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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.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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##### References:
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