Banach-Valued Modulation Invariant Carleson Embeddings and Outer-$$L^p$$ Spaces: The Walsh Case

Amenta, Alex; Uraltsev, Gennady

doi:10.1007/s00041-020-09768-0

Cited by 5 publications

(17 citation statements)

References 33 publications

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“…This takes a considerable amount of technical work, particularly in the tail estimates. A similar argument was implicitly used to prove [2,Theorem 5.3], the discrete version of this result, but in that result the tails are not present and life is simpler.…”

Section: Theorem 71 Fix R ∈ [2 ∞) and Let X Be An R -Intermediate Umentioning

confidence: 90%

“…Our analysis roughly follows the path laid out in our previous work [2] on the tritile operator. We recommend that readers new to time-frequency analysis start by reading that article, as it contains many of the core ideas of our arguments without most of the annoying technicalities.…”

Section: Theorem 13 Fix R ∈ [2 ∞) and Let X Be R -Intermediate Umdmentioning

confidence: 98%

“…Finally, we list a convenient way of bounding operators between γ -spaces: the γextension theorem. This says that operators that are essentially 'scalar' and bounded on a scalar-valued L 2…”

Section: Corollary 211mentioning

confidence: 99%

“…First we introduce R-bounds with respect to a trilinear form; we used this previously in [2], but to our knowledge it was first used by Di Plinio and Ou in [18].…”

Section: Novelties In the Trilinear Settingmentioning

confidence: 99%

“…The wave packets ϕ that are used all satisfy spt ϕ ⊂ B b , so ϕ θ has Fourier support vanishing in B b and in particular´R ϕ θ (z)dz = 0. Thus F R s out 12 We used a discrete version with Rademacher sums in place of γ -norms in [2]; this discrete version was also used in [18].…”

Section: Remark 311 To Illustrate the Definition Of The Local Sizementioning

confidence: 99%

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The bilinear Hilbert transform in UMD spaces

Amenta

Uraltsev

2020

Math. Ann.

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We prove $$L^p$$ 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 $$L^p(\mathbb {R};X)$$ L p ( R ; X ) into outer Lebesgue spaces on the time-frequency-scale space $$\mathbb {R}^3_+$$ R + 3 .

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Section: Theorem 71 Fix R ∈ [2 ∞) and Let X Be An R -Intermediate Umentioning

confidence: 90%

Section: Theorem 13 Fix R ∈ [2 ∞) and Let X Be R -Intermediate Umdmentioning

confidence: 98%

Section: Corollary 211mentioning

confidence: 99%

“…First we introduce R-bounds with respect to a trilinear form; we used this previously in [2], but to our knowledge it was first used by Di Plinio and Ou in [18].…”

Section: Novelties In the Trilinear Settingmentioning

confidence: 99%

Section: Remark 311 To Illustrate the Definition Of The Local Sizementioning

confidence: 99%

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The bilinear Hilbert transform in UMD spaces

Amenta

Uraltsev

2020

Math. Ann.

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Duality for Outer $$L^p_\mu (\ell ^r)$$ Spaces and Relation to Tent Spaces

Fraccaroli

2021

J Fourier Anal Appl

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We study the outer $$L^p$$ L p spaces introduced by Do and Thiele on sets endowed with a measure and an outer measure. We prove that, in the case of finite sets, for $$1< p \leqslant \infty , 1 \leqslant r < \infty $$ 1 < p ⩽ ∞ , 1 ⩽ r < ∞ or $$p=r \in \{ 1, \infty \}$$ p = r ∈ { 1 , ∞ } , the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) quasi-norms are equivalent to norms up to multiplicative constants uniformly in the cardinality of the set. This is obtained by showing the expected duality properties between the corresponding outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces uniformly in the cardinality of the set. Moreover, for $$p=1, 1 < r \leqslant \infty $$ p = 1 , 1 < r ⩽ ∞ , we exhibit a counterexample to the uniformity in the cardinality of the finite set. We also show that in the upper half space setting the desired properties hold true in the full range $$1 \leqslant p,r \leqslant \infty $$ 1 ⩽ p , r ⩽ ∞ . These results are obtained via greedy decompositions of functions in the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces. As a consequence, we establish the equivalence between the classical tent spaces $$T^p_r$$ T r p and the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces in the upper half space. Finally, we give a full classification of weak and strong type estimates for a class of embedding maps to the upper half space with a fractional scale factor for functions on $$\mathbb {R}^d$$ R d .

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Banach-Valued Multilinear Singular Integrals with Modulation Invariance

Plinio

Martikainen

et al. 2020

International Mathematics Research Notices

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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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Banach-Valued Modulation Invariant Carleson Embeddings and Outer-$$L^p$$ Spaces: The Walsh Case

Cited by 5 publications

References 33 publications

The bilinear Hilbert transform in UMD spaces

The bilinear Hilbert transform in UMD spaces

Duality for Outer $$L^p_\mu (\ell ^r)$$ Spaces and Relation to Tent Spaces

Banach-Valued Multilinear Singular Integrals with Modulation Invariance

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