889 research outputs found

### 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

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