Higher-Order Riesz Transforms in the Inverse Gaussian Setting and UMD Banach Spaces

Betancor, Jorge J.; Rodríguez-Mesa, Lourdes

doi:https://doi.org/10.1155/2021/6899603

Journal of Function Spaces

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Research Article | Open Access

Volume 2021 | Article ID 6899603 | https://doi.org/10.1155/2021/6899603

Higher-Order Riesz Transforms in the Inverse Gaussian Setting and UMD Banach Spaces

Jorge J. Betancor¹and Lourdes Rodríguez-Mesa¹

Academic Editor: Dumitru Motreanu

Received11 Jun 2021

Accepted14 Sept 2021

Published10 Nov 2021

Abstract

In this paper, we study higher-order Riesz transforms associated with the inverse Gaussian measure given by on . We establish -boundedness properties and obtain representations as principal values singular integrals for the higher-order Riesz transforms. New characterizations of the Banach spaces having the UMD property by means of the Riesz transforms and imaginary powers of the operator involved in the inverse Gaussian setting are given.

1. Introduction

Our setting is endowed with the measure whose density with respect to the Lebesgue measure is , . The measure is called the inverse Gaussian measure. The study of harmonic analysis operators in was began by Salogni [1]. The principal motivation for Salogni’s studies was the connection with the Gaussian setting. However, as Bruno and Sjögren [2] pointed out, can be seen as a model of a variety of settings where a theory of singular integrals has not been developed. Also, the natural Laplacian on , that we will denote by , can be interpreted as a restriction of the Laplace-Beltrami operator associated with a warped-product manifold whose Ricci tensor is unbounded from below. A complete exposition of the theory of this kind of manifolds can be found in [3].

The aim of this paper is to study -boundedness properties of higher-order Riesz transforms in the inverse Gaussian setting. Also, we characterize the UMD Banach spaces by using these Riesz transforms.

We consider the second-order differential operator defined by where , the space of the smooth functions with compact support in . Here, and denote the usual Euclidean Laplacian and gradient, respectively.

is essentially selfadjoint in . denotes the closure of in .

For every by , we represent the -th Hermite polynomial given by , , where for every ,

We have that, for every , where and , . The spectrum of in is the discrete set .

The operator generates a diffusion semigroup (in the Stein sense [4]) in where for every , we have that for every , , and being

The maximal operator defined by was studied by Salogni ([1]). She proved that is bounded from into . From the general results in [4], it can be deduced that is bounded from into itself, for every . Recently, Betancor et al. [5] have characterized the Köethe function spaces with the Hardy-Littlewood property by using the maximal operators

In [1], -boundedness properties with for some spectral multipliers associated with the operator were proved. The imaginary power , , of is a special case of the multipliers studied in [1]. Bruno ([6]) established endpoints results for , , proving that is bounded from into . Also, he showed that, for , the shifted first-order Riesz transform is bounded from into . These operators are studied on new Hardy type -spaces.

Higher-order Riesz transforms associated with the operator were studied by Bruno and Sjögren [2]. For every , the -th Riesz transform is defined by , where and . In ([2], Theorem 1.1), it was established that is bounded from into if and only if .

In ([6], Remark 2.6), Bruno proved that, for every with , is bounded from into itself, for every . In [2], Bruno and Sjögren say that they do not know whether is bounded from into itself for every and , , though they expect so. In our first result, we prove that, as they expected, is bounded from into itself when and . We also obtain a representation of as a principal value singular integral. In order to prove our result, we need to use some properties of the negative power , , of . In Section 2, we analyze , . We obtain that, for every , the operator is bounded from into . This result contrasts with the one in ([7], Proposition 6.2) where it is proved that , , is not bounded from into , where represents the Ornstein-Uhlenbeck operator and denotes the Gaussian measure () on .

Theorem 1. Let . For every , the derivative exists for almost all and there exists such that where if is odd for some .

Furthermore, when and is even, the last integral is actually absolutely convergent for every , and in this case, no principal value is needed. Here,

Let . Since, for every , we have that, for every ,

Let . We can write in , where for every ,

We define

For every ,

Then where the constant depends on and . Hence, is bounded from into itself.

If , then , .

Theorem 2. Let and . The Riesz transform can be extended from to as a bounded operator from into itself. By denoting again to this extension, we have that, there exists such that, for every , where if is odd for some .

When and is even the integral defining is absolutely convergent.

As it was mentioned, Bruno and Sjögren ([2], Theorem 1.1]) proved that is bounded from into if and only if . This property also holds in the Gaussian setting (see [8, 9]). Aimar et al. ([10]) introduced Riesz type operators , , related to the Ornstein-Uhlenbeck operator. is bounded from into for every . Motivated by the results in [10], we define Riesz transform in the inverse Gaussian setting whose behavior in is different from the one for .

We can write , where for every , . We consider the operator . We have that . Let the closure of in . For every , , and the spectrum of in is the set of nonnegative integers. For every , we have that , . Here, we understand when . Then, for every , by denoting , we get

If and , with , ,

In other cases, (see Section 4 for details).

Let . We define the Riesz transform on as follows

Thus, is bounded from into itself. If and then .

Theorem 3. Let . The Riesz transform can be extended from to as a bounded operator from (i) into itself, for every (ii) into , provided that or , when By denoting again to the extension we have that, for every , , where when is odd for some . Here

Let be a Banach space. Suppose that is a -valued martingale. The sequence , where is understood as 0, is called the martingale difference associated with . We say that is a -martingale difference sequence when it is the difference sequence associated with a -martingale. If , is said to be a -space when there exists such that for all -valued -martingale difference sequence and for all ,

is an abbreviation of unconditional martingale difference. If is for some , then is for every . This fact justifies to call to the property without any reference to . Burkholder [11] and Bourgain [12] proved that the property of is necessary and sufficient for the boundedness of the Hilbert transform in , . The property is a central notion in the development of the harmonic analysis when the functions are taking values in infinite-dimensional spaces. Banach spaces have been characterized by using other singular integrals that can be seen as Riesz transforms associated to orthogonal systems (see [13–16], for instance). In the following result, we characterize the Banach spaces with the property by using Riesz transforms in the inverse Gaussian setting. For every , we define where , , and .

Theorem 4. Let be a Banach space. The following assertions are equivalent. (i) is (ii)For every , can be extended from to as a bounded operator from into itself, for every (iii)For every , can be extended from to as a bounded operator from into itself, for some (iv)For every , can be extended from to as a bounded operator from into Also, the equivalences hold when in the properties (ii), (iii), and (iv), we replace by the maximal operator defined by

Theorem 5. Let be a Banach space. The following assertions are equivalent. (i) is (ii)For every there exists, for each , the limit for every .(iii)For some , there exists, for every , the limitfor each . (iv)For every , and , , for almost all (v)For some and for every and , , for almost all

The properties stated in Theorems 4 and 5 can also be established when we replace -Riesz transforms by -Riesz transforms.

Next aim is to state characterizations of the Banach spaces with the property by using imaginary powers , , of . Salogni ([1], Theorem 3.4.3) proved that, for every and , as , when . Actually, is a Laplace transform type multiplier associated with defined by the function , , for every . Then, since is a Stein diffusion semigroup, the -boundedness of , , follows from the general results established in ([4], Chapter III). Recently, Bruno ([6], Theorem 4.1) proved that , , is bounded from into .

Let . We have that

It is immediate to see that is bounded from into itself. For every , we define in the obvious way when is a Banach space. In order to the operator is bounded from into itself as subspace of , we need to impose some additional property to the Banach space . For instance, if is isomorphic to a Hilbert space, then can be extended from to as a bounded operator from into itself. We are going to characterize the Banach spaces as those Banach spaces for which can be extended from to as a bounded operator from into itself, when , and from into . Our result is motivated by the one in ([17], p. 402) where Banach spaces are characterized by the -boundedness properties of the imaginary power , , of . Guerre-Delabriére’s result was extended to higher dimensions by considering imaginary powers of the Laplacian in ([18], Proposition 1). In [19], this kind of characterization for Banach spaces is obtained in Hermite and Laguerre settings. As far as we know, this property has not been proved for the Ornstein-Uhlenbeck operator in the Gaussian framework.

Theorem 6. Let . For every , we have that where and being , .
Let be a Banach space. The following assertions are equivalent. (i) is (ii)For every , can be extended from to as a bounded operator from into itself(iii)For some , can be extended from to as a bounded operator from into itself(iv) can be extended from to as a bounded operator from into We define the maximal operator by The following assertions are equivalent to (i). (v)For every , is bounded from into itself(vi)For some , is bounded from into itself(vii) is bounded from into (viii)For every and every there exists the limit(ix)For some and every , there exists the limit(x)For every and every , , for almost all (xi)For some and every , , for almost all

This paper is organized as follows. In Section 2 we study the negative power , , of . Higher order Riesz transforms in the inverse Gauss setting are considered in Section 3 where we prove Theorems 1 and 2. Theorem 3 is established in Section 4 and Theorems 4 and 5 in Section 5. Finally, Section 6 is devoted to show the proof of Theorem 6.

Throughout this paper, and denote positive constants that can change in each occurrence.

2. Negative Powers of

In this section, we prove -boundedness properties of the negative powers , , of . These properties are different than the ones of the negative powers of the Ornstein-Uhlenbeck operator . We prove that, for every , defines a bounded operator from into . However, in ([7], Proposition 6.2), it was proved that if , is not bounded from into .

Let . We define

is bounded from into itself. Moreover, when , the series also converges pointwisely in . Indeed, let . We have that

Also, for every , (see (14)), and according to ([20], p. 324),

Then, and if it follows that

The series in (34) converges pointwise absolutely for each when . Indeed, let . Partial integration allows us to see that, for every , there exists such that

Then,

We also consider the operator defined, for every , by where the integral is understood in the -Bochner sense.

For each , we can write

We obtain

Then,

Suppose now that . By using (40), we obtain that there exists such that

We can write

Since is dense in , , .

According to ([6], Theorem 2.5), we have that, for every for all outside the support of , where

Proposition 7. Let . The operator can be extended from to as a bounded operator from into itself, when , and from into .

Proof. We use the method consisting in decomposing the operator in two parts called local and global parts. This procedure of decomposition was employed by Muckenhoupt ([21, 22]) in the Gaussian setting.
From now on, we consider the function given by , , and and the region defined by We decompose as follows where , .
According to ([1], Lemma 3.3.1), we get By choosing , we obtain In the last inequality, we have taken into account that , , and that , provided that .
We have that Also, since , , Hence, the operator defined by is bounded from into itself, for every . Since is a local operator, by using ([1], Proposition 3.2.5), we deduce that is bounded from into itself, for every .
Suppose that . As above, we take and we get Then, We deduce that , for almost all . Since and are bounded from into itself, , . It follows that, for every , can be extended from to as a bounded operator from into itself.
We now study the operator defined by By making the change of variables , , and taking into account that , , , we obtain We now use some notations that were introduced in [23] and proceed as in the proof of ([23], Proposition 2.2). For every , we define Assume that . Suppose first that . In this case, , . Furthermore, , . Then, By making the change of variable , , and taking into account that , we obtain Suppose now . We write . By using that (see [24], p. 850) we have that Since we conclude that, when , Let . Since , , and , when and , as in ([25], p. 501), we obtain Also, we have that We deduce that is bounded from into itself, for every .
Next, we are going to see that is bounded from into . We decompose , , as follows For every , we denote by the angle between and (we understand , when ). By using ([1], Lemma 3.3.3), we get that, for every , , On the other hand, by proceeding as in ([2], Proposition 5.1), we estimate , . We first observe that If then , , and it follows that If we define and we write , where is parallel to and is orthogonal to .
By making the change of variables and since , it follows that, if , In the last inequality, we have used that when and .
By combining the above estimates, we obtain According to ([1], Lemma 3.3.4) and ([2], Lemmas 4.2 and 4.3), we can see that the operator is bounded from into .
The estimation (74) allows us to prove that, for every , , for almost all . Then, since and are bounded from into itself, , . Hence, can be extended from to as a bounded operator from into itself, when , and from into .
Thus, the proof is finished.

3. Higher-Order Riesz Transforms Associated with the Operator

In this section, we prove Theorems 1 and 2 concerning to the higher-order Riesz transforms in the inverse Gaussian setting.

Proof of Theorem 1. Let . If , we have that and then Suppose that and such that . Then,

Indeed, by considering the function defined in Proposition 7 and using (76) and ([1], Lemma 3.3.1), we obtain, for ,

On the other hand, by reading the proof of ([1], Lemma 3.3.3), we deduce that

Note that this estimation also holds when .

Since , (77) holds. Hence, according to ([15], Lemma 4.2), we have that, when , and ,

We assume , so we can suppose without loss of generality that . Let us take . According to (80), we can write

Assume first and write where and

Here, , , denotes the classical heat kernel , .

Next, we show that

By taking into account (76), we get

Also by using (79), we obtain

Now we are going to estimate

We can write

Then

By proceeding as in the proof of ([1], Lemma 3.3.1), we can see that

Then, by taking into account that it follows by using (91) that

Let us analyze the term , . For every , , we write

Let , , and consider and . By taking into account that , it follows that and then, for each , and, by considering also (1) and using the mean value theorem, we get

Then, and thus,

We deduce that

This estimation, jointly with (86) and (87), leads to

where we have used that has compact support. According to ([15], Lemma 4.2), (85) is then established.

We are going to evaluate . We write , , where

Since , we have that

According to ([15], Lemma 4.2), we can derivate under the integral sign obtaining where the last integral is absolutely convergent.

For every , we define . Partial integration leads to

where

Let us estimate , . We recall that can be written as follows

Suppose now that is odd. Then,

We have that, for every and ,

On the other hand, by performing the change of variable , we get

It follows that

Then, by using the dominated convergence theorem, we obtain

Suppose now that is even. Then, and proceeding as above, we get

It follows that

We note that if is odd for some , then . Thus, we conclude that

where when is odd for some .

By putting together (85) and (115), we obtain

We now deal with the case . We have , . According to (80), we can write where and

Note that if is even, then .

By proceeding as in the case of and taking into account that we can see that the integral defining is absolutely convergent for every and being also this integral absolutely convergent for almost .

On the other hand, by considering we can write and according to ([15], Lemma 4.2), we get

By partial integration, we obtain

Then where

By taking into account (10) and that , , it follows that

On the other hand, we have that and,

Let . Since , we get

If is odd, then is even and, for every ,

Hence, if is odd, .

If is even, then is odd, , and

We obtain provided that is even.

We conclude that, for certain

Then, we get where when is odd.

Finally, we observe that when is even

By taking into account the arguments developed in this proof, we can see that for every . Then,

The proof of Theorem 1 is completed.

Proof of Theorem 2. For every , we have that and according to (36), (37), and (40), the last series is pointwise absolutely convergent, it defines a smooth function on and Then, according to Theorem 1, for each , Here, when is odd for some .
To establish our result, it is sufficient to show that for every the limit exists for almost all and the operator defined by is bounded from into itself. Thus, is the unique extension of from to as a bounded operator from into itself.
For every we define the set Observe that and that if , and , then that is, . In particular, we have that if then provided that .
Let . We consider the operators and defined on by

We recall that

Then,

for every . From now on we consider and . Since , , , by making the change of variables in the last integral, we get

Assume first that . Then, . By proceeding as in the proof of (66), it follows that, when ,

Suppose now that . Again, as in the estimation in (66), since , we obtain

On the other hand, proceeding as in ([23], p. 862) and considering the notation in (61) and ([23], Lemma 2.3), we have

From the above estimates, we conclude that, when ,

On the other hand, we consider the kernel

Let us show that

For every , we have that

The same proof of (99) allows us to obtain that

Also, we get and thus, (156) is established.

From (154), we obtain that can be extended to as a bounded operator from into itself, and the extension is given by (147). Indeed, when , we have that . By taking into account also that , we get

Since it follows that and in a similar way from which we deduce that is bounded on .

On the other hand, by using (156) and that , , we have that

Also, since , , we get

Hence, the operator defined by is a bounded operator from into itself. Since is a local operator, by ([1], Proposition 3.2.5) is bounded from into itself.

We now observe that the kernel is a standard Calderón-Zygmund kernel. Indeed, we get

Let and denote . We have that

Then,

The Euclidean -order Riesz transform is bounded from into itself, for every . According to ([1], Proposition 3.2.5), the operator defined by is bounded from into itself, for every .

We can write on . Then, can be extended from to as a bounded operator from into itself.

Since , we conclude that can be extended from to as a bounded operator from into itself.

Let us consider the maximal operator

Let , . For every , by using the above estimates, we can write

We also have that

Since the maximal operator is bounded from into itself, by using a vector-valued version of ([1], Proposition 3.2.5) (see ([16], Proposition 2.3)), we deduce that the local maximal operator is bounded from into itself.

By using the same arguments as above, we conclude that is bounded from into itself.

From (142) and since is dense in and is bounded in , by using a standard procedure, we can conclude that the limit exists for almost all and (defined in (144)) is a bounded operator from into itself.

Remark 10. The -boundedness of the local part of can be proved also by using Calderón-Zygmund theory. We have preferred to do it by comparing with the classical local Riesz transform because in this way we can know how the singularity of is. Furthermore, these comparative results will be useful in the proof of Theorems 1.4 and 1.5.

4. Riesz Transform Associated with the Operator

Our objective in this section is to prove Theorem 3.

We define as the space that consists of all those such that . Let . For every , is defined by

We have that

We introduce the operator defined by

Let . We have that

Then,

Hence, the integral defining converges in the -Bochner sense.

Let . According to (36), (37), and (40), we get

It follows that the series that defines converges pointwisely and absolutely and where , .

On the other hand, since is compact, we have that and, taking ,

Then, we obtain where and

By denoting the projection from to , we have proved that, for every ,

Let . Next, we show that for almost all . Here, and, when is odd for some , .

For every ,

Then, for each ,

Since , , when , by proceeding as in the proof of (80), we get that for every and being ,

Without loss of generality we can assume that and consider . When , we can proceed as in the proof of Theorem 1. For , we write

We observe that, for every ,

By considering the decomposition We can argue as in the proof of (85) to obtain that

On the other hand, to deal with , we consider and , , and write

We proceed as in the proof of (115) to get where and if is odd for some . Thus, (189) is established when .

If , we can also follow the proof of Theorem 1 by using the decomposition where and

When , that is, , we can replace in (193) and in (199) by and respectively, and proceed as above to obtain (189).

According to (36), (37), and (40), we get and then, with , when is odd for some .

We are going to show the -boundedness properties of . We recall that

Consider first and choose . By making the change of variables , , we obtain

Let and consider the local and global operators defined on by

By proceeding as in the proof of (154) it follows that, for each ,

We have that

Then, since ,

Also, we get

We conclude that is bounded from into itself.

We are going to study the operator . We write

By taking into account that we get (see (86))

Also, from (195) and proceeding as in the proof of (99), we can see that

Since when , we obtain that

On the other hand, we observe that where is the classical kernel considered in (155). The result can be established by proceeding as in the proof of Theorem 2 by taking into account that the Euclidean Riesz transform is bounded from into itself.

To deal with the case , we consider the local and global operators and defined above with instead of . Since the classical Riesz transform is bounded from into , we can use (217) and argue as in the proof of Theorem 2 to obtain that defines a bounded operator from into .

In order to prove that defines a bounded operator from into itself, we make the change of variables , , and write , where

Suppose that or when . By using ([1], Lemma 3.3.3), it follows that

On the other hand, by proceeding as in the estimation of in ([2], proof of Proposition 5.1), we obtain, for every ,

From ([1], Lemma 3.3.4) and ([2], Lemma 4.2), we deduce that defines a bounded operator from into .

Thus, we conclude that can be extended to as a bounded operator from into .

5. UMD Spaces and Riesz Transforms in the Inverse Gaussian Setting

Proof of Theorem 4. For every , by , we denote the -th Euclidean Riesz transform defined, for every , , by where Observe that

Let be a Banach space. For every , we define on , , in the obvious way.

The UMD-property for can be characterized by using , . The properties stated in Theorems 4 and 5 hold when is replaced by , . The estimations established in the proofs of Theorems 1 and 2 allow us to pass from to , .

Let and . We are going to see that the following two assertions are equivalent: (i) can be extended from to as a bounded operator from into itself(ii) can be extended from to as a bounded operator from into itself

We choose and consider the global and local operators as in the previous sections according to the region , with .

Suppose that (ii) holds. We can write Since is a Calderón-Zygmund operator, by using a vectorial version of ([1], Proposition 3.2.5) (see ([16], Proposition 2.3)), we deduce that can be extended from to as a bounded operator from into itself.

According to (154) and (156), we have that

and the integral operators are bounded from into itself. Then, and define bounded operators from into itself, and the same property holds for the operators and . We conclude that (i) holds.

Suppose now that (i) holds. By (149), we get

In a similar way, we get, for each ,

Then, according to a vector-valued version of ([1], Propositions 3.2.5 and 3.2.7) (see ([16], Propositions 2.3 and 2.4), we deduce that defines a bounded operator from into itself. Also, defines a bounded operator from into itself. We conclude that defines a bounded operator from into itself. Since is dilatation invariant, by proceeding as in the proof of ([16], Theorem 1.10, (ii)(i)), it follows that can be extended from to as a bounded operator from into itself.

The same arguments allow us to prove that the following assertions are equivalent. (i) can be extended from to as a bounded operator from into (ii) can be extended from to as a bounded operator from into

Furthermore, in a similar way, we can see that (i)(ii) and (iii)(iv) when and are replaced by and , respectively.

The proof of Theorem 4 is thus finished.

Proof of Theorem 5. Let and . We are going to see that the following two assertions are equivalent. (a)For every , there exists(b)For every , there exists

We consider again and . Suppose that is true. Let . We can write

Since (225) holds and the operator is bounded from into itself, there exists the limit

On the other hand, we get

Then, there exists the limit

Suppose that . It was seen in the proof of Theorem 2 that

Then,

Since (a) holds, there also exists the limit

Let . We have that provided that and . Then, for every ,

Since , and then there exists

Hence, we get that there exists

We conclude that there exists

In a similar way, we can see that (a) holds provided that (b) is true. Note that .

As it was proved in [2] the operator defined by is bounded from into . Then, for every , there exists

The same arguments allow us to prove that (a) (b) when .

By using a -dimensional version of ([26], Theorem D), we deduce that the properties (i), (ii), and (iii) in Theorem 5 are equivalent.

By proceeding as above, we can see that the properties in (a) and (b) continue being equivalent when we replace the principal values by the maximal operators and in (a) and (b), respectively. Then, the property (b) is equivalent to the property UMD for (see the comments before the proof of ([16], Theorem 1.10, p. 19).

Thus, the proof of Theorem 5 is finished.

6. UMD Spaces and the Imaginary Powers of

In this section, we prove Theorem 6.

According to ([18], (11)), we have that, for every , where and

Note that the limit does not exist.

Let . By proceeding as in ([18], p. 213), we can see that

We have that

and writing , we get

Also, we have that

We can write, for every and ,

The derivative under the integral sign is justified and we get

To ensure the change on the order of integration, we are going to see that and

We observe that, since , it is sufficient to see that , , for certain continuous function .

We consider the decomposition

Since by (252), we have that

We now estimate , . We take into account that, if , with , then

where , . By considering the decomposition (89) for , the estimations (92) and the proof of (97), we can see that

Then, according to (252), we have that

Since , , we get when ,

We deduce that

Then, , and (254) and (255) are established.

Note that the estimations that we have just proved are depending on .

By interchanging the order of integration, we get

It follows that

for almost all .

We conclude that where

Salogni ([1], Theorem 3.4.3) proved that is bounded from into itself, for every , and Bruno ([6], Theorem 4.1, (i)) established that is bounded from into . In order to extend to functions taking values in a Banach space, we need to prove these results in a different way by making a comparation between and .

Let . We define the local and global part as follows and

The operators and are defined in analogous way.

We are going to see that

We can write

First, we observe that

By using (250) and since when , we have that

Finally, from (252), proceeding as above for the estimation of , but now by taking into account that , we obtain that

Thus, (269) is proved.

As it was established in Section 3.2,

Then, the operator defined by is bounded from into itself for every . From ([1], Proposition 3.2.5), it follows that is bounded from into itself, for every .

Note that

We now study the global operator . We recall that is the integral operator defined by

We have, by making , , that

For every , there exists a polynomial with degree 4 and a positive function such that

Then, for every , the function changes the sign at most four times in . We deduce that

If , with , we have that

Also, , . Then, if , then .

We take . Since when , according to ([24], Proposition 2.1), we get, for ,

By proceeding as in the proof of Theorem 2, we can see that, for every , and

Hence, the operator defined by is bounded from into itself, for every .

On the other hand, according to ([1], Lemma 3.3.3)

We recall that represents the angle between the nonzero vectors and when and , , when . By ([1], Lemma 3.3.4) the operator is bounded from into .

Note that .

By taking into account that is a Calderón-Zygmund operator, the arguments developed in the proofs of Theorems 2 and 3 allow us to finish the proof of this theorem.

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Jorge J. Betancor and Lourdes Rodríguez-Mesa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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