Bilinear Rubio de Francia inequalities for collections of non-smooth squares
Abstract
Let $\Omega$ be a collection of disjoint dyadic squares $\omega$, let $\pi_\omega$ denote the non-smooth bilinear projection onto $\omega$
\[ \pi_\omega (f,g)(x):=\int\int \mathbf{1}_{\omega}(\xi,\eta) \widehat{f}(\xi) \widehat{g}(\eta) e^{2\pi i (\xi + \eta) x} d \xi d\eta \]
and let $r>2$. We show that the bilinear Rubio de Francia operator
\[ \Big(\sum_{\omega\in\Omega} |\pi_{\omega} (f,g)|^r \Big)^{1/r} \]
is $L^p \times L^q \rightarrow L^s$ bounded with constant at most $O_{\varepsilon}({\#\Omega}^{\varepsilon})$ for any $\varepsilon>0$ whenever $1/p + 1/q = 1/s$, $r'
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https://hal.science/hal-01670742
Submitted on : Thursday, December 21, 2017-4:02:16 PM
Last modification on : Thursday, April 18, 2024-4:27:45 PM
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- HAL Id : hal-01670742 , version 1
Cite
Frederic Bernicot, Marco Vitturi. Bilinear Rubio de Francia inequalities for collections of non-smooth squares. Publicacions Matemàtiques, 2020, 64 (1), pp.43-73. ⟨hal-01670742⟩
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