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arXiv · 2607.01069

Bounds for the maximal and Riesz potential operators with variable fractionality

Francisco Gonçalves·Tuomas Oikari
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Abstract

We prove $L^{p(\cdot)}$-to-$L^{q(\cdot)}$ bounds for variable versions of the fractional maximal $M^{\alpha(\cdot)}$ and Riesz potential $I^{\alpha(\cdot)}$ operators. The changing fractionality in these operators is given by averaging the function $\alpha(\cdot)$ over balls. The bounds for $M^{\alpha(\cdot)}$ are in terms of a three-exponent Muckenhoupt condition relating $p(\cdot),q(\cdot),$ and $\alpha(\cdot)$, while the bounds for $I^{\alpha(\cdot)}$ are in terms of the boundedness of $M^{\alpha(\cdot)}$ and a packing condition on $\alpha(\cdot).$ These bounds hold under Hardy--Littlewood maximal function boundedness and Muckenhoupt conditions on the individual exponents $p(\cdot),q(\cdot),\alpha(\cdot).$ The proofs are based on an adaptation of sparse domination to variable fractionality and an embedding into variable sequential spaces.

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Francisco Gonçalves, Tuomas Oikari. 2026-07-01. Bounds for the maximal and Riesz potential operators with variable fractionality. https://arxiv.org/abs/2607.01069

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