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Zeitschrift für Analysis und ihre Anwendungen


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Volume 33, Issue 4, 2014, pp. 369–383
DOI: 10.4171/ZAA/1517

Published online: 2014-10-15

Convolution in Rearrangement-Invariant Spaces Defined in Terms of Oscillation and the Maximal Function

Martin Křepela[1]

(1) Karlstad University, Sweden

We characterize boundedness of a convolution operator with a fixed kernel between the classes $S^p(v)$, defined in terms of oscillation, and weighted Lorentz spaces $\Gamma^q(w)$, defined in terms of the maximal function, for $0 < p,q \le \infty$. We prove corresponding weighted Young-type inequalities of the form $$\|f\ast g\|_{\Gamma^q(w)} \le C \|f\|_{\S^p(v)}\|g\|_Y$$ and characterize the optimal rearrangement-invariant space $Y$ for which these inequalities hold.

Keywords: Convolution, Young inequality, weighted Lorentz spaces, oscillation

Křepela Martin: Convolution in Rearrangement-Invariant Spaces Defined in Terms of Oscillation and the Maximal Function. Z. Anal. Anwend. 33 (2014), 369-383. doi: 10.4171/ZAA/1517