Mixed inequalities of Fefferman-Stein type for singular integral operators

Abstract

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

17 pages