Heat kernels for time-dependent non-symmetric stable-like operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:465

Article

Chen, Zhen-Qing¹  Zhang, Xicheng² 

[1] Univ Washington, Dept Math, Seattle, WA 98195 USA

[2] Wuhan Univ, Sch Math & Stat, Wuhan 430072, Hubei, Peoples R China

关键词: Heat kernel estimates;    Non-symmetric nonlocal operator;    Levi's method;    Riesz's transform;   

DOI  :  10.1016/j.jmaa.2018.03.054

来源: Elsevier

【 摘 要 】

When studying non-symmetric nonlocal operators on R-d: Lf(x) = integral(Rd) (f(x + z) - f(x) - del f(x) . z1({vertical bar z vertical bar <= 1})) kappa(x, z)/vertical bar z vertical bar(d+alpha) dz, where 0 < alpha < 2, d >= 1, and kappa (x, z) is a function on R-d x R-d that is bounded between two positive constants, it is customary to assume that kappa(x, z) is symmetric in z. In this paper, we study heat kernel of L and derive its two-sided sharp bounds without the symmetric assumption kappa(x, z) = kappa(x, -z). In fact, we allow the kernel kappa to be time-dependent and x -> kappa(t, x, z) to be only locally beta-Holder continuous with Holder constant possibly growing at a polynomial rate in vertical bar z vertical bar. We also derive gradient estimate when beta is an element of (0 V (1 - alpha), 1) as well as fractional derivative estimate of order 0 is an element of (0, (alpha + beta) Lambda 2) for the heat kernel. Moreover, when alpha is an element of (1, 2), drift perturbation of the time-dependent non-local operator L-t with drift in Kato's class is also studied in this paper. As an application, when kappa(x, z) = kappa(z) does not depend on x, we show the boundedness of nonlocal Riesz's transformation: for any p > 2d/(d + alpha), parallel to L(1/2)f parallel to(p) asymptotic to parallel to Gamma(f)(1/2)parallel to(p), where Gamma(f) := 1/2L(f(2)) - f L f is the carre du champ operator associated with L, and.L-1/2 is the square root operator of L defined by using Bochner's subordination. Here asymptotic to means that both sides are comparable up to a constant multiple. (C) 2018 Elsevier Inc. All rights reserved.

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