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 B α [ L p ( ⋅ ) ( 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 D α f in the space L p ( ⋅ ) ( R n ) is equivalent to the existence of the limit 1 ε α ( I − P ε ) α f . In the pre-limiting case sup x p ( x ) < n α we show that the Bessel potential space is characterized by the condition ‖ ( I − P ε ) α f ‖ p ( ⋅ ) ≦ C ε α .