Variable Besov Spaces Associated with Heat Kernels
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Variable Besov Spaces Associated with Heat Kernels Ciqiang Zhuo1 · Dachun Yang2 Received: 23 December 2019 / Revised: 3 May 2020 / Accepted: 14 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let (X , ρ, μ) be a space of homogeneous type. Suppose that p(·), q(·) : X → (0, ∞] are such that both 1/ p(·) and 1/q(·) satisfy the globally log-Hölder continuous condition, and s(·) : X → R is a bounded function satisfying the locally log-Hölder continuous condition. In this article, the authors introduce the variable Besov space s(·),L B p(·),q(·) (X ), associated with a nonnegative self-adjoint operator L whose heat kernels satisfy small time Gaussian upper bound estimates, the Hölder continuity, and the Markov property, which is new even on the sphere and the ball of Rd . Equivalent characterizations of this space, in terms of Peetre maximal functions and the heat semigroup, are established. Moreover, under the additional assumptions that μ satisfies the reverse doubling condition and the non-collapsing condition, its frame characterization is obtained. When L is the Laplacian operator on Rd , this space coincides with the existing variable Besov space. Keywords Besov space · Metric measure · Variable exponent · Heat kernel · Peetre maximal function Mathematics Subject Classification Primary 46E36 · Secondary 42B35 · 42B25
Communicated by Pencho Petrushev. This project is supported by the National Natural Science Foundation of China (Grant Nos. 11701174, 11831007, 11871100, 11971058, 11761131002, and 11671185). The first author is also supported by the Construct Program of the Key Discipline in Hunan Province, Hunan Natural Science Foundation (Grant No. 2018JJ3321) and China Scholarship Council (Grant No. 201906725036).
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Dachun Yang [email protected] Ciqiang Zhuo [email protected]
1
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, Hunan, People’s Republic of China
2
Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
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Constructive Approximation
1 Introduction In recent decades, function spaces on Euclidean spaces or spaces of homogeneous type have played a prominent role in various analysis areas such as harmonic analysis, partial differential equations, approximation theory, and probability. Particularly, function spaces with variable exponents have attracted more and more attention recently. Indeed, they have been the subject of intensive research since the early work [49] of Kováˇcik and Rákosník, as well as [17] of Cruz-Uribe and [22] of Diening because of their applications in harmonic analysis (see, for instance, [18,19, 23,56]), in partial differential equations and variation calculus (see, for instance, [3,37, 60]), and in fluid dynamics and image processing (see, for instance, [2,59]). Recall that the variable Leb
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