Some Extensions of E. Stein’s Work on Littlewood–Paley Theory Applied to Symmetric Diffusion Semigroups

Abstract

Discrete diffusion semigroups have proven to be highly effective tools for machine learning and data analysis due to the interplay between diffusion processes (of inferences), and their naturally associated geometries. Inspired by the harmonic analysis machinery that E. Stein developed for symmetric diffusion semigroups acting on $$L_p$$ spaces, we show that a correspondence between the rate of diffusion approximation and a Besov-type version of smoothness exists in the general continuous case, even without a local kernel representation. Specifically, let $$\left\{ A_t\right\} _{t\ge 0}$$ be a symmetric diffusion semigroup on $$L_p(X)$$, for X a complete positive $$\sigma $$-finite measure space. We first establish that for $$1<p<\infty $$ the semigroup $$\left\{ A_t\right\} _{t\ge 0}$$ acting on $$L_p(X)$$ is strongly continuous and holomorphic. We then show that for $$f\in L_p(X)$$ and $$0<\alpha <1$$, the following are equivalent: We also present some extensions, including results for the $$p=\infty $$ case.