Browsing by Author "Rafeiro, Humberto"
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- A note on vanishing Morrey -> VMO result for fractional integrals of variable orderPublication . Rafeiro, Humberto; Samko, StefanIn the limiting case of Sobolev-Adams theorem for Morrey spaces of variable order we prove that the fractional operator of variable order maps the corresponding vanishing Morrey space into VMO.
- Addendum to “On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato space”, Math Meth Appl Sci. 2020; 1–8Publication . Rafeiro, Humberto; Samko, StefanIn the paper mentioned in the title, it is proved the boundedness of the Riesz potential operator of variable order 𝛼(x) from variable exponent Morrey space to variable exponent Campanato space, under certain assumptions on the variable exponents p(x) and 𝜆(x) of the Morrey space. Assumptions on the exponents were different depending on whether 𝛼(x)p(x)−n+𝜆(x) p(x) takes or not the critical values 0 or 1. In this note, we improve those results by unifying all the cases and covering the whole range 0 ⩽ 𝛼(x)p(x)−n+𝜆(x) p(x) ⩽ 1. We also provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper.
- BMO-VMO results for fractional integrals in variable exponent Morrey spacesPublication . Rafeiro, Humberto; Samko, StefanWe prove the boundedness of the fractional integration operator of variable order alpha(x) in the limiting Sobolev case alpha(x)p(x) = n - lambda(x) from variable exponent Morrey spaces L-p(.),L-lambda(.) (Omega) into BMO (Omega), where Omega is a bounded open set. In the case alpha(x) (math) const, we also show the boundedness from variable exponent vanishing Morrey spaces VLp(.),lambda (.) (Omega) into VMO (Omega). The results seem to be new even when p and A are constant. (C) 2019 Elsevier Ltd. All rights reserved.
- Boundedness of the Bergman projection and some properties of Bergman type spacesPublication . Karapetyants, Alexey; Rafeiro, Humberto; Samko, StefanWe give a simple proof of the boundedness of Bergman projection in various Banach spaces of functions on the unit disc in the complex plain. The approach of the paper is based on the idea of Zaharyuta and Yudovich (Uspekhi Mat Nauk 19(2):139-142, 1964) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderon-Zygmund operators. We exploit this approach and treat the cases of variable exponent Lebesgue space, Orlicz space and variable exponent generalized Morrey spaces. In the case of variable exponent Lebesgue space the boundedness result is known, so in that case we provide a simpler proof, whereas the other cases are new. The major idea of this paper is to show that the approach can be applied to a wide range of function spaces. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollified dilations.
- Characterization of the variable exponent Bessel potential spaces via the Poisson semigroupPublication . Rafeiro, Humberto; Samko, StefanUnder 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.
- Coincidence of variable exponent Herz spaces with variable exponent Morrey type spaces and boundedness of sublinear operators in these spacesPublication . Rafeiro, Humberto; Samko, StefanWe introduce generalized local and global Herz spaces where all their characteristics are variable. As one of the main results we show that variable Morrey type spaces and complementary variable Morrey type spaces are included into the scale of these generalized variable Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized variable Herz spaces with application to variable Morrey type spaces and their complementary spaces, based on the mentioned inclusion.
- Corrigendum to "hardy type inequality in variable lebesgue spaces"Publication . Rafeiro, Humberto; Samko, StefanIn 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].
- Fractional integrals and derivatives: mapping propertiesPublication . Rafeiro, Humberto; Samko, StefanThis 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.
- Fractional operators of variable order from variable exponent Morrey spaces to variable exponent Campanato spaces on quasi-metric measure spaces with growth conditionPublication . Rafeiro, Humberto; Samko, StefanWe study fractional potential of variable order on a bounded quasi-metric measure space (X, d, mu) as acting from variable exponent Morrey space L-p(center dot),L-lambda(center dot)(X) to variable exponent Campanato space L-p(center dot),L-lambda(center dot)(X). We assume that the measure satisfies the growth condition mu B(x,r) <= Cr-gamma, the distance is theta-regular in the sense of Macias and Segovia, but do not assume that the space (X, d, mu) is homogeneous. We study the situation when gamma - lambda(x) <= alpha(x)p(x) <= gamma-lambda(x) + theta p(x), and pay special attention to the cases of bounds of this interval. The left bound formally corresponds to the BMO target space. In the case of right bound a certain "correcting factor" of logarithmic type should be introduced in the target Campanato space.
- Grand Lebesgue space for p = ∞ and its application to Sobolev–Adams embedding theorems in borderline casesPublication . Rafeiro, Humberto; Samko, Stefan; Umarkhadzhiev, SalaudinWe 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.