Multiple Vector Valued Inequalities via the Helicoidal Method

By C. Benea, Camil Muscalu. 2015

61 citations recorded by the source corpus.

We develop a new method of proving vector-valued estimates in harmonic analysis, which we like to call "the helicoidal method". As a consequence of it, we are able to give affirmative answers to some questions that have been circulating for some time. In particular, we show that the tensor product $BHT \otimes \Pi$ between the bilinear Hilbert transform $BHT$ and a paraproduct $\Pi$ satisfies the same $L^p$ estimates as the $BHT$ itself, solving completely a problem introduced in a paper of Muscalu, Pipher, Tao and Thiele. Then, we prove that for "locally $L^2$ exponents" the corresponding vector valued $\overrightarrow{BHT}$ satisfies (again) the same $L^p$ estimates as the $BHT$ itself. Before the present work there was not even a single example of such exponents. Finally, we prove a bi-parameter Leibniz rule in mixed norm $L^p$ spaces, answering a question of Kenig in nonlinear dispersive PDE.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8c9d12c9b642687edf038983db7798c85b76a0b4

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In Applied Mathematics

1. Dachun Yang
2. Kangwei Li
3. Long Huang
4. S. Ombrosi
5. J. M. Martell
6. Andrei K. Lerner
7. Wen Yuan
8. P. Auscher

In Mathematical Physics

1. Long Huang
2. Dachun Yang
3. J. M. Martell
4. Kangwei Li
5. S. Ombrosi
6. Wen Yuan
7. Luong Dang Ky
8. Xuan Thinh Duong

Citations in the corpus

• On Function Spaces with Mixed Norms — A Survey (2019, background): “Another recent interest in developing the theory of function spaces with mixed norms comes from bilinear estimates and their vector-valued extensions which have proved useful in partial differential equations involving functions in n dimension space variable x and one dimension time variable t; see,”
• Extrapolation for multilinear Muckenhoupt classes and applications (2018, methodology): “Here it is important to emphasize that [11] did not recover the full range of vector-valued estimates from [3, 4] in the unweighted situation.”