Mixed inequalities of Fefferman-Stein type for singular integral operators

We give Feffermain-Stein type inequalities related to mixed estimates for Calderón-Zygmund operators. More precisely, given $δ>0$, $q>1$, $φ(z)=z(1+\log^+z)^δ$, a nonnegative and locally integrable function $u$ and $v\in \mathrm{RH}_\infty\cap A_q$, we prove that the inequality $uv\left(\left\{x\in \mathbb{R}^n: \frac{|T(fv)(x)|}{v(x)}>t\right\}\right)\leq \frac{C}{t}\int_{\mathbb{R}^n}|f|\left(M_{φ, v^{1-q'}}u\right)M(Ψ(v))$ holds with $Ψ(z)=z^{p'+1-q'}\mathcal{X}_{[0,1]}(z)+z^{p'}\mathcal{X}_{[1,\infty)}(z)$, for every $t>0$ and every $p>\max\{q,1+1/δ\}$. This inequality provides a more general version of mixed estimates for Calderón-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 $L^Φ-$Hörmander condition, generalizing some previously known results which involve mixed estimates and Fefferman-Stein inequalities for these operators.

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