A note on weighted bounds for rough singular integrals

We show that the $L^2(w)$ operator norm of the composition $M\!\circ T_{\Omega}$, where $M$ is the maximal operator and $T_{\Omega}$ is a rough homogeneous singular integral with angular part $\Omega\in L^{\infty}(S^{n-1})$, depends quadratically on $[w]_{A_2}$, and this dependence is sharp

Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals

We prove sharp $L^p(w)$ norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the $A_p$ characteristic of $w$ for all $1<p<\infty$. This implies the same sharp inequalities for the classical Lusin area integral $S(f)$, the Littlewood-Paley $g$-function, and their continuous analogs $S_{\psi}$ and $g_{\psi}$. Also, as a corollary, we obtain sharp weighted inequalities for any convolution Calder\'on-Zygmund operator for all $1<p\le 3/2$ and $3\le p<\infty$, and for its maximal truncations for $3\le p<\infty$

On weighted norm inequalities for the Carleson and Walsh-Carleson operators

We prove $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$, its lacunary version $\mathcal C_{lac}$, and its analogue for the Walsh series \W in terms of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show that, exactly as for the Hilbert transform, $\|{\mathcal C}\|_{L^p(w)}$ is bounded linearly by $[w]_{A_q}$ for $1\le q<p$. We also obtain $L^p(w)$ bounds in terms of $[w]_{A_p}$, whose sharpness is related to certain conjectures (for instance, of Konyagin \cite{K2}) on pointwise convergence of Fourier series for functions near $L^1$. Our approach works in the general context of maximally modulated Calder\'on-Zygmund operators.Comment: A major revision of arXiv: 1310.3352. In particular, the main result is proved under a different assumption, and applications to the lacunary Carleson operator and to the Walsh-Carleson operator are give