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

Francesco Di Plinio; Andrei K. Lerner

doi:10.1112/jlms/jdu049

paper

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

arXiv:1312.0833 · doi:10.1112/jlms/jdu049

Abstract

We prove $L^{p} (w)$ bounds for the Carleson operator $C$ , its lacunary version $C_{l a c}$ , and its analogue for the Walsh series $\W$ in terms of the $A_{q}$ constants $[w]_{A_{q}}$ for $1 \leq q \leq p$ . In particular, we show that, exactly as for the Hilbert transform, $∥ C ∥_{L^{p} (w)}$ is bounded linearly by $[w]_{A_{q}}$ for $1 \leq q < p$ . 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 Calderón-Zygmund operators.

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 given

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

Abstract

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