2021/08/20 by Fabio Berra, Berra, Fabio
Mathematics · #Advanced Harmonic Analysis Research #Nonlinear Partial Differential Equations #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.2108.09296
We prove mixed inequalities for the generalized maximal operator M_\Φ\nwhen the function v is a radial power function that fails to be locally\nintegrable. Concretely, let u be a weight, v(x)=|x|^\β with \β<-n\nand r\≥ 1. If \Φ is a Young function with certain properties, then the\ninequality \uvr
left(
left
x
in
mathbbRn:
fracM_
Phi\n(fv)(x)v(x)gt;t
right
right)
leq\nC
int_
mathbbRn
Phi
left(
frac|f(x)|t
right)vr(x)Mu(x)
,dx holds\nfor every t>0 and every bounded function. This improves a similar mixed\nestimate proved in citeBCP-M.\n As an application, we give mixed estimates for the generalized fractional\nmaximal operator M\γ,\Φ, where 0<\γ<n and \Φ is of L\log\nL type. A special case involving the fractional maximal operator M_\γ\nallows to obtain a similar estimate for the fractional integral operator\nI_\γ through an extrapolation result. Furthermore, we also give mixed\nestimates for commutators of singular integral Calder 'on-Zygmund operators and\nof I_\γ, both with Lipschitz symbol.\n