Sobolev‐type inequalities for potentials in grand variable exponent Lebesgue spaces

David E. Edmunds, Vakhtang Kokilashvili and Alexander Meskhi

Mathematische Nachrichten, 2019, vol. 292, issue 10, 2174-2188

Abstract: We introduce a new scale of grand variable exponent Lebesgue spaces denoted by L∼p(·),θ,ℓ. These spaces unify two non‐standard classes of function spaces, namely, grand Lebesgue and variable exponent Lebesgue spaces. The boundedness of integral operators of Harmonic Analysis such as maximal, potential, Calderón–Zygmund operators and their commutators are established in these spaces. Among others, we prove Sobolev‐type theorems for fractional integrals in L∼p(·),θ,ℓ. The spaces and operators are defined, generally speaking, on quasi‐metric measure spaces with doubling measure. The results are new even for Euclidean spaces.

Date: 2019
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https://doi.org/10.1002/mana.201800239

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