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Abstract We prove that the class of trilinear multiplier forms with singularity over a one-dimensional subspace, including the bilinear Hilbert transform, admits bounded $L^p$-extension to triples of intermediate $$\operatorname{UMD}$$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $$\operatorname{UMD}$$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $$\textrm{UMD}$$-valued setting. This is then employed to obtain appropriate single-tree estimates by appealing to the $$\textrm{UMD}$$-valued bound for bilinear Calderón–Zygmund operators recently obtained by the same authors.more » « less
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Abstract We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $$H_\Omega $$ along finite subsets of a finite order lacunary set of directions $$\Omega \subset \mathbb{R}^3$$, answering a question of Parcet and Rogers in dimension $n=3$. Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2. The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of 2D angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting.more » « less
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The article arXiv:1309.0945 by Do and Thiele develops a theory of Carleson embeddings in outer Lp spaces for the wave packet transform of functions in L^p, in the 2≤p≤∞ range referred to as local L^2. In this article, we formulate a suitable extension of this theory to exponents 1<2, answering a question posed in arXiv:1309.0945. The proof of our main embedding theorem involves a refined multi-frequency Calder\'on-Zygmund decomposition. We apply our embedding theorem to recover the full known range of Lp estimates for the bilinear Hilbert transforms without reducing to discrete model sums or appealing to generalized restricted weak-type interpolation.more » « less
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