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

We prove L-p (w) bounds for the Carleson operator C, its lacunary version C-lac, and its analogue for the Walsh series W in terms of the A(q) constants [w]A(q) for 1 <= q <= p. In particular, we show that, exactly as for the Hilbert transform, parallel to C parallel to L-p(w) is bounded linearly by [w]A(q) for 1 <= 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 [International Congress of Mathematicians, vol. II (European Mathematical Society, Zurich, 2006) 1393-1403]) 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.