Bi-parameter paraproducts

By Camil Muscalu, Jill Pipher, Terence Tao, C. Thiele. 2003

185 citations recorded by the source corpus.

In the first part of the paper we prove a bi-parameter version of a well known multilinear theorem of Coifman and Meyer. As a consequence, we generalize the Kato-Ponce inequality in nonlinear PDE, obtaining a fractional Leibnitz rule for derivatives in the $x_1$ and $x_2$ directions simultaneously. Then, we show that the double bilinear Hilbert transform does not satisfy any $L^p$ estimates.

View Paper: https://www.semanticscholar.org/paper/f1385854c4a4a0c85158bfb6b17d52387564d37d

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Citations in the corpus

Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces (2017, background): “ p = 1 p1 + 1 p2 . The authors also state that the result hold in higher dimensions. In the case of Lebesgue spaces the analog mixed derivative version of (18) was previously studied by Muscalu et al [50]. Using diﬀerent methods, Di Plinio and Ou [27] prove some multiplier results which implicitly a”
Multiparameter Riesz commutators (2007, background): “For a cone in C ⊂ R d , we ﬁx a Calder´on–Zygmund kernel K C which satisﬁes the size and smoothness assumptions above, and in addition, Moreover, we choose the cone operator to make a sufﬁciently smooth transition from 0 to 1 that the operator T with symbol given by K deﬁnes a Calder´on– Zygmund…”
Multiple Vector Valued Inequalities via the Helicoidal Method (2015, background): “Muscalu, Pipher, Thiele, and Tao proved the following theorem: Theorem 3 [Muscalu et al. 2004a].”
On the two dimensional Bilinear Hilbert Transform (2008, methodology): “In order to discretize Λ(F1, F2, F3), we first perform a ”cone decomposition” of K, that is we decompose smoothly K̂ into pieces localized in (finitely many) cones(7) centered at the origin (see for example [13]).”
On Multilinear Fourier Multipliers of Limited Smoothness (2013, unknown): “All these advances are too numerous to be included in this introduction; here we only mention the articles of Bényi and Torres [1], Grafakos and Torres [11], Kenig and Stein [14], Lacey and Thiele [15], Lerner, Ombrosi, Pérez, Torres and Trujillo-González [16] and Muscalu, Pipher, Tao and Thiele [18”
Hörmander type theorems for multi-linear and multi-parameter Fourier multiplier operators with limited smoothness (2014, methodology): “In the bilinear and bi-parameter case, Muscalu, Pipher, Tao, and Thiele [18] proved the following Theorem 1.6.”
Weighted Norm Inequalities for Paraproducts and Bilinear Pseudodifferential Operators with Mild Regularity (2009, unknown): “Further applications now include the study of compensated compactness, see [11, 12], and [49], and, as bilinear DOs also model expressions of the type ∑ α,β cα,β∂ α x f ∂ β x g, they are useful in generalizing Leibnitz’s rule in the spirit of the Kato–Ponce inequality, see [4] and [36].”
Quasi-Banach Valued Inequalities via the Helicoidal method (2016, background): “e indices satisfy the natural H¨older-type conditions. That s2 should be greater than both 1 1+α and 1 1+β is sensible, since this is also the natural condition in the case when s1 = s2, as proved in [MPTT04]. On the other hand, for s1 we only have one constraint: 1 1+α <s1 <∞. Paraproducts co”
The Cauchy problem for the 3D Zakharov-Kuznetsov equation (2009, background): “The estimate (67) was proved by Muscalu, Pipher, Tao and Thiele in [17], see also [10].”
The Kato-Ponce Inequality (2013, background): “[17] to allow for partial fractional derivatives in R.”
Representation of bi-parameter singular integrals by dyadic operators (2011, unknown): “There is a wide body of more recent developments of which we here only mention the papers by Ferguson and Lacey [6], Lacey and Metcalfe [12] and Muscalu, Pipher, Tao and Thiele [13].”