Regularity estimates of fractional heat semigroups related with uniformly elliptic operators

TL;DR

This work develops a framework for the space–time regularity of fractional heat semigroups associated with uniformly elliptic divergence-form operators $L=-\mathrm{div}(A(x)\nabla)+V(x)$. By leveraging a subordinated representation through the parabolic equation $\partial_tu+Lu=0$, the authors obtain pointwise, gradient, and time-derivative kernel estimates for $K_{\alpha,t}^{L}$ (and related kernels) under Gaussian upper bounds and mild smoothness on $A$ and $V$. They establish $L^p$–type bounds and Lipschitz continuity for the kernels, including $D_{\alpha,t}^{L,\beta}$ and $\widetilde{D}_{\alpha,t}^{L,\beta}$, and derive sharp regularity results for the spatial variables via parabolic arguments and perturbation theory. As an application, they characterize Campanato-type spaces $\Lambda_{L,\gamma}(\mathbb{R}^n)$ and Campanato–Sobolev spaces $\Lambda^{\kappa}_{L,\gamma}(\mathbb{R}^n)$ in terms of fractional heat semigroups and associated Carleson/area function criteria, linking these spaces to Hardy spaces $H^p_L$ and their duals. The results generalize known Schrödinger-operator bounds to more general uniformly elliptic operators, offering a robust semigroup approach to regularity in nonlocal settings.

Abstract

Let $L = -{\rm div}( A(x) \cdot \nabla ) + V(x)$ be a second-order uniformly elliptic operator on $\mathbb{ R }^{n}$ $(n\geq 3)$, where $A(x)$ is a real symmetric matrix satisfying standard ellipticity conditions, and $V$ is a nonnegative potential belonging to the reverse Hölder class. For $ α\in (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 $ \partial_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 $Λ_{ L , γ} \left( \mathbb{R}^n \right)$ via the fractional heat semigroups $\{exp ( - t L ^ {α} ) \} _ { t > 0 } $.

Theorems & Definitions (43)

• Remark 1
• Lemma 1
• Lemma 2
• Lemma 3
• Proposition 4
• Proposition 5
• Proposition 6
• Proposition 7
• Lemma 8
• proof