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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.
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This content will become publicly available on April 4, 2026
Zygmund dilations: bilinear analysis and commutator estimates
We develop both bilinear theory and commutator estimates in the context of entangled dilations, specifically Zygmund dilations $$(x_1, x_2, x_3) \mapsto (\delta_1 x_1, \delta_2 x_2, \delta_1 \delta_2 x_3)$$ in $$\mathbb{R}^3$$. We construct bilinear versions of recent dyadic multiresolution methods for Zygmund dilations and apply them to prove a paraproduct free $T1$ theorem for bilinear singular integrals invariant under Zygmund dilations. Independently, we prove linear commutator estimates even when the underlying singular integrals do not satisfy weighted estimates with Zygmund weights. This requires new paraproduct estimates.
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- Award ID(s):
- 2247234
- PAR ID:
- 10591619
- Publisher / Repository:
- American Mathematical Society
- Date Published:
- Journal Name:
- Transactions of the American Mathematical Society
- ISSN:
- 0002-9947
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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