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- On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato spacePublication . Rafeiro, Humberto; Samko, StefanFor the Riesz potential of variable order over bounded domains in Euclidean space, we prove the boundedness result from variable exponent Morrey spaces to variable exponent Campanato spaces. A special attention is paid to weaken assumptions on variability of the Riesz potential.
- 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.
- 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.
- Commutators of fractional maximal operator in variable Lebesgue spaces over bounded quasi‐metric measure spacesPublication . Guliyev, Vagif S.; Samko, StefanWe study the fractional maximal commutators Mb,𝜂 and the commutators[b, M𝜂] of the fractional maximal operator with b ∈ BMO(X) in the variable Lebesgue spaces Lp(·)(X) over bounded quasi-metric measure spaces. We give necessary and sufficient conditions for the boundedness of the operators Mb,𝜂 and [b, M𝜂] on the spaces Lp(·)(X) when b ∈ BMO(X). Furthermore, we obtain some new characterizations for certain subspaces of BMO(X).