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- Corrigendum to "hardy type inequality in variable lebesgue spaces"Publication . Rafeiro, Humberto; Samko, Stefan
Expand In this note, we correct a gap made in the assumptions of the final results of the paper [1]. We give these results with corrected formulations and provide a proof of some auxiliary result which allows us to simultaneously admit regions more general than used in [1].Expand - Fractional integrals and derivatives: mapping propertiesPublication . Rafeiro, Humberto; Samko, Stefan
Expand This survey is aimed at the audience of readers interested in the information on mapping properties of various forms of fractional integration operators, including multidimensional ones, in a large scale of various known function spaces.As is well known, the fractional integrals defined in this or other forms improve in some sense the properties of the functions, at least locally, while fractional derivatives to the contrary worsen them. With the development of functional analysis this simple fact led to a number of important results on the mapping properties of fractional integrals in various function spaces.In the one-dimensional case we consider both Riemann-Liouville and Liouville forms of fractional integrals and derivatives. In the multidimensional case we consider in particular mixed Liouville fractional integrals, Riesz fractional integrals of elliptic and hyperbolic type and hypersingular integrals. Among the function spaces considered in this survey, the reader can find Holder spaces, Lebesgue spaces, Morrey spaces, Grand spaces and also weighted and/or variable exponent versions.Expand - Grand Lebesgue space for p = ∞ and its application to Sobolev–Adams embedding theorems in borderline casesPublication . Rafeiro, Humberto; Samko, Stefan; Umarkhadzhiev, Salaudin
Expand We define the grand Lebesgue space corresponding to the case p=infinity$ p = \infty$ and similar grand spaces for Morrey and Morrey type spaces, also for p=infinity$ p = \infty$, on open sets in Rn$ \mathbb {R}<^>n$. We show that such spaces are useful in the study of mapping properties of the Riesz potential operator in the borderline cases alpha p=n$ \alpha p = n$ for Lebesgue spaces and alpha p=n-lambda$ \alpha p = n-\lambda$ for Morrey and Morrey type spaces, providing the target space more narrow than BMO. While for Lebesgue spaces there are known results on the description of the target space in terms better than BMO, the results obtained for Morrey and Morrey type spaces are entirely new. We also show that the obtained results are sharp in a certain sense.Expand - Characterization of the variable exponent Bessel potential spaces via the Poisson semigroupPublication . Rafeiro, Humberto; Samko, Stefan
Expand 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.Expand - Hardy type inequalityin variable lebesgue spacesPublication . Rafeiro, Humberto; Samko, Stefan
Expand We prove that in variable exponent spaces where L-p(.)(Omega), where p(.) satisfies the log-condition and Omega is a bounded domain in R-n with the property that R-n\(Omega) over bar has the cone property, the validity of the Hardy type inequality parallel to 1/delta(x)(alpha)integral(Omega)phi(y)/vertical bar x-y vertical bar(n-alpha)dy parallel to(p(.)) <= C parallel to phi parallel to(p(.)), 0 < alpha < min (1, n/p(+)), where delta(x) is approximately equal to dist(x, partial derivative Omega), is equivalent to a certain property of the domain Omega expressed in terms of alpha and chi(Omega).Expand - Inversion of the Bessel potential operator in weighted variable Lebesgue spacesPublication . Almeida, Alexandre; Rafeiro, Humberto
Expand We study the inversion problem of the Bessel potential operator within the frameworks of the weighted Lebesgue spaces with variable exponent. The inverse operator is constructed by using pproximative inverse operators. This generalizes some classical results to the variable exponent setting. (c) 2007 Elsevier Inc. All rights reserved.Expand