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Title: Lp maximal bounds and Sobolev regularity of two-parameter averages over tori

Abstract: We investigate L-p boundedness of the maximal function defined by the averaging operator fbar right arrowA(t)(s)f over the two-parameter family of tori T-t(s):={((t+scos theta)cos phi,(t+scos theta)sin phi,ssin theta): theta, phi is an element of[0,2 pi)} with c(0)t>s>0 for some c(0)is an element of(0,1). We prove that the associated (two-parameter) maximal function is bounded on L-p if and only if p>2. Also, we obtain L-p-L-q bounds for the local maximal operator on a sharp range of p,q. Furthermore, sharp smoothing estimates are obtained including the local smoothing estimates for the operators fbar right arrowA(t)(s)f and fbar right arrowA(t)(c0t)f. For these purposes, we make use of Bourgain-Demeter's decoupling inequality and Guth-Wang-Zhang's local smoothing estimate for the 2-dimensional wave operator. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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