Maximal functions associated to flat plane curves with Mitigating\n factors

2016/09/26 by Ramesh Manna, Manna, Ramesh
Computer Science · Mathematics · #42B25 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1609.08140

openalex publication_date 2016/09/26 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28

Abstract

We study the boundedness problem for maximal operators mathbbM_\σ\nassociated to flat plane curves with Mitigating factors, defined by\n
mathbbM
_sigmaf(x)
, :=
,
sup_{1
leq t
leq 2}
left|
int₀¹\nf(x-t
Gamma(s))
, (
kappa(s))
^sigma
, ds
right|, where \κ(s)\ndenotes the curvature of the curve \Γ(s)=(s, g(s)+1), ~g(s) \∈ C⁵[0,1]\nin \ℝ². Let triangle be the closed triangle with vertices\nP=(\(2)/(5), \(1)/(5)), ~ Q=(\(1)/(2), \(1)/(2)), ~ R=(0, 0).\n In this paper, we prove that for (\(1)/(p), \(1)/(q)) \∈\n\[(\(1)/(p), \(1)/(q)) :(\(1)/(p), \(1)/(q)) \∈ triangle\n\∖ P, Q \] \∩ \[(\(1)/(p), \(1)/(q)) :q >\nmax \σ^-1,2 \], there is a constant B such that \n\‖ mathbbMf\‖_L^q(\ℝ²) \≤ , B , \‖f\‖_L^p(\ℝ²). \n

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