Characterization of the variable exponent Bessel potential spaces via the Poisson semigroup

TL;DR

This paper characterizes variable exponent Bessel potential spaces using the Poisson semigroup, linking fractional derivatives to semigroup convergence, under specific conditions on the variable exponent function.

Contribution

It provides a novel characterization of variable exponent Bessel potential spaces via the Poisson semigroup and fractional derivatives, extending classical results to variable exponent settings.

Findings

Characterization of Bessel potential spaces via Poisson semigroup convergence.

Equivalence between Riesz fractional derivatives and semigroup limits.

Conditions under which the space is characterized by semigroup decay rates.

Abstract

Under the standard assumptions on the variable exponent $p(x)$ (log- and decay conditions), we give a characterization of the variable exponent Bessel potential space $\mathfrak B^\alpha[L^{p(\cdot)}(\mathbb R^n)]$ in terms of the rate of convergence of the Poisson semigroup $P_t$. We show that the existence of the Riesz fractional derivative $\mathbb{D}^\al f$ in the space $L^{p(\cdot)}(\rn)$ is equivalent to the existence of the limit $\frac{1}{\ve^\al}(I-P_\ve)^\al f$. In the pre-limiting case $\sup_x p(x)<\frac{n}{\al}$ we show that the Bessel potential space is characterized by the condition $\|(I-P_\ve)^\al f\|_{p(\cdot)}\leqq C \ve^\al$