2020/06/05 by Fabio Berra, Berra, Fabio
Mathematics · #26A33 #42B25 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Inequalities and Applications #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2006.03612
openalex publication_date 2020/06/05 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
In this article we prove mixed inequalities for maximal operators associated\nto Young functions, which are an improvement of a conjecture established in\n citeBerra. Concretely, given r\≥ 1, u\∈ A1, vr\∈ A_\∞ and a\nYoung function \Φ with certain properties, we have that inequality\n \uvr
left(
left
x
in
mathbbRn:
fracM_
Phi(fv)(x)M_
Phi\nv(x)gt;t
right
right)
leq\nC
int_
mathbbRn
Phi
left(
frac|f(x)|t
right)u(x)vr(x)
,dx holds for\nevery positive t. The involved operator \(M_\Φ(fv)(x))/(M_\Φ v(x))\nseems to be an adequate extension when vr\∈ A_\∞, since when we assume\nvr\∈ A1 we can replace M_\Φ v by v, yielding a mixed inequality for\nM_\Φ proved in citeBerra-Carena-Pradolini(MN).\n As an application, we furthermore exhibe and prove mixed inequalities for the\ngeneralized fractional maximal operator M\γ,\Φ, where 0<\γ<n\nand \Φ is a Young function of L\log L type.\n