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We give Feffermain-Stein type inequalities related to mixed estimates forCalder\'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 amore general version of mixed estimates for Calder\'on-Zygmund operators provedin \cite{CruzUribe-Martell-Perez}. It also generalizes the Fefferman-Steinestimates given in \cite{P94} for the same operators.  We further get similar estimates for operators of convolution type withkernels satisfying an $L^\Phi-$H\"ormander condition, generalizing somepreviously known results which involve mixed estimates and Fefferman-Steininequalities for these operators.

Mixed inequalities of Fefferman-Stein type for singular integral operators