L^p maximal bound and Sobolev regularity of two-parameter averages over tori

2022/10/24 by Lee, Juyoung, Lee, Sanghyuk
#35S30 #42B15 #42B25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2210.13377

Abstract

We investigate L^p boundedness of the maximal function defined by the averaging operator f→ A_t^s f over the two-parameter family of tori \mathbbT_t^s:=\ ( (t+scosθ)cosϕ, (t+scosθ)sinϕ, ssinθ): θ, ϕ∈ [0,2π) \ with c₀t>s>0 for some c₀∈ (0,1). We prove that the associated (two-parameter) maximal function is bounded on L^p if and only if p>2. We also obtain L^p--L^q estimates for the local maximal operator on a sharp range of p,q. Furthermore, the sharp smoothing estimates are proved including the sharp local smoothing estimates for the operators f→ \mathcal A_t^s f and f→ \mathcal A_t^c₀t f. For the purpose, we make use of Bourgain--Demeter's decoupling inequality for the cone and Guth--Wang--Zhang's local smoothing estimates for the 2 dimensional wave operator.

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