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arXiv · 1312.0833

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

Francesco Di Plinio·Andrei K. Lerner
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Abstract

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 Calderón-Zygmund operators.

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BibTeXRIS

Francesco Di Plinio, Andrei K. Lerner. 2013-12-03. On weighted norm inequalities for the Carleson and Walsh-Carleson operators. https://doi.org/10.1112/jlms%2Fjdu049

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