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.
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 operator can be defined for a suitable function by where the above integral makes sense for 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 and . There are some other results about the boundedness of Hausdorff operators on function spaces (see [7–10]).
The modulation spaces , 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 of 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 (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 on (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 plays important roles not only in the dilation property of , but also in the boundedness of Hausdorff operator on . Therefore, the method in [20] must be modified to cope with this new situation.
Remark 1 (basic assumptions on ). In order to establish the sharp conditions for the boundedness of , we need to add some suitable assumptions on .
Firstly, we assume in this article. To make the lower bound for , 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 for as follows:In fact, assumption (2) is the weakest conditions to ensure that the Schwartz function can be mapped into tempered distribution by Hausdorff operator ; one can see [20] for details. We also remark that (2) can be deduced by the boundedness of on modulation spaces, since .
We turn to give the definition of modulation space.
Let be the Schwartz space and be the space of tempered distributions. For , we define the Fourier transform and the inverse Fourier transform by
Definition 2. Let . The weighted Lebesgue space consists of all measurable functions such that is finite, where . We write for short if there is no confusion. If , we denote and for brevity.
The translation operator is defined as and the modulation operator is defined as , for . Fixing a nonzero function , the short-time Fourier transform of with respect to the window is given by and that can be written as if . We give the definitions of weighted modulation space .
Definition 3. Let . The weighted modulation space consists of all such that the (quasi-)norm is finite, with the usual modifications when or . This definition is independent of the choice of the window .
In particular, for , the modulation space (continuous version) is defined as follows.
Definition 4. Let . The modulation space consists of all such that the (quasi-) norm is finite, with the usual modifications when or . This definition is independent of the choice of the window .
Applying the frequency-uniform localization techniques, one can give an alternative definition of modulation spaces (see [13, 21] for details).
Let be the unit cube with the center at . Then the family constitutes a decomposition of . Let be a smooth function satisfying that for and for . Denote the translation of by Since in , we have that for all . Denote It is easy to know that constitutes a smooth partition of the unity, and . The frequency-uniform decomposition operators can be defined by for . Now, we give the (discrete) definition of modulation space .
Definition 5. Let . The modulation space consists of all such that the (quasi-) norm is finite. We write for short. We also recall that this definition is independent of the choice of and the definitions of and are equivalent [21] when and .
Now, we state our main result as follows.
Theorem 6. Let , and be a nonnegative function. Then is bounded on if and only if
In Section 2, 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 use to denote the statement that , with a positive constant that may depend on , but it might be different from line to line. The notation means the statement . We use to denote , meaning that the implied constant depends on the parameter . For , we denote
2. Proof of the Main Theorem
Lemma 7 ([20] the Fourier transform of ). Let be a nonnegative function satisfying the basic assumption (2). For , define Then(1) is a tempered distribution and that the map is continuous.(2) in the distribution sense.
Lemma 8 ([22] weight lifting). For any , , the map is a bijection from to .
Lemma 9 ([22] potential lifting). For any , the map is a bijection from to .
Lemma 10 (symmetry of time and frequency). .
Proof. By the fact that the conclusion follows by the definition of modulation space.
Lemma 11 ([19] dilation property of weighted modulation space). Let . Set . Then
Lemma 12 (embedding relations between modulation and Lebesgue spaces). The following embedding relations are right: (1) for and ;(2) for .
Proof. The first relationship can be obtained by , where the is a known inclusion relation (see [23]).
For the second one, recalling the definition of and the fact with , we use Lemma 8 to deduce that
In order to make the proof more clear, we give the following two technical propositions.
Proposition 13 (for technique). Let , and be a nonnegative function. Then if is bounded, one hasif is bounded, one has
Proof. We first give the proof of statement (1). Suppose is bounded. Let be a smooth bump function supported in the ball and be equal to 1 on the ball . Let . Then is a positive smooth function supported in the annulus , satisfying on a smaller annulus . Denote . We have and on . Thus, we have and on . Take to be a nonnegative smooth function satisfying that . Choose Therefore, we haveIndeed, the previous inclusion relation of (22) follows from the support condition of . To verify the latter inequality, we only need to prove it when the right hand is nonzero; that is, . For the nonnegative function satisfying , there exists a positive constant , such that when . By the triangle inequality and the properties of we have that so we prove (22). Recalling that is nonnegative, for , we have the following estimate: where we use the fact that for and . On the other hand, observing that , so for all and if . And then we have that Using the boundedness of and the above estimates for and , we have that Letting , we have By the arbitrariness of , we let and obtain that .
Now we turn to give the proof of (2). Suppose is bounded. As in the proof of conclusion (1), we take . A direction calculation yields that On the other hand, Using Lemma 10, we obtain that We deduce that Letting , we have By the arbitrariness of , we let and obtain that .
Next, we establish the following two propositions for reduction.
Proposition 14 (for reduction). Let , and be a nonnegative function. If the Hausdorff operator is bounded on , one has the following: (1) is bounded,(2) is bounded.
Proof. The first conclusion can be deduced by the embedding relation (see Lemma 12) directly. We turn to prove the second conclusion. For any Schwartz function , by the property of and Lemmas 8, 9, and 10, we have and similarly we obtain where we use Lemma 7 in the last equality. Thus, if is bounded on , we have Lemma 12, that is, the embedding relation , then yields that for all .
Now, we are ready to give the proof of Theorem 6.
Proof of Theorem 6. We divide this proof into two parts.
IF Part. Using the Minkowski inequality, we deduce that Recalling the dilation properties of modulation space (see Lemma 11), we obtain that This implies the boundedness of on .
ONLY IF Part. Using Proposition 14, we obtain and . Next, Proposition 13 yields that Hence, we have which is the desired conclusion.
Remark 15. For some technical reasons, our main theorems only characterize the boundedness of Hausdorff operator on for , . By dual argument, one can easily get the similar conclusion for the range , . 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 .
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Foundation of China (nos. 11601456, 11371316, 11771388).