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

Maximal L− regularity of sectorial operators is an important theory, which brings a powerful tool in investigating the evolution equations in L− spaces. Let X be a Banach space and A be a closed operator defined in X with the dense domain D(A) and dense range R(A), and let X1 � D(A) endowed with the graph norm. A is called a sectorial operator, if there are constants M0 > 0 and 0<ω< π/2, such that the sector


Introduction
Maximal L p − regularity of sectorial operators is an important theory, which brings a powerful tool in investigating the evolution equations in L p − spaces. Let X be a Banach space and A be a closed operator defined in X with the dense domain D(A) and dense range R(A), and let X 1 � D(A) endowed with the graph norm. A is called a sectorial operator, if there are constants M 0 > 0 and 0 < ω < π/2, such that the sector is contained in 9(− A), and the inequality holds for all λ ∈ Σ ω . Recall that for a sectorial operator A, its negative − A generates an analytic C 0 − semigroup e − tA (refer to [1], Section 2.5).
Let I � [0, b] with 0 < b < ∞ or I � [0, ∞), and consider the abstract differential equation We say that A satisfies the maximal L p − regularity on I, or A ∈ MR p (I) in symbol, if for all f ∈ L p (I, X), there is a unique solution u ∈ W 1,p (I, X) ∪ L 1,p (I, X 1 ) of equation (3) with the initial value u(0) � 0. Using the interpolation method for the convolution operators with singular kernels, we know that (see [2,3], etc.) if A ∈ MR q (I) for some 1 < q < ∞, then A ∈ MR p (I) for all 1 < p < ∞.
In [4,5], the authors gave a general introduction on the L p − regularity of sectorial operators and [6][7][8][9] investigated the maximal L p − regularity of the second order elliptic and Stokes operators, made some L p − or L p − L q − estimates for the parabolic evolution and nonstationary Navier-Stokes equations. Maximal regularity of sectorial operators in weighted L p − 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 L p(·) (I, X), so it is natural to consider the maximal regularity in such spaces. Since p(·) is a variable exponent, the interpolation method used in [2,3] is not suitable anymore. Because of lacking of translation 1,p(·) 0 (I, X) ∩ L 1,p(·) (I, X 1 ) can be imbedded in C − (J, X 1− 1/p(·),p(·) ), a range-varying function space established on the regular Banach space net X α,p(α) : α ∈ [0, 1] . is gives an affirmative answer to the question about the trace of the homogeneous maximal regularity space. is paper is organized as follows. As preliminaries, in this and the next sections, we make a brief review on the maximal L p − regularity of sectorial operators and the X θ(·) − valued function spaces. In Section 3, the main results on singular integral operators with operator-valued kernels with application to maximal regularity in L p(·) (I, X) 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. is example implies the wide application of our work in the study of parabolic partial differential equations with nonstandard growth.

Preliminaries
Given a Banach space X and a sectorial operator A which is densely defined in X. Let X 1 � D(A) endowed with the graph norm as above, and let I � [0, b] or I � [0, ∞).
By the inverse operator theorem of the closed operators, we can assert that, if A ∈ MR q (I), then there is a constant C q > 0 such that where f ∈ L q (I, X) and u ∈ E 1,q 0 (I) is the solution of equation (3). Furthermore, if A ∈ MR q (R + ), then C q is independent of the length of I, and − A generates an exponentially decaying analytic semigroups e − tA , i.e., there are constants M 0 ≥ 1 and ω > 0 such that for all t ≥ 0. In this case, the real interpolation space X 1− 1/q,q � (X, X 1 ) 1− 1/q,q has an equivalent norm (cf. [19], Section 5.1) It is well known that (cf. [4,5,13]) A has the maximal L q − regularity on the interval I if and only if the singular integral operator T defined through is well defined and can be extended onto L q (I, X) as a bounded linear operator.
As preparations for the discussions on the trace of the space E 1,p(·) 0 (I), 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 A is an ordered topological space with the order ≺ , in which every order-bounded subset has the order supremum and order infimum. Suppose also A is totally orderbounded, i.e., there are α ± in another order space containing A such that α − ≺ α ≺ α + for all α ∈ A. Under present situation, A is called a totally bounded lattice. Let α k ⊆ A and α ∈ A, we say that α k is approaching α, we mean that α k ≺ β for all k ∈ N and lim k⟶∞ α k � β at the same time.
Let X α : α ∈ A be a family of Banach spaces attached to A. We say it is a regular Banach space net, provided the hypotheses are both fulfilled: (1) If α ≺ β, then X β ↪ X α , and there is a constant C > 0 independent of α, β such that ‖x‖ α ≤ C‖x‖ β for all x ∈ X β .
Let I be an interval as above and Λ(I) be the collection of all bounded subintervals of I. Consider the map θ: I ⟶ A. When we say θ is order-continuous, we mean that for any nest of intervals J k ∈ Λ(I): k � 1, 2, . . . shrinking to t, the limit always holds, where θ − J and θ + J denote the order infimum and supremum of θ on J, respectively. Define is is a linear space according to the addition and scalar multiplication of functions. Moreover, for all ere are two types of range-varying function spaces derived from L 0 (I, X θ(·) ), one is of continuous type defined through which is a Banach space equipped with the norm . And the other is of an integral type defined through with the Luxemburg norm where p: I ⟶ [1, ∞) is a measurable variable exponent. If θ(t) ≡ 0, then we obtain the familiar Lebesgue-Bochner space of variable exponent type L p(·) (I, X 0 ).

Main Results and Proofs
We firstly focus on boundedness of the singular integral operator with operator-valued kernel on L p(·) (R N , X).
Let X and Y be two Banach spaces, Υ � (x, y): x, y ∈ R N , x ≠ y}, and let k: Υ ⟶ L(Y, X) is a locally integrable function. Define a linear operator T as follows: T is called a singular integral operator of strong (q, q) type, provided it can be extended onto L q (R N , X) to L q (R N , Y) for some 1 < q < ∞, and there is a C 1 > 0 such that then k is called a standard kernel. Here assumption (17) tells us that k(x, y) is a singular kernel, and (18) and (19) together imply that k(x, y) is locally Hölder continuous in some way. All of them are connected to the strong (q, q) boundedness of T in case that k is a scalar kernel. And under the strong (q, q) assumption of T, we only use (18) to deal with the strong (p(·), p(·)) property of T for the operator-valued kernel. We say k satisfies the Hörmander's integral condition, if there is another constant C 3 > 0 such that for every cube Q Journal of Function Spaces 3 with sides parallel to the coordinate axes and all y, z ∈ Q, we have where 2Q 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]. (20), a singular integral operator Tof strong (q, q) type is also of weak (1, 1) type in the sense that

Lemma 1. Under Hörmander's integral condition
e 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. (15) with the standard kernel k. Suppose that T can be extended as a weak (1, 1) type operator as above and 0 < s < 1, then for all

Lemma 2. Let T be an operator defined through
for all x ∈ R N .
Proof. Take any f ∈ C ∞ 0 (R N ; Y) and x 0 ∈ R N . Without loss of generality, assume that Mf(x 0 ) > 0. Let Q be a cube containing x 0 with sides parallel to the coordinate axes.
Take t � Mf(x 0 ), and we obtain For the second part f 2 , we have Notice that for all x ∈ Q, by (18), where r denotes the radius of Q, we obtain Putting the above two estimates together, we obtain 1 for some constant C 5 � C (N, s, δ, C 4 ), which means that ‖Tf‖ s Y ∈ BMO(R N ), and estimate (22) holds. Given a variable exponent p: R N ⟶ [1, ∞). We say p is log-Hölder continuous, or symbolically p ∈ P log (R N ), if there are constants C 0 > 0 and p ∞ ≥ 1 such that for all x, y ∈ R N .

Remark 1.
If Ω is a bounded domain of R N , then p is log-Hölder continuous on Ω, 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 P log (R N ) and p − > 1, then the maximal operator M is bounded from L p(·) (R N ) to L p(·) (R N ), i.e., there is a constant C 6 � C(N, p − , C 0 ) > 0 such that Furthermore, under the extra assumption p + < ∞, for the sharp operator M # , there is another constant for all f ∈ L q (R N ) (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
(q, q) type for some 1 < q < ∞ with the standard kernel k satisfying Hörmander's integral condition and p ∈ P log (R N ) be a variable exponent satisfying 1 < p − ≤ p + < ∞. en, T is bounded from L p(·) (R N ; Y) to L p(·) (R N ; X) with the bounds Proof. Take a constant exponent s such that 0 < s < 1, and then the variable exponent p(·)/s is also log-Hölder continuous the same constant C 0 , and (p(·)/s) − � p − /s > 1 and (p(·)/s) + � p + /s < ∞. us, combining (22), (30), and (31), we can deduce that where the constant is 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 L p(·) − regularity for the sectorial operator A. Define

Straight calculations show that k(t, s) � K(t − s) is a standard kernel satisfying Hörmander's integral condition, and Tf can be expressed by
is the zero extension of f. In this setting, A ∈ MR q (I) is equivalent to say that T is a singular integral operator of strong (q, q) type. Given a variable exponent p ∈ P(I) satisfying the log-Hölder condition (29) with R N replaced by I and 1 < p − ≤ p + < ∞. From [15], Section 4.1, we know that p has an extension p ∈ P(R) with the same constant C 0 and p ± � p ± . Analogous to E 1,q (I) and E 1,q 0 (I), define the maximal L p(·) − regularity space E 1,p(·) (I) and its closed subspace E 1,p(·) 0 (I) with the norm Applying eorem 1, we can drive the following.

Theorem 2.
Assume that A ∈ MR q (I) for some 1 < q < ∞ and p ∈ P(I) with 1 < p − ≤ p + < ∞. en, A satisfies the maximal L p(·) − regularity on I, that is, for all f ∈ L p(·) (I, X), there is a unique function u ∈ E 1,p(·) (I) solving (3) with u 0 � 0, and satisfying with the constant C > 0 depending on A, C 0 , C q , and p ± . In the following paragraphs, we turn our attention to the trace of E 1,p(·) (I). Here and after we need assumption (7) for the semigroup e − tA . Denote by cE 1,p(·) (I) the trace space of E 1,p(·) (I), that is, Journal of Function Spaces with the norm with the equivalent norms.

Conclusions and Discussion
In this paper, we study the maximal L p(·) − regularity for the sectorial operators. By extending the boundedness of singular integral operators from the scalar type to abstractvalued type; we see that, if a sectorial operator A lies in MR q (I) for some 1 < q < ∞, then it lies in MR p(·) (I) for every Hölder continuous variable exponent p(·) with 1 < p − < p + < ∞. We also prove that if − A generates an exponentially decaying analytic semigroup, then for the maximal regular space E 1,p(·) (R + ), its trace space is exactly X 1− 1/p(0),p(0) , and the homogeneous maximal regular space 0 E 1,p(·) (I) can be embedded continuously into the rangevarying function space C − (I, X 1− 1/p(·),p(·) ) with the embedding bounds independent of the length of the interval I. Different to the constant exponent type, translation series T s : L p(·) (R + , X) ⟶ L p(·+s) (R + , X), s ≥ 0 could not make up a C 0 semigroup on L p(·) (R + , X), since L p(·) (R + , X) 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 L p(·) − regularity can be preserved under time-dependent perturbation B(t).

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

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.