Mixed inequalities of Fefferman-Stein type for singular integral operators (2203.04360v1)

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.