∈Aκ(Rd), then f(x)=f^^(-x).

(iii) The Dunkl transform f⟶f^ is a topological automorphism on S(Rd).

(iv) For f∈S(Rd), then Djf^(ξ)=iMjf^(ξ) and Mjf^(ξ)=iDjf^(ξ), where Mjf(x)=xjf(x), j=1,2,…,d.

(v) For all f,g∈L1(Rd,hκ2), we have 〈f^,g〉κ=〈f,g^〉κ.

(vi) There exists a unique extension of the Dunkl transform to L2(Rd,hκ2) with f^κ,2=fκ,2.

2.2. Generalized Translation Operator and Generalized Convolution

Let y∈Rd be given. The generalized translation operator f⟼τyf is defined on L2(Rd,hκ2) by τyf^(x)=E(y,-ix)f^(x), x∈Rd.

For f,g∈L2(Rd,hκ2), the generalized convolution operator is defined by(6)f∗κgx=∫Rdfyτxg~yhκ2ydy,where g~(y)=g(-y). The main properties of the generalized translation operator and the generalized convolution are collected below [6, 9, 10].

Proposition 2.

(i) For f∈Aκ(Rd) and g∈L1(Rd,hκ2) being bounded, then (7)∫Rdτyfξgξhκ2ξdξ=∫Rdfξτ-ygξhκ2ξdξ.

(ii) For f∈S(Rd), (8)τyf^ξ=Eξ,iyf^ξ,E·,iyf·^x=τ-yf^x.When G=Z2d,

(iii) for f∈Lp(Rd,hκ2), 1≤p≤∞, τyfκ,p≤fκ,p

(iv) for f,g∈Lp(Rd,hκ2), we have f∗κg^=f^·g^ and f∗κg=g∗κf

(v) let p,q,r≥1 and 1/r=1/p+1/q-1. For f∈Lp(Rd,hκ2), g∈Lq(Rd,hκ2), f∗κgκ,r≤cfκ,pgκ,q.

2.3. Dunkl Transform of Distributions

References [5, 11, 12] study the actions of the Dunkl operators and Dunkl transform on the space S′(Rd). Reference [4] gives the definition of the Dunkl transform for the local integrable functions under the measure hκ2(x)dx.

Let f∈Lloc1(Rd,hκ2); the generalized function associated with f is defined by(9)f,φκ=∫Rdfxφxhκ2xdx,∀φ∈SRd. The Dunkl transform of f∈S′(Rd) is defined as(10)f^,φκ=f,φ^κ,∀φ∈SRd.Then, for f∈Lloc1(Rd,hκ), the Dunkl transform of f is (11)f^,φκ=∫Rdfxφ^xhκ2xdx,∀φ∈SRd.For f∈S′(Rd), the dilation transform εf is defined as (12)εf,φκ=f,ε-λκφε-1·κ,∀φ∈SRd.Let δ be the Dirac distribution associated with the measure hκ2(x)dx; that is, 〈δ,φ〉κ=φ(0), φ∈S(Rd).

Lemma 3.

Let φ∈L1(Rd,hκ2) satisfy ∫Rdφ(x)hκ2(x)dx=1. For any ε>0, define ϕϵ(x)=ε-λκϕ(x/ε); then ϕε is a δ-sequence; that is, (13)limε→0+ϕϵ,φκ=φ0=δ,φκ,∀φ∈SRd.Then by (9) and (10), we obtain the Dunkl transform of δ as δ^(ξ)=1.

Define the action of the Dunkl operators Dj, j=1,2,…,d on the space S′(Rd) as(14)Djf,φκ=-f,Djφκ,∀φ∈SRd.Denote β=(β1,β2,…,βd)∈Z+d and Dβ=D1β1∘D2β2∘⋯∘Ddβd. Combining (9), (14), and Proposition 1 (iv), we have(15)Dβδ^ξ=iξβ=iβξ1β1ξ2β2⋯ξdβd,xβ^=iβDβδ.

2.4. Dunkl Riesz Potentials

For simplicity, we call the Riesz potential for Dunkl transform as the Dunkl Riesz potential Iακf, which is defined on S(Rd) in [7] as(16)Iακfx=dακ-1∫Rdτyfx1yλκ-αhκ2ydy,where 0<α<λκ and dακ=2-λκ/2+αΓ(α/2)Γ((λκ-α)/2). The Dunkl transform and the Hardy-Littlewood-Sobolev theorem of Iακ are given in [7].

Proposition 4.

Let 0<α<λκ. The identity(17)Iακf^x=x-αf^xholds in the sense that (18)∫RdIακfxgxhκ2xdx=∫Rdf^xx-αg^xhκ2xdxwhenever f,g∈S(Rd).

Proposition 5.

Let G=Z2d and 0<α<λκ. Let p and q satisfy (19)1p-1q=αλκ,1≤p<q<∞.

For f∈Lp(Rd,hκ2), p>1, Iακfκ,q≤cfκ,p.

For f∈L1(Rd,hκ2), the mapping f⟼Iακf is of weak type (1,q).

3. Semyanistyi-Lizorkin Spaces

The following spaces were introduced by Semyanistyi and generalized by Lizorkin and Samko; see [13–15].

Let Ψ=Ψ(Rd) be the class of functions ψ in S(Rd) vanishing at the origin 0 with all their derivatives; that is,(20)Ψ=ψ∈SRd:∂βψ0=0 forallβ=0,1,2,….The space Ψ is a closed linear subspace of S(Rd). It can be regarded as a linear topological space with the induced topology generated by the sequence of norms (21)ψm=maxx1+xm∑j≤m∂βψx,m=0,1,2,….We claim that Dβψ(0)=0 for ψ∈Ψ, since ∂jψ(0)=0 implies Djψ(0)=0 for j=1,2,…,d.

Let Φ=Φ(Rd) be the image of Ψ under the Dunkl transform; that is, Φ={ψ∈Ψ:ψ^}. Since the Dunkl transform is an automorphism of S(Rd), the space is a closed linear subspace of S(Rd). We equip Φ with the induced topology of the ambient space S(Rd). Then Φ becomes a linear topology space which is isomorphic to Ψ under the action of the Dunkl transform. According to the definition of Φ, we conclude that the space Φ consists of all functions φ which are orthogonal to all polynomials as for the measure hκ2(x)dx; that is, (22)Φ=φ∈SRd:∫Rdxkφxhκ2xdx=0, forallk∈Z+d.In fact, if φ∈Φ, then ψ^∈Φ, and for any multi-index k, by Proposition 1 (iv), we have (23)∫Rdxkφxhκ2xdx=∫RdEx,-iξxkφxhκ2xdxξ=0=Mkφ^0=ikDkφ^0=0.

Denote that Φ′ and Ψ′ are the spaces of all semilinear functionals on Φ and Ψ, respectively. Some properties of Φ and Ψ are given in the following proposition.

Proposition 6.

(i) The spaces Φ and Ψ are not empty.

(ii) The space Φ does not contain compactly supported infinitely differentiable functions, rather than 0.

(iii) The space Φ is invariant under the generalized translations.

(iv) The space Φ is dense in Lp(Rd,hκ2), 1<p<∞.

(v) S′(Rd)-distributions that coincide in the Φ′-sense differ from each other by a polynomial.

(vi) Let f∈Lr(Rd,hκ2) and g∈Lp(Rd,hκ2), 1≤r,p<∞. If f=g in the Φ′-sense, then f≡g almost everywhere.

The proof of the this proposition is similar to the ones in [13–15] except with the reflection-invariant measure hκ2(x)dx. Now we sketch it below.

Proof.

(i) Choose ψ∈S(Rd) satisfying that suppψ={x∈Rd:|x|>1}. Then ψ∈Ψ.

(ii) Suppose φ∈Cc∞(Rd); then we have (24)φ^ξ=∑β=0∞ξββ!∂βφ^0.If φ∈Φ, φ^∈Ψ, and ∂βφ^(0)=0, so φ^(ξ)=0 for all ξ∈Rd. Then φ(x)≡0.

(iii) This conclusion can be obtained by Proposition 2 (iii).

(iv) It suffices to approximate a function f∈S(Rd) by functions fN∈Φ in the Lp(Rd,hκ2) norm. We introduce the functions (25)ψNx=μNxfxwhere μ∈C∞([0,∞]) such that μ(t)=1 for t≥2, μ(t)=0 for 0≤t≤1 and 0≤μ(t)≤1. We define fN(x)=ψ^N(x). Since ψN∈S(Rd) and ψN(x)=0 as |x|≤1/N, we have ψ∈Ψ and then fN∈Φ. In order to show that fN(x) approximate the function f(x), we represent them as (26)fNx=fx-∫RdkyτNyfxhκ2ydywhere k(y) is the inverse Dunkl transform of the function 1-μ(|x|). Then by Lemma 7, f-fNκ,p⟶0 as N⟶∞.

(v) Suppose f,g∈S(Rd) and f=g in the sense of Φ′; that is, for all φ∈Φ, 〈f-g,φ〉k=0. Then, for all ψ∈Ψ, (27)f-g^,ψκ=f-g,ψ^κ=0.This means that supp(f-g^)={0}, which implies that f-g^ is a finite linear combination of the derivatives of the delta function. Hence, by (15), f-g is a polynomial.

(vi) For γ>0, denote μf(γ) and μg(γ) as the distributions of f and g, respectively. Then (28)fκ,rr=∫Rdfxrhκ2xdx≥∫x∈Rd:fx≥γfxrhκ2xdx≥∫x∈Rd:fx≥γγrhκ2xdx=γr·μfγ.Then μf(γ) is finite since f∈Lr(Rd,hκ2). The same argument gives that μg(γ) is finite as well. We claim that f=g almost everywhere. In fact, f(x)=g(x)+P(x) by (v), where P(x) is a polynomial. Then, for all γ>0,(29)μP2γ=μf-g2γ≤μfγ+μgγ<+∞.Thus P(x)≡0 a.e. So we have f=g a.e. consequently.

Lemma 7.

Let ρ∈L1(Rd,hκ2) and f∈Lp(Rd,hκ2), 1<p<∞. Define (30)gNx=∫RdρyτNyf~xhκ2ydy.Then, gNκ,p⟶0 as N⟶∞.

Proof.

According to the definition, (31)gNx=N-2λκ∫RdρyNτyf~xhκ2ydy.If p=2, (32)gNκ,22=N-4λκ∫Rdρ^xNf^x2hκ2xdx.Then gNκ,2⟶0 as N⟶∞ by Lebesgue’s dominated theorem.

When p≠2, by Proposition 2 (v),(33)gNκ,p≤cρκ,1fκ,p.It suffices to verify gNκ,p⟶0 for f∈C0∞(Rd). Let r>1 be any number such that 2<p<r. By Hölder’s inequality, we have (34)gNκ,p≤gNκ,r1-νgNκ,2ν,where ν=2(r-p)/p(r-2). Then, by (33) (35)gNκ,p≤cρκ,1fκ,r1-νgNκ,2νtends to 0 as N→∞.

Theorem 8.

Let 0<α<λκ. The operators Iακ are the automorphisms on the space Φ.

Proof.

According to the last equation in the proof for Proposition 4.1 in [7], for φ∈Φ,(36)Iακφx=·-αφ^·^x. Since |y|-αφ^(y) belongs to Ψ and Dunkl transform maps Ψ isomorphically onto Φ, it follows that the map φ⟶·-αφ^(·)^ is continuous from Φ to itself. Owing to (36), Iακ is a linear continuous operator from Φ to Φ. Conversely, we claim that Iακ is surjective. In fact, for φ∈Φ, let φ0(x)=·-αφ^(·)^(x). Then φ0∈Φ and Iακφ0=φ. Furthermore, the map φ→φ0 is continuous in the topology of the space Φ. This completes the proof.

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Now we can give the main result of this paper, the inversion of the Dunkl Riesz potentials Iακ when the group G=Z2d. 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 f∈L2(Rd,hκ2)∩Lp(Rd,hκ2), p≥1, 0<α<max{λκ/2,λκ/p}. Suppose that w is a bounded radial function in L1(Rd,hκ2) and the integral (37)dwα=1σd-1∫Rdw^ξξλκ+αhκ2ξdξ=limε→01σd-1∫ξ>εw^ξξλκ+αhκ2ξdξis finite. Then (38)dwαfx=∫0∞WtIακfxt1+αdt=limε→0∫ε∞WtIακfxt1+αdtwhere(39)Wtfx=f∗κwtx=t-λκ∫Rdfyτyw·txhκ2ydy.

Proof.

Denote(40)Tεφx=∫ε∞Wtφxt1+αdt,ψεξ=1σd-1∫y>εξw^yyλκ+αhκ2ydy.We claim that the operator Tε is bounded on Lp(Rd,hκ2) for any 1≤p<∞. Indeed, according to Proposition 2 (iv), we have (41)Wtφκ,p=φ∗κwtκ,p≤φκ,pwκ,1.Then the generalized Minkowski’s inequality gives that(42)Tεφκ,p=∫Rd∫ε∞Wtφxt1+αdtphκ2xdx1/p=∫ε∞Wtφκ,pt1+αdt≤φκ,pwκ,1∫ε∞dtt1+α.If we can show that(43)TεIακf=ψεf^∨,where f∨ denotes the inverse Dunkl transform of f, then we have (44)TεIακf-dwαfκ,2=TεIακf^-dwαf^κ,2=ψεf^-dwαf^κ,2tending to 0 as ε⟶0, by Lebesgue’s theorem on dominated convergence.

Recall that f∈L2(Rd,hκ2)∩Lp(Rd,hκ2). Let first 0<α<λκ/p, p>1. Then Proposition 4 gives that Iακf∈Lq(Rd,hκ2) when 1/p-1/q=α/λκ. And therefore, TεIακf∈Lq(Rd,hκ2) by replacing φ with Iακf in (42).

On the other hand, (ψεf^)∨∈L2(Rd,hκ2). According to Proposition 6 (vii), it suffices to prove (43) in the Φ′-sense; that is,(45)TεIακf,uκ=ψεf^∨,uκforallu∈Φ.After changing the order of the integration, the left-hand side of (45) equals (46)IακTεf,uκ=Tεf,Iακuκ=f,TεIακuκ.Since the Dunkl transform of Iακu is |ξ|-αu^(ξ), then (47)f,TεIακuκ=f^ξ,ξ-αu^ξ∫ε∞w^tξt1+αdtκ.Since w is radial, then (48)∫ε∞w^tξt1+αdt=ξασd-1∫y>εξw^yyλκ+αhκ2ydy=ξαψεξ.Combining the above, we obtain (45) as desired.

When λκ/p<α<λκ/2, p>1, the argument is the same as the first case, except with q=2λκ/(λκ-2α).

For the case p=1, f∈L1(Rd,hκ2)∩L2(Rd,hκ2) and 0<α<λκ. By interpolation, f∈Ls(Rd,hκ2) for all 1≤s≤2. Choosing s in the interval (1,min{2,λκ/α}) and q=λκs/(λκ-αs) in the Hardy-Littlewood-Sobolev theorem, we can get the result by repeating the argument when p>1. 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).

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