On plate decompositions of cone multipliers

By Gustavo Garrigós, Andreas Seeger. Proceedings of the Edinburgh Mathematical Society, 2007

54 citations recorded by the source corpus.

Abstract An important inequality due to Wolff on plate decompositions of cone multipliers is known to have consequences for sharp Lp results on cone multipliers, local smoothing for the wave equation, convolutions with radial kernels, Bergman projections in tubes over cones, averages over finite-type curves in ℝ3 and associated maximal functions. We observe that the range of p in Wolff's inequality, for the conic and the spherical versions, can be improved by using bilinear restriction results. We also use this inequality to give some improved estimates on square functions associated to decompositions of cone multipliers in low dimensions. This gives a new L4 bound for the cone multiplier operator in ℝ3.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/228b289ebc7c176edc081391b0613cd5ad74e3c0

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Citations in the corpus

A sharp square function estimate for the cone in ℝ3 (2019, methodology): “hing conjecture for p≥ 74 in the full range2 of α. In that paper, Wolﬀ introduced the idea of decoupling. His method was extended to higher dimensions by Laba–Wolﬀ [16] and reﬁned by Garrig´os–Seeger [12][13] and Garrig´os–Schlag–Seeger [11]. Then in [2], Bourgain and Demeter proved a sharp decoupli”
Radial Fourier multipliers in high dimensions (2009, background): “proved to be rather hard and the first result was obtained by Wolff [26]; he established, in a deep and fundamental paper, the validity of Sogge’s conjecture in two dimension for the range q > 74. Versions of this result for the higher dimensional cases were obtained by Laba and Wolff [13] and furth”
The proof of the l 2 Decoupling Conjecture (2015, background): “Examples include progress on the “local smoothing conjecture for the wave equation” (see [30] and [36]), the regularity for convolutions with arclength measures on helices [27], and the boundedness properties of the Bergman projection in tube domains over full light cones; see [20] and [2].”
Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three (2015, background): “The relevant machinery that we now call decouplings showed its initial potential for applications in papers such as [17], [14], [16] and [4].”
Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds (2018, result): “nal work of Wolﬀ [46]. For this, Wolﬀ [46] introduced what have since become known as decoupling inequalities for the light cone. The results of [46] were improved and extended by a number of authors [27, 17, 18, 6] before the remarkable breakthrough of Bourgain– Demeter [8] established essentially ”