A local smoothing estimate in higher dimensions

By Izabella Laba, T. Wolff. 2002

73 citations recorded by the source corpus.

We prove the higher-dimensional analogue of Wolff's local smoothing estimate (Geom. Funct. Anal. 2001) for large p. As in the 2+1-dimensional case, the estimate is sharp for any given value of p, but it is likely that the range of p can be improved.

Subfields: Applied Mathematics, Mathematical Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/29cd522c467f93833f4ec9501bfaae7cc1adcae2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

In Numerical Analysis

Citations in the corpus

Radial Fourier multipliers in high dimensions (2009, background): “For the L q spaces even partial results proved to be rather hard and the first result was obtained by Wolff [31]; he established, in a deep and fundamental paper, the validity of Sogge’s conjecture in two dimensions for the range q > 74.”
A local smoothing estimate for the Schrödinger equation (2008, background): “Moving away from the critical exponent, but remaining in two spatial dimensions, Wolff [37] proved the (almost) sharp estimate in the range q > 74, and Łaba and Wolff [16] generalised this to higher dimensions.”
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 ”
Linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces (2006, result): “|) it is easy to verify that the condition on μ is sharp (see [23]) and some sharp Lp-local smoothing estimates were obtained by Wolff [28] in R3, and Łaba and Wolff [8] in higher dimensions.”
Some Problems in Harmonic Analysis (2017, methodology): “This follows from the validity of the Wolff lp ′ -decoupling inequalities ([52], [32]) in the optimal range p′ > 2(d+1) d−1 which is a consequence of the more recent results by Bourgain and Demeter [5].”
A sharp square function estimate for the cone in ℝ3 (2019, methodology): “In [26], Wolﬀ proved the local smoothing 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”
Littlewood–Paley Decompositions Related to Symmetric Cones and Bergman Projections in Tube Domains (2004, background): “In particular, we establish a link between our problem and WolL’s conjecture for k kp;mic-norms (see [17, 12]).”
L p regularity of averages over curves and bounds for associated maximal operators (2005, background): “In particular they generalize Wolff’s inequality for decompositions of light cones in higher dimensions ([12]) to more general elliptical cones generated by convex hypersurfaces with nonvanishing curvature.”