Mixed inequalities of Fefferman-Stein type for singular integral operators

TL;DR

This paper establishes generalized Fefferman-Stein inequalities with mixed estimates for Calderón-Zygmund operators and convolution-type operators, extending previous results and providing new bounds involving weighted functions and Orlicz norms.

Contribution

It introduces a broader class of mixed inequalities for singular integral operators, extending classical Fefferman-Stein estimates and incorporating new conditions on kernels and weights.

Findings

Proved mixed inequalities for Calderón-Zygmund operators involving weights and Orlicz functions.

Extended Fefferman-Stein inequalities to convolution operators with $L^\

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Abstract

We give Feffermain-Stein type inequalities related to mixed estimates for Calder\'on-Zygmund operators. More precisely, given $\delta>0$, $q>1$, $\varphi(z)=z(1+\log^+z)^\delta$, a nonnegative and locally integrable function $u$ and $v\in \mathrm{RH}_\infty\cap A_q$, we prove that the inequality \[uv\left(\left\{x\in \mathbb{R}^n: \frac{|T(fv)(x)|}{v(x)}>t\right\}\right)\leq \frac{C}{t}\int_{\mathbb{R}^n}|f|\left(M_{\varphi, v^{1-q'}}u\right)M(\Psi(v))\] holds with $\Psi(z)=z^{p'+1-q'}\mathcal{X}_{[0,1]}(z)+z^{p'}\mathcal{X}_{[1,\infty)}(z)$, for every $t>0$ and every $p>\max\{q,1+1/\delta\}$. This inequality provides a more general version of mixed estimates for Calder\'on-Zygmund operators proved in \cite{CruzUribe-Martell-Perez}. It also generalizes the Fefferman-Stein estimates given in \cite{P94} for the same operators. We further get similar estimates for operators of convolution type with kernels satisfying an $L^\Phi-$H\"ormander condition, generalizing some previously known results which involve mixed estimates and Fefferman-Stein inequalities for these operators.