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Restriction theorems and maximal operators related to oscillatory integrals in \(\mathbf R^3\). (English) Zbl 0946.42011
Let \(d\sigma\) be surface measure on a compact surface \(S\) with boundary in \(R^3\). Assume that \(S\) has positive Gaussian curvature at every point. A classical estimate of Sjölin (or of Tomas and Stein in general dimensions) states that \[ \|\widehat{f d\sigma} \|_{L^4(\mathbb R^2)} \lesssim \|f\|_{L^2(S)} \] for all \(L^2\) functions \(f\) on \(S\). The exponent \(L^2\) on the right-hand side cannot be lowered (without also lowering the \(L^4\) exponent on the left-hand side). However, improvements are possible if one leaves the regime of Lebesgue spaces. In this paper, the authors show that the \(L^2\) norm can be replaced by the more complicated norm \(X^p\) defined by \[ \|f\|_{X^p} = \Biggl(\sum_Q |Q|^2 \biggl(\frac{1}{|Q|} \int_Q |f|^p \biggr)^{4/p} \Biggr)^{1/4} \] where \(Q\) ranges over all dyadic “squares” in \(S\), for all \(p \geq 4(\sqrt{2}-1)\), and that this is sharp. (Sjölin’s estimate corresponds to \(p=4\); previous work by Bourgain established this for \(p \geq 16/9\). For the paraboloid or sphere the authors had already established this for \(p > 4(\sqrt{2}-1)\)). The authors then apply this to the problem of pointwise convergence of solutions to the two-dimensional Schrödinger equation to the initial data, showing that pointwise convergence obtains for all \(H^s\), \(s > (164 + \sqrt{2})/339\). This result improves upon earlier work of Bourgain; the result for \(s>1/2\) is straightforward. The results are closely related to the bilinear restriction estimates which appear later in [T. Tao, A. Vargas and L. Vega, J. Am. Math. Soc. 11, No. 4, 967-1000 (1998; Zbl 0924.42008); T. Tao and A. Vargas, Geom. Funct. Anal. 10, No. 1, 216-258 (2000; Zbl 0949.42013)] and a comparison between the two types of estimates can be found in these papers.

42B25 Maximal functions, Littlewood-Paley theory
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Full Text: DOI
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