# Hausdorff Operators on Modulation SpacesMp,ps

Authors: Guoping Zhao; Qiong Lou
Journal: Journal of Function Spaces
 (2018)

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## Abstract

The sharp conditions are given for the boundedness of Hausdorff operators on modulation spaces with potential. By this, we extend some previous results in this topic.


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## Body

## 1. Introduction and Preliminary

As we know, Hausdorff operator has a deep root in the study of the Fourier analysis and it has a long history in the study of real and complex analysis. The reader is referred to [1, 2] for a survey of some historic background and recent developments on Hausdorff operator.The Hausdorff operatorHΦ can be defined for a suitable function Φ by (1)HΦfx=∫RnΦyfxydy,where the above integral makes sense for f belongs to some classes of nice functions. There are researches concerning the boundedness of Hausdorff operators on function spaces. Among them, the sharp conditions for the boundedness of Hausdorff operator can be characterized in only few cases. One can see [3, 4] for the characterization for the boundedness of Hausdorff operators on Lebesgue spaces, and see [5, 6] for that on Hardy spaces H1 and h1. There are some other results about the boundedness of Hausdorff operators on function spaces (see [7–10]).The modulation spacesMp,qs, first introduced by Feichtinger [11] in 1983, are closely related to the topic of time-frequency analysis (see [12]). As function spaces associated with the uniform decomposition on frequency (see [13]), they have been regarded as appropriate function spaces for the study of partial differential equations (see [14]). We refer the reader to [15] for some motivations and historical remarks. One is also referred to our recent papers [16–18] for the properties of modulation spaces.As a frequency decomposition space, the norm off in the modulation space cannot be completely determined by the absolute value of the function. On the other hand, the scaling property of modulation spaces is not as simple as that of Lp (see [19]). Based on the above two observations, the characterization of bounded Hausdorff operator on modulation space is quite different from that on Lebesgue space. By introducing new technique, we established the sharp conditions for the boundedness of HΦ on Mp,q (see [20]).Note that in [20], only the modulation spaces without potential were taken into consideration. Since the Hausdorff operator is not an operator of convolution type, and the dilation properties of modulation spaces are more complicated with potential, it is quite interesting to establish the sharp conditions for the boundedness of Hausdorff operator on modulation spaces with potential. As in the simple case in [20], the classical method used for the Lebesgue space is also not applicable here. As we will see, the potential index s plays important roles not only in the dilation property of Mp,qs, but also in the boundedness of Hausdorff operator on Mp,qs. Therefore, the method in [20] must be modified to cope with this new situation.Remark 1 (basic assumptions onΦ).
In order to establish the sharp conditions for the boundedness ofHΦ, we need to add some suitable assumptions on Φ.
Firstly, we assumeΦ⩾0 in this article. To make the lower bound for HΦ, this assumption appears in most of the previous papers which consider the characterizations for the boundedness of Hausdorff operator (see [4–6]).
Secondly, we make another assumption forΦ as follows:(2)∫B0,1ynΦydy<∞,∫B0,1cΦydy<∞.In fact, assumption (2) is the weakest conditions to ensure that the Schwartz function can be mapped into tempered distribution by Hausdorff operator HΦ; one can see [20] for details. We also remark that (2) can be deduced by the boundedness of HΦ on modulation spaces, since S⊂Mp,ps⊂S′.We turn to give the definition of modulation space.LetS≔S(Rn) be the Schwartz space and S′≔S′(Rn) be the space of tempered distributions. For f∈S(Rn), we define the Fourier transform Ff and the inverse Fourier transform F-1f by (3)Ffξ=f^ξ=∫Rnfxe-2πix·ξdx,F-1fx=f˘x=f^-x=∫Rnfξe2πix·ξdξ.Definition 2.
Lets∈R,0<p,q⩽∞. The weighted Lebesgue space Lx,ps consists of all measurable functions f such that (4)fLx,ps=∫Rnfxpxpsdx1/p,p<∞esssupx∈Rn⁡fxxs,p=∞is finite, where 〈x〉≔(1+|x|2)1/2. We write Lps for short if there is no confusion. If s=0, we denote Lp≔Lp0 and ·p≔·Lp for brevity.The translation operator is defined asTx0f(x)≔f(x-x0) and the modulation operator is defined as Mξf(x)≔e2πiξ·xf(x), for x,x0,ξ∈Rn. Fixing a nonzero function ϕ∈S, the short-time Fourier transform of f∈S′ with respect to the window ϕ is given by (5)Vϕfx,ξ≔f,MξTxϕ,and that can be written as (6)Vϕfx,ξ=∫Rnfyϕy-x¯e-2πiy·ξdyif f∈S. We give the definitions of weighted modulation space Mp,qs,t.Definition 3.
Lets,t∈R,0<p,q⩽∞. The weighted modulation space Mp,qs,t consists of all f∈S′(Rn) such that the (quasi-)norm (7)fMp,qs,t≔Vϕfx,ξLx,ptLξ,qs=∫Rn∫RnVϕfx,ξpxtpdxq/pξsqdξ1/qis finite, with the usual modifications when p=∞ or q=∞. This definition is independent of the choice of the window ϕ.In particular, fort=0, the modulation space (continuous version) Mp,qs is defined as follows.Definition 4.
Lets∈R,0<p,q⩽∞. The modulation space Mp,qs consists of all f∈S′(Rn) such that the (quasi-) norm (8)fMp,qs≔Vϕfx,ξLx,pLξ,qs=∫Rn∫RnVϕfx,ξpdxq/pξsqdξ1/qis finite, with the usual modifications when p=∞ or q=∞. This definition is independent of the choice of the window ϕ∈S.Applying the frequency-uniform localization techniques, one can give an alternative definition of modulation spaces (see [13, 21] for details).LetQk be the unit cube with the center at k. Then the family {Qk}k∈Zn constitutes a decomposition of Rn. Let η:Rn→[0,1] be a smooth function satisfying that η(ξ)=1 for ξ∈Q0 and η(ξ)=0 for ξ∉(3/2)Q0. Denote the translation of η by (9)ηkξ=ηξ-k,k∈Zn.Since ηk(ξ)=1 in Qk, we have that ∑k∈Znηk(ξ)⩾1 for all ξ∈Rn. Denote (10)σkξ=ηkξ∑l∈Znηlξ-1,k∈Zn.It is easy to know that {σk}k∈Zn constitutes a smooth partition of the unity, and σk(ξ)=σ(ξ-k). The frequency-uniform decomposition operators can be defined by (11)□k≔F-1σkFfor k∈Zn. Now, we give the (discrete) definition of modulation space Mp,qs.Definition 5.
Lets∈R,0<p,q⩽∞. The modulation space Mp,qs consists of all f∈S′ such that the (quasi-) norm (12)fMp,qs≔∑k∈Znksq□kfpq1/qis finite. We write Mp,q≔Mp,q0 for short. We also recall that this definition is independent of the choice of {σk}k∈Zn and the definitions of Mp,qs and Mp,qs are equivalent [21] when 0<p<∞ and 0<q⩽∞.Now, we state our main result as follows.Theorem 6.
Let1⩽p⩽2, s⩾0 and Φ be a nonnegative function. Then HΦ is bounded on Mp,ps if and only if(13)∫Rnyn/p+yn/p′·1+y-sΦydy<∞.In Section2, we will collect some basic properties of modulation spaces. The proof of Theorem 6 will be also given in Section 2.Throughout this paper, we will adopt the following notations. We useX≲Y to denote the statement that X⩽CY, with a positive constant C that may depend on n,p, but it might be different from line to line. The notation X~Y means the statement X≲Y≲X. We use X≲λY to denote X⩽CλY, meaning that the implied constant Cλ depends on the parameter λ. For x=(x1,x2,…,xn)∈Rn, we denote 〈x〉≔(1+|x|2)1/2.


## 2. Proof of the Main Theorem

Lemma 7 ([20] the Fourier transform of HΦ).
LetΦ be a nonnegative function satisfying the basic assumption (2). For f,g∈S(Rn), define HΦ~(14)HΦ~fx=∫RnΦyynfyxdy.Then(1)
HΦ~f is a tempered distribution and that the map HΦ~:S→S′ is continuous.(2)
HΦf^=HΦ~f^ in the distribution sense.Lemma 8 ([22] weight lifting).
For anys,t,t0∈R, p,q∈[1,∞], the map (15)f⟼·t0fis a bijection from Mp,qs,t+t0 to Mp,qs,t.Lemma 9 ([22] potential lifting).
For anyt,s,s0∈R,p,q∈[1,∞], the map (16)f⟼F-1·s0Ffis a bijection from Mp,qs+s0,t to Mp,qs,t.Lemma 10 (symmetry of time and frequency).
F
-
1
f
M
p
,
p
~
F
f
M
p
,
p.Proof.
By the fact that(17)Vϕfx,ξ=Vϕ^f^ξ,-x,the conclusion follows by the definition of modulation space.Lemma 11 ([19] dilation property of weighted modulation space).
Let1⩽p⩽∞. Set fλ(x)=f(λx). Then (18)fλMp,ps≲max⁡λ-n/p,λ-n/p′·max⁡1,λs·fMp,ps.Lemma 12 (embedding relations between modulation and Lebesgue spaces).
The following embedding relations are right:(1)
Mp,ps↪Lp for 1⩽p⩽2 and s⩾0;(2)
Mp,p0,s↪Lps for 1⩽p⩽2.Proof.
The first relationship can be obtained byMp,ps↪Mp,p↪Lp, where the Mp,p↪Lp is a known inclusion relation (see [23]).
For the second one, recalling the definition ofLps and the fact Mp,p↪Lp with 1⩽p⩽2, we use Lemma 8 to deduce that (19)fLps=·sf·Lp≲·sf·Mp,p~fMp,p0,s.In order to make the proof more clear, we give the following two technical propositions.Proposition 13 (for technique).
Let1⩽p⩽2, s⩾0 and Φ be a nonnegative function. Then (1)
ifHΦ:Mp,ps→Lp is bounded, one has(20)∫Rnyn/pΦydy<∞;2
ifHΦ~:Mp,p0,s→Lps is bounded, one has(21)∫Rnyn/p′-sΦydy<∞.Proof.
We first give the proof of statement (1). SupposeHΦ:Mp,ps→Lp is bounded. Let ψ:Rn→[0,1] be a smooth bump function supported in the ball {ξ:|ξ|<3/2} and be equal to 1 on the ball {ξ:|ξ|⩽4/3}. Let ρ(ξ)=ψ(ξ)-ψ(2ξ). Then ρ is a positive smooth function supported in the annulus {ξ:2/3<|ξ|<3/2}, satisfying ρ(ξ)=1 on a smaller annulus {ξ:3/4⩽|ξ|⩽4/3}. Denote ρj(ξ)≔ρ(ξ/2j). We have supp⁡ρj⊂{ξ:2/3·2j⩽|ξ|⩽3/2·2j} and ρj(ξ)=1 on {ξ:3/4·2j⩽|ξ|⩽4/3·2j}. Thus, we have supp⁡∑j=1Nρj(ξ)⊂{ξ:4/3⩽|ξ|⩽3/2·2N} and ∑j=1Nρj(ξ)=1 on {ξ:3/2⩽|ξ|⩽4/3·2N}. Take φ to be a nonnegative smooth function satisfying that supp⁡φ^⊂B(0,1/2),φ(0)=1. Choose fN(x)=∑j=0N+1ρj(x)·|x|-n/p∗φ. Therefore, we have(22)supp⁡fN^⊂B0,12,fNx≳∑j=1Nρjx·x-n/p.Indeed, the previous inclusion relation of (22) follows from the support condition of φ^. To verify the latter inequality, we only need to prove it when the right hand is nonzero; that is, x∈{4/3⩽|x|⩽3/2·2N}. For the nonnegative function φ satisfying φ(0)=1, there exists a positive constant δ<1/12, such that φ(x)>1/2 when |x|<δ. By the triangle inequality and the properties of φ we have that (23)fNx=∑j=0N+1ρjx·x-n/p∗φ=∫Rn∑j=0N+1ρjx-y·x-y-n/pφydy≳∫4/3≤x-y≤3/2·2Ny<δ∑j=0N+1ρjx-y·x-n/pdy≳∑j=1Nρjx·x-n/p,so we prove (22). Recalling that Φ is nonnegative, for 0<M≪N, we have the following estimate: (24)HΦfNLp=∫RnΦyfNxydyLp≳∫RnΦyyn/p·∑j=1Nρjxy·x-n/pdyLp⩾∫B0,2/3·2M∖B0,3/4·2-MΦyyn/p·∑j=1Nρjxy·x-n/pdyLp⩾∫B0,2/3·2M∖B0,3/4·2-MΦyyn/p·χ2M<x<2N-Mx·x-n/pdyLp=∫B0,2/3·2M∖B0,3/4·2-MΦyyn/pdy·x-n/pχ2M<x<2N-MxLp≳∫B0,2/3·2M∖B0,3/4·2-MΦyyn/pdy·lg⁡2N-2M1/p,where we use the fact that ∑j=1Nρj(x/|y|)=1 for y∈B(0,2/3·2M)∖B(0,3/4·2-M) and x∈B(0,2N-M)∖B(0,2M). On the other hand, observing that suppfN^⊂B(0,1/2), so #{k:σkfN^≠0}≲1 for all N and |l|⩽1 if l∈{k:σkfN^≠0}. And then we have that (25)fNMp,ps=∑σkfN^≠0k∈ZnkspF-1σkfN^Lpp1/p≲∑σkfN^≠0k∈ZnkspfNLpp1/p≲fNLp≲∑j=0N+1ρjx·x-n/pLp~ln⁡2N1/p.Using the boundedness of HΦ and the above estimates for HΦfN and fN, we have that (26)HΦMp,ps→Lp⩾HΦfNLpfNMp,ps≳∫B0,2/3·2M∖B0,3/4·2-MΦyyn/pdylg⁡2N-2Mlg⁡2N1/p.Letting N→∞, we have (27)∫B0,2/3·2M∖B0,3/4·2-MΦyyn/pdy≲HΦMp,ps→Lp.By the arbitrariness of M, we let M→∞ and obtain that ∫RnΦ(y)|y|n/pdy≲HΦMp,ps→Lp.
Now we turn to give the proof of (2). SupposeHΦ~:Mp,p0,s→Lps is bounded. As in the proof of conclusion (1), we take gN(x)=∑j=1Nρj(x)·|x|-n/p-s. A direction calculation yields that (28)HΦ~gNLps=∫RnΦyyngNyxdyLps=∫RnΦyyn/p′-s·∑j=1Nρjyx·x-n/p-sdyLps⩾∫B0,4/3·2M∖B0,3/2·2-MΦyyn/p′-s·∑j=1Nρjyx·x-n/p-sdyLps⩾∫B0,4/3·2M∖B0,3/2·2-MΦyyn/p′-sdy·x-n/p-sχ2M<x<2N-MxLps≳∫B0,4/3·2M∖B0,3/2·2-MΦyyn/p′-sdy·lg⁡2N-2M1/p.On the other hand, (29)F-1σkgNLp=F-1σk∑j=1Nρjx·x-n/p-sLp≲k-n/p-sF-1σk∑j=1NρjxLp⩽k-n/p-sF-1σkLp·∑1⩽j⩽N:σkρj≠0F-1ρjxL1≲k-n/p-s.
Using Lemma10, we obtain that (30)gNMp,p0,s=·sgN·Mp,p=F-1ξsgNξ·Mp,p=F-1gNMp,ps=∑k∈ZnkspF-1σkgNLpp1/p≲∑k<2N+1k∈Znk-n1/p~lg⁡2N1/p.We deduce that (31)HΦ~Mp,p0,s→Lps⩾HΦ~gNLpsgNMp,p0,s≳∫B0,4/3·2M∖B0,3/2·2-MΦyyn/p′-sdylg⁡2N-2Mlg⁡2N1/p.Letting N→∞, we have (32)∫B0,4/3·2M∖B0,3/2·2-MΦyyn/p′-sdy≲HΦ~Mp,p0,s→Lps.By the arbitrariness of M, we let M→∞ and obtain that ∫RnΦ(y)|y|n/p′-sdy≲HΦ~Mp,p0,s→Lps.Next, we establish the following two propositions for reduction.Proposition 14 (for reduction).
Let1⩽p⩽2, s⩾0 and Φ be a nonnegative function. If the Hausdorff operator HΦ is bounded on Mp,ps, one has the following: (1)
HΦ:Mp,ps→Lp is bounded,(2)
HΦ~:Mp,p0,s→Lps is bounded.Proof.
The first conclusion can be deduced by the embedding relationMp,ps↪Lp (see Lemma 12) directly. We turn to prove the second conclusion. For any Schwartz function f, by the property of HΦ and Lemmas 8, 9, and 10, we have (33)fMp,ps=F-1ξsFfMp,p=ξsFfMp,p=FfMp,p0,sand similarly we obtain (34)HΦfMp,ps=FHΦfMp,p0,s=HΦ~f^Mp,p0,s,where we use Lemma 7 in the last equality. Thus, if HΦ is bounded on Mp,ps, we have (35)HΦ~f^Mp,p0,s≲f^Mp,p0,s.Lemma 12, that is, the embedding relation Mp,p0,s↪Lps, then yields that (36)HΦ~fLps≲fMp,p0,sfor all f∈S(Rn).Now, we are ready to give the proof of Theorem6.Proof of Theorem6.
We divide this proof into two parts.
IF Part. Using the Minkowski inequality, we deduce that (37)HΦfMp,ps≲∫RnΦyfxydyMp,ps≲∫RnΦyfxyMp,psdy.Recalling the dilation properties of modulation space (see Lemma 11), we obtain that (38)HΦfMp,ps≲∫RnΦy·max⁡yn/p,yn/p′·max⁡1,y-sdyfMp,ps≲∫Rnyn/p+yn/p′1+y-sΦydyfMp,ps.This implies the boundedness of HΦ on Mp,ps.
ONLY IF Part. Using Proposition 14, we obtain HΦ:Mp,ps→Lp and HΦ~:Mp,p0,s→Lps. Next, Proposition 13 yields that (39)∫Rnyn/pΦydy<∞,∫Rnyn/p′-sΦydy<∞.Hence, we have (40)∫Rnyn/p+yn/p′·1+y-sΦydy=∫B0,1+∫Bc0,1yn/p+yn/p′·1+y-sΦydy⩽4∫B0,1yn/p′-sΦydy+4∫Bc0,1yn/pΦydy≲∫Rnyn/p′-sΦydy+∫Rnyn/pΦydy<∞,which is the desired conclusion.Remark 15.
For some technical reasons, our main theorems only characterize the boundedness of Hausdorff operator onMp,ps for 1⩽p⩽2, s⩾0. By dual argument, one can easily get the similar conclusion for the range 2⩽p⩽∞, s⩽0. Moreover, similar results for Wiener amalgam spaces can be also established as in [20]. Our theorem remains an open problem for the characterization of Hausdorff operator on the full range 1⩽p,q⩽∞,s∈R.



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