Abstract
This paper is devoted to the maximal regularity of sectorial operators in Lebesgue spaces Lp⋅ 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 Lp⋅−regularity of sectorial operators is established. This paper also investigates the trace of the maximal regularity space E01,p⋅I, together with the imbedding property of E01,p⋅I into the range-varying function space C−I,X1−1/p⋅,p⋅. Finally, a type of semilinear evolution equations with domain-varying nonlinearities is taken into account.
Highlights
Maximal Lp− regularity of sectorial operators is an important theory, which brings a powerful tool in investigating the evolution equations in Lp− spaces
We firstly focus on boundedness of the singular integral operator with operator-valued kernel on Lp(·)(RN, X)
T is called a singular integral operator of strong (q, q) type, provided it can be extended onto Lq(RN, X) to Lq(RN, Y) for some 1 < q < ∞, and there is a C1 > 0 such that
Summary
Maximal Lp− regularity of sectorial operators is an important theory, which brings a powerful tool in investigating the evolution equations in Lp− spaces. Is method is associated with the maximal operator M, the sharp maximal operator M# and the singular integral operator T attached with A in Lp− 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 A ∈ MRq(I) for some 1 < q < ∞, A ∈ MRp(·)(I) for all log-Holder continuous exponents p(·) with 1 < p− < p+ < ∞, where p+ and p− denote the supremum and infimum of p(·) on the interval I, respectively. As preliminaries, in this and the sections, we make a brief review on the maximal Lp− regularity of sectorial operators and the Xθ(·)− valued function spaces. 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
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