L^p-improving bounds of maximal functions along planar curves

2023/09/05 by Liu, Naijia, Yu, Haixia
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.01992

Abstract

In this paper, we study the L^p(ℝ²)-improving bounds, i.e., L^p(ℝ²)→ L^q(ℝ²) estimates, of the maximal function M_γ along a plane curve (t,γ(t)), where M_γf(x₁,x₂):=sup_{u∈ [1,2]}|∫₀¹f(x₁-ut,x₂-u γ(t)) \textrmdt|, and γ is a general plane curve satisfying some suitable smoothness and curvature conditions. We obtain M_γ : L^p(ℝ²)→ L^q(ℝ²) if ((1)/(p),(1)/(q))∈ Δ∪ \(0,0)\ and ((1)/(p),(1)/(q)) satisfying 1+(1 +ω)((1)/(q)-(1)/(p))>0, where Δ:=\((1)/(p),(1)/(q)): (1)/(2p)<(1)/(q)≤ (1)/(p), (1)/(q)>(3)/(p)-1 \ and ω:=\limsup_{t→ 0⁺}(ln|γ(t)|)/(ln t). This result is sharp except for some borderline cases. As Hickman stated in [J. Funct. Anal. 270 (2016), pp. 560--608], this is a very different situation.

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