Borderline weak-type estimates for singular integrals and square functions (Q2788654)

The authors consider weak-type estimates for singular integral operators, extending some of the results found in [\textit{T. Hytönen} and \textit{C. Pérez}, J. Math. Anal. Appl. 428, No. 1, 605--626 (2015; Zbl 1323.42021)]. In particular, if \(\varphi\) is a Young function for which NEWLINE\[NEWLINE c_\varphi=\sum_{k=1}^\infty\frac1{\psi^{-1}(2^{2^k})}<\infty, NEWLINE\]NEWLINE where \(\psi\) is the complementary function of \(\varphi\), and if \(M_{\varphi(L)}\) denotes the Hardy-Littlewood maximal operator associated to the Orlicz space \(\varphi(L)\), then for every Calderón-Zygmund operator \(T\) and any weight \(w\) in \(\mathbb R^n\), it holds that NEWLINE\[NEWLINE \sup_{\lambda>0}\lambda w\{T^*f>\lambda\}\lesssim c_\lambda\int_{\mathbb R^n}|f(x)|M_{\varphi(L)}w(x)\,dx, NEWLINE\]NEWLINE where \(T^*\) is the maximal truncation of \(T\).NEWLINENEWLINEA second main result deals with estimates for the square functions \(G_\alpha\) defined in [\textit{M. Wilson}, Rev. Mat. Iberoam. 23, No. 3, 771--791 (2007; Zbl 1213.42072)]. If \(1\leq p<\infty\) and \(w\in A_p\) (the Muckenhoupt class), then NEWLINE\[NEWLINE \|G_\alpha f\|_{L^{p,\infty}(w)}\leq C_p(w)\|f\|_{L^p(w)}, NEWLINE\]NEWLINE where \(C_p(w)= [w]^{1/p}_{A_p}\) if \(1\leq p<2\) and \(C_p(w)= [w]^{1/2}_{A_p}(1+\log_+[w]_{A_\infty})^{1/2}\) if \(2\leq p<\infty\). Previous results were proved, for strong type estimates, in [\textit{A. K. Lerner}, Adv. Math. 226, No. 5, 3912--3926 (2011; Zbl 1226.42010)].