Boundedness of Singular Integral Operators with Operator-Valued Kernels and Maximal Regularity of Sectorial Operators in Variable Lebesgue Spaces

Zhang, Qinghua; Zhu, Yueping; Wang, Feng

doi:https://doi.org/10.1155/2020/3672892

Journal of Function Spaces

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Abstract Introduction Preliminaries Proofs Conclusions and Discussion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 3672892 | https://doi.org/10.1155/2020/3672892

Boundedness of Singular Integral Operators with Operator-Valued Kernels and Maximal Regularity of Sectorial Operators in Variable Lebesgue Spaces

Qinghua Zhang,¹Yueping Zhu,¹and Feng Wang¹

Academic Editor: Yongsheng S. Han

Received29 Sept 2019

Accepted11 Nov 2019

Published10 Mar 2020

Abstract

This paper is devoted to the maximal regularity of sectorial operators in Lebesgue spaces with a variable exponent. By extending the boundedness of singular integral operators in variable Lebesgue spaces from scalar type to abstract-valued type, the maximal regularity of sectorial operators is established. This paper also investigates the trace of the maximal regularity space , together with the imbedding property of into the range-varying function space . Finally, a type of semilinear evolution equations with domain-varying nonlinearities is taken into account.

1. Introduction

Maximal regularity of sectorial operators is an important theory, which brings a powerful tool in investigating the evolution equations in spaces. Let X be a Banach space and A be a closed operator defined in X with the dense domain and dense range , and let endowed with the graph norm. A is called a sectorial operator, if there are constants and , such that the sectoris contained in , and the inequalityholds for all . Recall that for a sectorial operator A, its negative generates an analytic semigroup (refer to [1], Section 2.5).

Let with or , and consider the abstract differential equation

We say that A satisfies the maximal regularity on I, or in symbol, if for all , there is a unique solution of equation (3) with the initial value . Using the interpolation method for the convolution operators with singular kernels, we know that (see [2, 3], etc.) if for some , then for all .

In [4, 5], the authors gave a general introduction on the regularity of sectorial operators and [6–9] investigated the maximal regularity of the second order elliptic and Stokes operators, made some or estimates for the parabolic evolution and nonstationary Navier–Stokes equations. Maximal regularity of sectorial operators in weighted spaces was established in [10] and applied in quasilinear equations in [11, 12]. During the same period, Chill and Fiorenza [13] dealt with the maximal regularity of sectorial operators in Orlicz spaces of rearrangement invariant Banach functions.

In some concrete situations, the nonlinear term f attached to (3) may be lying in , so it is natural to consider the maximal regularity in such spaces. Since is a variable exponent, the interpolation method used in [2, 3] is not suitable anymore. Because of lacking of translation invariance, space is not arrangement invariant, hence tools developed in [13] are not applicable directly yet. In order to establish the maximal regularity of A, a recently developed method for the maximal operator and singular integral operators can be employed. This method is associated with the maximal operator M, the sharp maximal operator and the singular integral operator T attached with A in spaces with variable exponents (refer to [14, 15]). By employing this method, with the aid of the estimate obtained in [13], in this paper, we will prove that if for some , then for all log-Hölder continuous exponents with , where and denote the supremum and infimum of on the interval I, respectively.

In order to apply the maximal regularity theory to the quasilinear evolution equations, in this paper, we also make some investigations on the trace of the maximal regularity space . As we know that, for all subintervals J of I, can be imbedded into the space (refer to [5, 10]), where , and is the real interpolation space between X and , a question arises naturally, that is, for arbitrary , whether or not the trace space of is exactly. This question was raised in [16] and has had not an answer until now. The main obstacle is that the imbedding bounds of ↪ depend on the length of J, and it could not be controlled as the interval J shrinks to the point t. Here, by using the properties of the log-Hölder function, together with the theory of range-varying function spaces developed in [16, 17], we give this question an affirmative answer. We will show that, in case that generates a exponentially decaying semigroup and is a log-Hölder function, then the homogeneous maximal regularity space can be imbedded in , a range-varying function space established on the regular Banach space net . This gives an affirmative answer to the question about the trace of the homogeneous maximal regularity space.

This paper is organized as follows. As preliminaries, in this and the next sections, we make a brief review on the maximal regularity of sectorial operators and the valued function spaces. In Section 3, the main results on singular integral operators with operator-valued kernels with application to maximal regularity in and time-varying trace of the maximal regularity space are derived. All the results will be applied to a semilinear evolution equation with the time-dependent nonlinearity at the end of the paper. This example implies the wide application of our work in the study of parabolic partial differential equations with nonstandard growth.

2. Preliminaries

Given a Banach space X and a sectorial operator A which is densely defined in X. Let endowed with the graph norm as above, and let or .

Given , define the maximal regularity spaceendowed with the norm and the homogeneous subspace

Under present situations, ↪ for and ↪ for with the imbedding bounds independent of (refer to [18], Section 3.4.10).

By the inverse operator theorem of the closed operators, we can assert that, if , then there is a constant such thatwhere and is the solution of equation (3). Furthermore, if , then is independent of the length of I, and generates an exponentially decaying analytic semigroups , i.e., there are constants and such thatfor all . In this case, the real interpolation space has an equivalent norm (cf. [19], Section 5.1)

It is well known that (cf. [4, 5, 13]) A has the maximal regularity on the interval I if and only if the singular integral operator T defined throughis well defined and can be extended onto as a bounded linear operator.

As preparations for the discussions on the trace of the space , let us recall the definition and construction of the abstract-valued function space of the range-varying type. For the detailed discussions, please refer to [16, 17].

Suppose that is an ordered topological space with the order , in which every order-bounded subset has the order supremum and order infimum. Suppose also is totally order-bounded, i.e., there are in another order space containing such that for all . Under present situation, is called a totally bounded lattice. Let and , we say that is approaching α, we mean that for all and at the same time.

Let be a family of Banach spaces attached to . We say it is a regular Banach space net, provided the hypotheses are both fulfilled:(1)If , then ↪ , and there is a constant independent of such that for all .(2)If approaches β, then for all . Moreover, if for all and , then and .

Let I be an interval as above and be the collection of all bounded subintervals of I. Consider the map . When we say θ is order-continuous, we mean that for any nest of intervals shrinking to t, the limitalways holds, where and denote the order infimum and supremum of θ on J, respectively.

Define

This is a linear space according to the addition and scalar multiplication of functions. Moreover, for all , the composite function is measurable.

There are two types of range-varying function spaces derived from , one is of continuous type defined throughwhich is a Banach space equipped with the norm or equivalently . And the other is of an integral type defined throughwith the Luxemburg normwhere is a measurable variable exponent. If , then we obtain the familiar Lebesgue–Bochner space of variable exponent type .

Discussions in [16] tell us that, if we take as the totally bounded lattice, then is a regular Banach space net. Hence, for the continuous exponent , we obtain the linear space and the Banach space . We can also construct the maximal regularity space with variable exponent with the norm and homogeneous subspace . All of them will be applied in the coming arguments.

3. Main Results and Proofs

We firstly focus on boundedness of the singular integral operator with operator-valued kernel on .

Let X and Y be two Banach spaces, , and let is a locally integrable function. Define a linear operator T as follows:

T is called a singular integral operator of strong type, provided it can be extended onto to for some , and there is a such thatfor all .

If there are constants , such thatthen k is called a standard kernel. Here assumption (17) tells us that is a singular kernel, and (18) and (19) together imply that is locally Hölder continuous in some way. All of them are connected to the strong boundedness of T in case that k is a scalar kernel. And under the strong assumption of T, we only use (18) to deal with the strong property of T for the operator-valued kernel.

We say k satisfies the Hörmander’s integral condition, if there is another constant such that for every cube Q with sides parallel to the coordinate axes and all , we havewhere represents the cube with the same center and double sides of Q.

Similar to the scalar case, for the operator-valued kernel, we have [13].

Lemma 1. Under Hörmander’s integral condition (20), a singular integral operator T of strong type is also of weak type in the sense thatfor all and some constant .

The following lemma is a natural extension of [20, 21] of the standard kernel from the scalar type to the operator-value type. For the convenience of the reader, we state it here and give it a complete proof.

Lemma 2. Let T be an operator defined through (15) with the standard kernel k. Suppose that T can be extended as a weak type operator as above and , then for all , the scalar function lies in the space , and there is a constant such thatfor all .

Proof. Take any and . Without loss of generality, assume that . Let Q be a cube containing with sides parallel to the coordinate axes. Consider the split , . For the first part , we haveTake , and we obtainFor the second part , we haveNotice that for all , by (18),where r denotes the radius of Q, we obtainPutting the above two estimates together, we obtainfor some constant , which means that , and estimate (22) holds.
Given a variable exponent . We say p is log-Hölder continuous, or symbolically , if there are constants and such thatfor all .

Remark 1. If is a bounded domain of , then p is log-Hölder continuous on , if and only if the first inequality of (29) is satisfied.
Next lemma is an important result in harmonic analysis, and it was first proved in [14] for bounded exponents and later extended to general cases in some literatures. For the complete proof with detail discussions, please refer to [15].

Lemma 3. Assume that and , then the maximal operator M is bounded from to , i.e., there is a constant such that

Furthermore, under the extra assumption , for the sharp operator , there is another constant for which estimatefor all (refer to [22], P.148).

Putting all the facts together, we obtain the following.

Theorem 1. Let T be a singular integral operator of strong type for some with the standard kernel k satisfying Hörmander’s integral condition and be a variable exponent satisfying . Then, T is bounded from to with the bounds .

Proof. Take a constant exponent s such that , and then the variable exponent is also log-Hölder continuous the same constant , and and . Thus, combining (22), (30), and (31), we can deduce thatwhere the constant .

Remark 2. This conclusion is a natural extension of that in [15], Section 1.6.3 for the singular integral operator from the scalar type to the abstract-valued type. For another treatment of the extension, please refer to [23].
Now we can establish the maximal regularity for the sectorial operator A. DefineStraight calculations show that is a standard kernel satisfying Hörmander’s integral condition, and can be expressed by , where is the zero extension of f. In this setting, is equivalent to say that T is a singular integral operator of strong type.
Given a variable exponent satisfying the log-Hölder condition (29) with replaced by I and . From [15], Section 4.1, we know that p has an extension with the same constant and . Analogous to and , define the maximal regularity space and its closed subspace with the normApplying Theorem 1, we can drive the following.

Theorem 2. Assume that for some and with . Then, A satisfies the maximal regularity on I, that is, for all , there is a unique function solving (3) with , and satisfyingwith the constant depending on , and .

In the following paragraphs, we turn our attention to the trace of . Here and after we need assumption (7) for the semigroup . Denote by the trace space of , that is,with the norm

Proposition 1. Suppose that with , thenwith the equivalent norms.

Proof. Firstly, for all with , we have for some constant independent of , and . Consider the splitFor the second part, we haveDefine , let and , then both of and are log-Hölder continuous with the same constant . Thus, for the first part , we havewherePutting all the parts together, we obtain with the constant independent of . And by scaling arguments, we have , which in turn yields .
Conversely, suppose that with and . By the unit ball property, we have and . By imbedding ↪ , we get the estimate for some constant independent of u. Notice thatwe haveSincewhereby the unit ball property of and boundedness of the maximal operator, we then have with the constant independent of u. By the scaling arguments and definition of , we can also derive that . Thus, equivalence of and has been reached.

Proposition 2. Under the same assumptions upon p as above, we have

Proof. irstly, for every bounded subinterval J of , imbeddingholds with the imbedding bounds depending on J, from which we obtainwhere the range-varying function space comes from [17].
For each , consider the translation operator . Obviously, is a bounded linear operator from to , andAnother fact is that for every , the translated exponent is also log-Hölder continuous with the same constant and . Thus, for all and all , using Proposition 1, we haveTherefore, andwith a constant independent of u (refer to [17]).

Theorem 3. Suppose that with , thenwith the imbedding bounds independent of .

Proof. LetIt is easy to check that with and , , andBy invoking Proposition 2, we know that , andNotice that and for all , we obtainThus, the proof has been proved.
At the end of the paper, we will use the maximal regularity results to deal with the semilinear evolution equation:where for some , , , and is a nonlinear map fulfilling assumptions as follows:(1)For almost all , can be defined and locally Lipschitz continuous on (2)For all , the compound function is strongly measurable on (3), and

Theorem 4. Under present situations, for every , there is a number , such that for each , and semilinear equation (59) has a unique solution on the interval with the initial condition . Here,is a ball in . Moreover, for every two points , , the corresponding solutions , , satisfy

Proof of this theorem is much similar to that completed in [16] where embedding is replaced by .

Remark 3. Using the maximal regularity theory, hypothesis used in [16] is dropped. Instead, here, a weaker assumption : is applied. In this sense, Theorem 4 is an improvement of that in [16].

4. Conclusions and Discussion

In this paper, we study the maximal regularity for the sectorial operators. By extending the boundedness of singular integral operators from the scalar type to abstract-valued type; we see that, if a sectorial operator A lies in for some , then it lies in for every Hölder continuous variable exponent with . We also prove that if generates an exponentially decaying analytic semigroup, then for the maximal regular space , its trace space is exactly , and the homogeneous maximal regular space can be embedded continuously into the range-varying function space with the embedding bounds independent of the length of the interval I. Different to the constant exponent type, translation series could not make up a semigroup on , since does not have the translation-invariant property. Consequently, whether or not the following estimates:still hold for the variable exponents remains unknown. We also wonder that under what situations maximal regularity can be preserved under time-dependent perturbation .

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11771223) and Preresearch Project of Nantong University (17ZY01).

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Copyright © 2020 Qinghua Zhang et al. 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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