Fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations. (June 2020) - Huang, Jizheng; Li, Pengtao; Liu, Yu; Shi, Shaoguang
Abstract: Let ( M, d, μ ) be a metric measure space with upper and lower densities: | | | μ | | | β ≔ sup ( x, r ) ∈ M × ( 0, ∞ ) μ ( B ( x, r ) ) r − β < ∞ ; | | | μ | | | β ⋆ ≔ inf ( x, r ) ∈ M × ( 0, ∞ ) μ ( B ( x, r ) ) r − β ⋆ > 0, where β, β ⋆ are two positive constants which are less than or equal to the Hausdorff dimension of M . Assume that p t ( ⋅, ⋅ ) is a heat kernel on M satisfying Gaussian upper estimates and L is the generator of the semigroup associated with p t ( ⋅, ⋅ ) . In this paper, via a method independent of Fourier transform, we establish the decay estimates for the kernels of the fractional heat semigroup e − t L α t > 0 and the operators L θ ∕ 2 e − t L 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 Radon measure ν on M × ( 0, ∞ ) such that R α L p ( M ) ⊆ L q ( M × ( 0, ∞ ), ν ) under ( α, p, q ) ∈ ( 0, 1 ) × ( 1, ∞ ) × ( 1, ∞ ), where u = R α f is 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 .