L^p-L^q estimates for maximal operators associated to families of finite type curves

2017/02/22 by Manna, Ramesh
#42B25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1702.06754

Abstract

We study the boundedness problem for maximal operators \mathbbM associated to averages along families of finite type curves in the plane, defined by \mathbbMf(x) := sup_{1 ≤ t ≤ 2} |∫_ℂ f(x-ty) ρ(y) dσ(y)|, where dσ denotes the normalised Lebesgue measure over the curves ℂ. Let \triangle be the closed triangle with vertices P=((2)/(5), (1)/(5)), ~ Q=((1)/(2), (1)/(2)), ~ R=(0, 0). In this paper, we prove that for ((1)/(p), (1)/(q)) ∈ (\triangle ∖ \P, Q\) ∩ \((1)/(p), (1)/(q)) :q > m \, there is a constant B such that ‖\mathbbMf‖_L^q(ℝ²) ≤ B ‖f‖_L^p(ℝ²). Furthermore, if m <5, then we have ‖\mathbbMf‖_{L^{5, ∞}(ℝ²)} ≤ B ‖f‖_{L^{(5)/(2) ,1} (ℝ²)}. We shall also consider a variable coefficient version of maximal theorem and we obtain the L^p-L^q boundedness result for ((1)/(p), (1)/(q)) ∈ \triangle^∘ ∩ \((1)/(p), (1)/(q)) :q > m \, where \triangle^∘ is the interior of the triangle with vertices (0,0), ~((1)/(2), (1)/(2)), ~((2)/(5), (1)/(5)). An application is given to obtain L^p-L^q estimates for solution to higher order, strictly hyperbolic pseudo-differential operators.

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