Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradients
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https://doi.org/10.1007/s11425-018-9472-x
Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradients Renming Song1 , Longjie Xie2,∗ & Yingchao Xie2 1Department 2School
of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA; of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221000, China Email: [email protected], [email protected], [email protected] Received August 31, 2018; accepted October 6, 2018
Abstract
Using Duhamel’s formula, we prove sharp two-sided estimates for the spectral fractional Laplacian’s
heat kernel with time-dependent gradient perturbation in bounded C 1,1 domains. In addition, we obtain a gradient estimate as well as the H¨ older continuity of the heat kernel’s gradient. Keywords MSC(2010)
spectral fractional Laplacian, Dirichlet heat kernel, Kato class, gradient estimate 60J35, 60J50
Citation: Song R M, Xie L J, Xie Y C. Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradients. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-018-9472-x
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Introduction
Let Wt be a Brownian motion in Rd (d > 1) with the generator ∆, and let Tt be an independent α/2stable subordinator where α ∈ (0, 2). Then, the subordinate process Xt := WTt is an isotropic α-stable process, and its infinitesimal generator is the fractional Laplacian operator −(−∆α/2 ), given by ∫ cd,α α/2 −(−∆ )f (x) := [f (x + z) − f (x) − 1|z|61 z · ∇f (x)] d+α dz |z| d R for f ∈ Cc2 (Rd ), where cd,α is a positive constant. It is well known that the heat kernel p(t, x, y) of −(−∆α/2 ), which is also the transition density of X := (Xt )t>0 , has the following estimate: for every t > 0 and x, y ∈ Rd , ( ) t p(t, x, y) ≍ t−d/α ∧ . (1.1) |x − y|d+α Here and in the following sections, for two non-negative functions, f and g, the notation f ≍ g expresses the presence of positive constants, c1 and c2 , such that c1 g(x) 6 f (x) 6 c2 g(x) in the common domain of the definitions of f and g. In [2], Bogdan and Jakubowski used Duhamel’s formula to study the following gradient perturbation of −(−∆α/2 ): L b := −(−∆α/2 ) + b(x) · ∇, α ∈ (1, 2), * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Song R M et al.
Sci China Math
where b = (b1 , . . . , bd ) : Rd → Rd with bj , j = 1, . . . , d, belonging to the Kato class Kα−1 , which is defined d as follows: for γ > 0, { } ∫ |f (y)| Kγd := f ∈ L1loc (Rd ) : lim sup dy = 0 . (1.2) r↓0 x∈Rd B(x,r) |x − y|d−γ Here and below, B(x, r) denotes the open ball centered at x ∈ Rd with the radius denoted as r. Let pb (t, x, y) be the heat kernel of L b . Small-time sharp two-sided estimates for pb (t, x, y) of the form (1.1) have been established in [2, Theorems 1 and 2]. In [2], the authors’ perturbation method includes two key components. First, an accurate estimate for ∇x p(t, x, y) is known, and second, the following 3-P inequality concerning p(t, x, y) holds: there exists C0 > 0 such that
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