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

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

2009-04-22

arxiv.org/abs/0904.3567v1

Journal of Mathematical Analysis and Applications, Volume 365, Issue 2, 2010, Pages 483-497

Mathematics

Functional Analysis

22 pages

Scientific paper

10.1016/j.jmaa.2009.11.008

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

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