Banach-Valued Modulation Invariant Carleson Embeddings and Outer- $$L^p$$ L p Spaces: The Walsh Case
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(2020) 26:53
Banach-Valued Modulation Invariant Carleson Embeddings and Outer-Lp Spaces: The Walsh Case Alex Amenta1 · Gennady Uraltsev2 Received: 12 August 2019 © The Author(s) 2020
Abstract We prove modulation invariant embedding bounds from Bochner spaces L p (W; X ) on the Walsh group to outer-L p spaces on the Walsh extended phase plane. The Banach space X is assumed to be UMD and sufficiently close to a Hilbert space in an interpolative sense. Our embedding bounds imply L p bounds and sparse domination for the Banach-valued tritile operator, a discrete model of the Banach-valued bilinear Hilbert transform. Keywords Tritile operator · Bilinear Hilbert transform · time-frequency analysis · Walsh group · UMD Banach spaces · Outer Lebesgue spaces · Interpolation spaces Mathematics Subject Classification Primary: 42B20; Secondary: 42B25; 47A56
1 Introduction The bilinear Hilbert transform (BHT) of two complex-valued Schwartz functions f 0 , f 1 ∈ S (R; C) is given by ˆ BHT( f 0 , f 1 )(x) := p. v.
R
f 0 (x − t) f 1 (x + t)
Communicated by Hans G. Feichtinger.
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 0123456789().: V,-vol
dt . t
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Journal of Fourier Analysis and Applications
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The L p bounds BHT( f 0 , f 1 ) L p (R) p0 , p1 f 0 L p0 (R) f 1 L p1 (R)
∀ f 0 , f 1 ∈ S (R; C), (1.1)
with p0 , p1 ∈ (1, ∞] and p ∈ (2/3, ∞) such that p −1 = p0−1 + p1−1 , were first proven by Lacey and Thiele [28,29]. Their proof extended techniques developed by Carleson and Fefferman in their proofs of Carleson’s theorem on the almost-everywhere convergence of Fourier series [8,17]. These techniques are now referred to as ‘time-frequency’ or ‘wave packet’ analysis. In order to streamline and modularise these techniques, Do and Thiele developed a theory of ‘outer-L p ’ spaces, yielding proofs of L p bounds for the BHT in which the key difficulties are cleanly compartmentalised [16]. The outer-L p technique is not applied directly to the BHT, but rather to its associated trilinear form BHF, given by dualising with a third function f 2 ∈ S (R; C): ˆ ˆ dt p. v. f 0 (x − t) f 1 (x + t) f 2 (x) dx. BHF( f 0 , f 1 , f 2 ) := (1.2) t R R For p0 , p1 , p ∈ [1, ∞], the estimate (1.1) is equivalent to the bound | BHF( f 0 , f 1 , f 2 )| p0 , p1 , p
2
f u L pu (R)
∀ f 0 , f 1 , f 2 ∈ S (R; C). (1.3)
u=0
The trilinear form BHF is a nontrivial linear combination of the Hölder form (which satisfies the desired L p bounds by Hölder’s inequality) and another trilinear form: ˆ i f 0 (x) f 1 (x) f 2 (x) dx − BHF( f 0 , f 1 , f 2 ) π R ˆ = E[ f 0 ](x, η − t −1 , t) E[ f 1 ](x, η + t −1 , t) E[ f 2 ](x, −2η, t) dx dη dt (1.4) R3
ˆ+ =:
R3+
E0 [ f 0 ](x, η, t) E1 [ f 1 ](x, η, t) E2 [ f 2 ](x, η, t) dx dη dt.
Here R3+ = R × R × (0, ∞) is the extended phase plane, which parametrises the underlying translation, modulation, and
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