Abstract
We prove theorems on the boundedness of commutators \([a,H_w^\alpha ]\) of the weighted multidimensional Hardy operator \(H^\alpha _w:= w H^\alpha \frac{1}{w}\) from a generalized local Morrey space \(\mathcal {L}^{p,\varphi ;0}({\mathbb {R}^n})\) to local or global space \(\mathcal {L}^{q,\psi }({\mathbb {R}^n})\). The main impacts of these theorems are
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1.
the use of CMO\(_s\)-class of coefficients a for the commutators;
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2.
the general setting when the function \(\varphi \) defining the Morrey space and the weight w are independent of one another and the weight w is not assumed to be in \(A_p\);
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3.
recovering the Sobolev–Adams exponent q instead of Sobolev–Spanne type exponent in the case of classical Morrey spaces
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4.
boundedness from local to global Morrey spaces;
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5.
the obtained estimates contain the parameter \(s > 1\) which may be arbitrarily chosen. Its choice regulates in fact an equilibrium between assumptions on the coefficient a and the characteristics of the space.
The obtained results are new also in non-weighted case.
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Samko, N. Commutators with Coefficients in CMO of Weighted Hardy Operators in Generalized Local Morrey Spaces. Mediterr. J. Math. 14, 64 (2017). https://doi.org/10.1007/s00009-017-0872-3
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DOI: https://doi.org/10.1007/s00009-017-0872-3