Heat Kernel for Fractional Diffusion Operators with Perturbations
Abstract
Let L be an elliptic differential operator on a complete connected Riemannian manifold M such that the associated heat kernel has two-sided Gaussian bounds as well as a Gaussian type gradient estimate. Let L() be the -stable subordination of L for ∈ (1,2). We found some classes K, (,∈ [0,)) of time-space functions containing the Kato class, such that for any measurable b: [0,∞)× M TM and c: [0,∞)× M M with |b|, c∈ K1,1, the operator Lb,c()(t,x):= L()(x)+ <b(t,x), ·> +c(t,x),\ \ (t,x)∈ [0,∞)× M has a unique heat kernel pb,c()(t,x;s,y), 0 s<t, x,y∈ M, which is jointly continuous and satisfies &t-sC\(x,y) (t-s)1\d+ pb,c()(t,x;s,y) C(t-s)(x,y) (t-s)1d+, & |x pb,c()(t,x; s,y)| C(t-s)-1(x,y) (t-s)1d+, 0 s<t,\ x,y∈ M for some constant C>1, where is the Riemannian distance. The estimate of ∇yp()b,c and the H\"older continuity of x pb,c() are also considered. The resulting estimates of the gradient and its H\"older continuity are new even in the standard case where L= on d and b,c are time-independent.
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