Papers › Regularity estimates of fractional heat semigroups related with uniformly elliptic operators
Regularity estimates of fractional heat semigroups related with uniformly elliptic operators
Honglei Shi, Pengtao Li, Kai Zhao
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Let L = -div( A(x) ·∇) + V(x) be a second-order uniformly elliptic operator on ℝ ⁿ (n≥3), where A(x) is a real symmetric matrix satisfying standard ellipticity conditions, and V is a nonnegative potential belonging to the reverse H\"older class. For α∈(0,1), we study regularity estimates of the fractional heat semigroups { exp (-tL^α )} _(t > 0), via the subordination formula and the fundamental solution of the associated uniformly parabolic equation ∂ₜ u + Lu = 0. This approach avoids the use of Fourier transforms and is applicable to second-order differential operators whose heat kernels satisfy Gaussian upper bounds. As an application, we characterize the Campanato-type space Λ_(L , γ) ( ℝⁿ ) via the fractional heat semigroups {exp ( - t L ^α ) } _(t > 0).
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