Local smoothing type estimates on \(L^p\) for large \(p\). (English) Zbl 0972.42005
This is one of Wolff’s last papers, and in the reviewer’s opinion, one of his greatest. In this paper he combines his results on the circular maximal function [presented here in a simplified and improved format from that in Am. J. Math. 119, No. 5, 985-1026 (1997; Zbl 0892.52003)], with oscillatory integral techniques and the “induction on scales” argument to obtain a sharp local smoothing estimate for the wave equation in \(R^{2+1}\) in \(L^p\) for all \(p > 74\). In other words, given a solution to the wave equation with initial data in \(L^p_\alpha\), the solution is shown to locally be in \(L^p\) with the sharp loss of derivatives \(\alpha\), namely \(\alpha > 1/2 - 2/p\).
Actually Wolff proves a stronger square function estimate, or, more precisely, a \(p\)-function estimate. Let \(f\) be a spacetime function of frequency \(\sim N\) which lives within distance 1 of the light cone (i.e., \(f\) is a solution to the wave equation localized in spacetime). We have the standard decomposition \[ f = \sum_\Theta f_\Theta \] where \(\Theta\) is a partition of the above neighbourhood of the light cone into \(1 \times \sqrt{N} \times N\) sectors and \(f_\Theta\) has Fourier support in \(\Theta\). The main result of the paper is then \[ \|f \|_p \lesssim N^{1/2-2/p+} ( \sum_\Theta \|f_\Theta\|_p^p)^{1/p}. \]
The exponent \(1/2 - 2/p\) is sharp. The condition \(p > 74\) is not sharp, and Wolff conjectures it should be improved to \(p \geq 6\).
The argument proceeds by decomposing the wave into wave packets. The above \(L^p\) estimate is in some sense controlling the number of points where one has a maximal focussing of wave packets. Using an induction on scales argument (assuming the claim has already been proven at the smaller scale of \(\sqrt{N}\)) and a Lorentz transform rescaling, the claim then reduces to a combinatorial statement about the overlap of wave packet boxes, which by a standard duality argument corresponds to a statement about the number of tangencies of a collection of circles. At this point Wolff recalls the machinery used for the circular maximal function (cell decomposition, circles of Appolonius, bilinear reductions) to obtain the required estimate. As a by-product he obtains a somewhat cleaner proof of his own circular maximal theorem.
In the final section some variants of the results are presented, for instance a Strichartz inequality for certain fractal measures.
Actually Wolff proves a stronger square function estimate, or, more precisely, a \(p\)-function estimate. Let \(f\) be a spacetime function of frequency \(\sim N\) which lives within distance 1 of the light cone (i.e., \(f\) is a solution to the wave equation localized in spacetime). We have the standard decomposition \[ f = \sum_\Theta f_\Theta \] where \(\Theta\) is a partition of the above neighbourhood of the light cone into \(1 \times \sqrt{N} \times N\) sectors and \(f_\Theta\) has Fourier support in \(\Theta\). The main result of the paper is then \[ \|f \|_p \lesssim N^{1/2-2/p+} ( \sum_\Theta \|f_\Theta\|_p^p)^{1/p}. \]
The exponent \(1/2 - 2/p\) is sharp. The condition \(p > 74\) is not sharp, and Wolff conjectures it should be improved to \(p \geq 6\).
The argument proceeds by decomposing the wave into wave packets. The above \(L^p\) estimate is in some sense controlling the number of points where one has a maximal focussing of wave packets. Using an induction on scales argument (assuming the claim has already been proven at the smaller scale of \(\sqrt{N}\)) and a Lorentz transform rescaling, the claim then reduces to a combinatorial statement about the overlap of wave packet boxes, which by a standard duality argument corresponds to a statement about the number of tangencies of a collection of circles. At this point Wolff recalls the machinery used for the circular maximal function (cell decomposition, circles of Appolonius, bilinear reductions) to obtain the required estimate. As a by-product he obtains a somewhat cleaner proof of his own circular maximal theorem.
In the final section some variants of the results are presented, for instance a Strichartz inequality for certain fractal measures.
Reviewer: Terence Tao (Los Angeles)
MSC:
| 42B15 | Multipliers for harmonic analysis in several variables |
| 35L05 | Wave equation |
| 28A80 | Fractals |
