Rafeiro, HumbertoSamko, Stefan2018-12-072018-12-072010-030022-247Xhttp://hdl.handle.net/10400.1/11664Under 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(alpha)[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(alpha) f in the space L(p(-))(R(n)) is equivalent to the existence of the limit 1/epsilon(alpha)(I - P(epsilon))(alpha) f. In the pre-limiting case sup(x) p(x) < n/alpha we show that the Bessel potential space is characterized by the condition parallel to(I - P(epsilon))(alpha) f parallel to p((.)) <= C epsilon(alpha). (C) 2009 Elsevier Inc. All rights reserved.engL-P SpacesMaximal-FunctionLebesgue SpacesOperatorsjournal article10.1016/j.jmaa.2009.11.008