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 $\text{W}$ 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, $\mathrm{\parallel }\mathcal{C}{\mathrm{\parallel }}_{L^p(w)}$ 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.