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

Rafeiro, Humberto; Samko, Stefan

http://hdl.handle.net/10400.1/11664

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Authors

Rafeiro, Humberto

Samko, Stefan

Abstract(s)

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(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.