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RESEARCH PRODUCT

REGULARITY OF THE FRACTIONAL MAXIMAL FUNCTION

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The purpose of this work is to show that the fractional maximal operator has somewhat unexpected regularity properties. The main result shows that the fractional maximal operator maps -spaces boundedly into certain first-order Sobolev spaces. It is also proved that the fractional maximal operator preserves first-order Sobolev spaces. This extends known results for the Hardy–Littlewood maximal operator.