arXiv subjects
Publications and source records attributed to Miguel F. Barea-Fernández.
For $1<p<\infty$, we characterize the boundedness of the segment multiplier on $L^p(w)$ by a large-scale Muckenhoupt condition, obtained by restricting the classical $A_p$ condition to intervals of length at least one. The proof uses a discretization involving the discrete Hilbert transform on an associated weighted sequence space.
Given a rearrangement-invariant (r.i.) space X, we show that the segment multiplier, the truncated Hilbert transform, and the discrete Hilbert transform (on the associated discretized space) are bounded on X simultaneously. Moreover, this boundedness is characterized by a condition on a pair of Boyd-type indices. We also exhibit an r.i. space on which the segment multiplier is bounded while the (non-truncated) Hilbert transform fails to be bounded.