Heat kernel and Lipschitz–Besov spaces

Abstract

On a metric measure space .M; ; / we consider a family of Lipschitz-Besov spaces ƒ s p;q that is defined only using the metric and measure , and a family of Besov spaces B s p;q that is defined using an auxiliary self-adjoint operator L and the associated heat semigroup. Under certain assumptions about the heat kernel of L, we prove the identity of the two families of the function spaces.

Hölder–Lipschitz Norms and Their Duals on Spaces with Semigroups, with Applications to Earth Mover’s Distance

“…The conditions are highly non-restrictive, and in Sect. 3 we show that they hold for a broad class of semigroups, such as those considered in [17]. Examples include heat kernels on closed Riemannian manifolds, heat kernels on certain fractals, and subordinated heat kernels in R n (including the Poisson kernel), as well as the non-symmetric example of shifted heat kernels on R n .…”

“…In contrast to [2,17], we do not assume the existence of a metric on the space X . Rather, we will use the kernels a t (x, y) to define a metric from scratch.…”

“…Specifically, we consider two very general conditions (conditions 1 and 2 formulated below) that are found in the paper [17]. These conditions assume the existence of another metric ρ(x, y) on X and posit that the kernels a t (x, y) exhibit a certain regularity with respect to ρ(x, y).…”

“…Such a distance-that is, a distance of the form ρ(x, y) δ , where 0 < δ < 1-is called a snowflake of ρ(x, y) [18]. By imposing even stronger regularity on a t (x, y) in the form of condition 3 below (also found in [17]), we will show that the distance D α (x, y) is in fact equivalent to a thresholded snowflake of ρ(x, y). We will use our analysis to establish conditions (S), (C), (I) and (R) for many examples of semigroups and compute the distance D α (x, y) for these examples.…”

“…It is wellknown that many function spaces, such as the Besov spaces on R n , can be characterized using, for example, the heat kernel [3,31]. Recent work has generalized these results to function spaces on more general domains; for instance, see [2,17]. More abstractly, a substantial amount of classical harmonic analysis in R n can be pushed through to the setting of a measure space equipped with a diffusion semigroup, as found in Stein's book [29].…”