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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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[1] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction , Grundlehren Math. Wiss., vol. 223, Springer-Verlag, Berlin, 1976. · Zbl 0344.46071
[2] Jean Bourgain, On the restriction and multiplier problems in \(\mathbf R^ 3\) , Geometric aspects of functional analysis (1989-90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 179-191. · Zbl 0792.42004
[3] J. Bourgain, A remark on Schrödinger operators , Israel J. Math. 77 (1992), no. 1-2, 1-16. · Zbl 0798.35131
[4] J. Bourgain, Estimates for cone multipliers , Geometric aspects of functional analysis (Israel, 1992-1994), Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 41-60. · Zbl 0833.43008
[5] Anthony Carbery, Radial Fourier multipliers and associated maximal functions , Recent progress in Fourier analysis (El Escorial, 1983), North-Holland Math. Stud., vol. 111, North-Holland, Amsterdam, 1985, pp. 49-56. · Zbl 0632.42012
[6] Lennart Carleson, Some analytic problems related to statistical mechanics , Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979), Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 5-45. · Zbl 0425.60091
[7] Lennart Carleson and Per Sjölin, Oscillatory integrals and a multiplier problem for the disc , Studia Math. 44 (1972), 287-299. (errata insert). · Zbl 0215.18303
[8] Björn E. J. Dahlberg and Carlos E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation , Harmonic analysis (Minneapolis, Minn., 1981), Lecture Notes in Math., vol. 908, Springer, Berlin, 1982, pp. 205-209. · Zbl 0519.35022
[9] Charles Fefferman, Inequalities for strongly singular convolution operators , Acta Math. 124 (1970), 9-36. · Zbl 0188.42601
[10] Carlos E. Kenig and Alberto Ruiz, A strong type \((2,\,2)\) estimate for a maximal operator associated to the Schrödinger equation , Trans. Amer. Math. Soc. 280 (1983), no. 1, 239-246. JSTOR: · Zbl 0525.42011
[11] A. Moyua, A. Vargas, and L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform , Internat. Math. Res. Notices (1996), no. 16, 793-815. · Zbl 0868.35024
[12] D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms. I , Acta Math. 157 (1986), no. 1-2, 99-157. · Zbl 0622.42011
[13] Alberto Ruiz and Luis Vega, Corrigenda to: “Unique continuation for Schrödinger operators with potential in Morrey spaces” , Publ. Mat. 39 (1995), no. 2, 405-411. · Zbl 0849.47022
[14] Per Sjölin, Regularity of solutions to the Schrödinger equation , Duke Math. J. 55 (1987), no. 3, 699-715. · Zbl 0631.42010
[15] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals , Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. · Zbl 0821.42001
[16] Luis Vega, Schrödinger equations: pointwise convergence to the initial data , Proc. Amer. Math. Soc. 102 (1988), no. 4, 874-878. JSTOR: · Zbl 0654.42014
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