The bilinear Hilbert transform in UMD spaces
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Mathematische Annalen
The bilinear Hilbert transform in UMD spaces Alex Amenta1
· Gennady Uraltsev2
Received: 16 September 2019 / Revised: 17 July 2020 © The Author(s) 2020
Abstract We prove L p -bounds for the bilinear Hilbert transform acting on functions valued in intermediate UMD spaces. Such bounds were previously unknown for UMD spaces that are not Banach lattices. Our proof relies on bounds on embeddings from Bochner spaces L p (R; X ) into outer Lebesgue spaces on the time-frequency-scale space R3+ . Mathematics Subject Classification 42B20 · 42B25 · 47A56
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Analysis on the time-frequency-scale space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Local size-Hölder / the single-tree estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Local size domination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Embeddings into non-iterated outer Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Embeddings into iterated outer Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Applications to bilinear Hilbert transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Communicated by Loukas Grafakos.
B
Alex Amenta [email protected] Gennady Uraltsev [email protected]
1
Mathematisches Institut, Universität Bonn, Bonn, Germany
2
Department of Mathematics, Cornell University, Ithaca, NY, USA
123
A. Amenta, G. Uraltsev
1 Introduction The bilinear Hilbert transform is the bilinear singular integral operator defined on scalar-valued Schwartz functions f 1 , f 2 ∈ S (R; C) by ˆ BHT( f 1 , f 2 )(x) := p. v.
R
f 1 (x − y) f 2 (x + y)
dy y
∀x ∈ R.
(1)
This operator was introduced by Calderón in the 1960s in connection with the first Calderón commutator. The first L p -bounds for BHT were proven by Lacey and Thiele [27,28] using a newly-developed form of time-frequency analysis, extending techniques introduced by Carleson and Fefferman [11,22]. Since then the bilinear Hilbert transform has served as the fundamental example of a modulation-invariant singular integral (see for example [37]) and as a proving ground for time-frequency techniques. Since we are mainly interested in the vector-valued theory we direct the reader to [36,43] as starting points for further reading. Consider three complex Banach spaces X 1 , X 2 , X 3 and a bounded trilinear form Π : X 1 × X 2 × X 3 → C. With respect to this data one can define a trilinear singular integral form on Schwartz functions f i ∈ S (R; X i ) by ˆ BHFΠ ( f 1 , f 2 , f 3 ) :=
R
ˆ p. v.
dy dx. Π f 1 (x − y), f 2 (x + y), f 3 (x) y R
This is the dual form to a bilinear operator BH
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