Fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations

Abstract

Let (M, d, µ) be a metric measure space with upper and lower densities:where β, β ⋆ are two positive constants which are less than or equal to the Hausdorff dimension of M. Assume that pt(·, ·) is a heat kernel on M satisfying Gaussian upper estimates and L is the generator of the semigroup associated with pt(·, ·). In this paper, via a method independent of Fourier transform, we establish the decay estimates for the kernels of the fractional heat semigroup {e −tL α }t>0 and the operators {L θ/2 e −tL α }t>0, respectively. By these estimates, we obtain the regularity for the Cauchy problem of the fractional dissipative equation associated with L on (M, d, µ). Moreover, based on the geometric-measure-theoretic analysis of a new L p -type capacity defined in M × (0, ∞), we also characterize a nonnegative Randon measureis the weak solution of the fractional diffusion equation (∂t + L α )u(t, x) = 0 in M × (0, ∞) subject to u(0, x) = f (x) in M.

A Capacity Associated with the Weighted Lebesgue Space and Its Applications

“…(a)-(c) follow immediately from the special case of Proposition 4.2 of [10] (M � R n and dμ � w(x)dx).…”

“…In this section, we characterize (33) under the lower case 1 < p ≤ q < ∞ by C w αp (•). In eorem 5.3 of [10], the authors obtained the following result.…”

“…Based on the theory of the Hedberg-Wolff potential, a noncapacitary characterization of the extension R α L p (R n )⊆L q μ (R 1+n + ) was given in [9]. In [10], the authors considered the fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations. As a complement of [10], when the measure coincided with the Muckenhoupt class, this article addresses a weighted version of that extension.…”

“…In [10], the authors considered the fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations. As a complement of [10], when the measure coincided with the Muckenhoupt class, this article addresses a weighted version of that extension.…”

“…which is a larger class than the A p class. Condition (10) was first introduced to study weighted nonlinear potential theory by Adams in [13]. e plan of this paper is as follows.…”