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

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 $\|(I-P_\ve)^\al f\|_{p(\cdot)}\leqq C \ve^\al$

22 pages

References in corpus (1)

• Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces

Cited by in corpus (1)

• Explicit Formula for the n-th Derivative of a Quotient