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

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 B^α [L^{p (ṡ)} (Rⁿ)] in terms of the rate of convergence of the Poisson semigroup P_t. We show that the existence of the Riesz fractional derivative D^α f in the space L^{p (ṡ)} (Rⁿ) is equivalent to the existence of the limit frac(1, ε^α) (I - P_ε)^α f. In the pre-limiting case sup_x p (x) < frac(n, α) we show that the Bessel potential space is characterized by the condition {norm of matrix} (I - P_ε)^α f {norm of matrix}_{p (ṡ)} ≦ C ε^α.