Convolution of a measure with itself and a restriction theorem

Bak, Jong-Guk; McMichael, David

doi:10.1090/S0002-9939-97-03569-7

Convolution of a measure with itself and a restriction theorem
HTML articles powered by AMS MathViewer

by Jong-Guk Bak and David McMichael PDF

Proc. Amer. Math. Soc. 125 (1997), 463-470 Request permission

Abstract:

Let $S_{k}=\left \{ (y,|y|^{k})\colon y \in \mathbf {R}^{n-1} \right \} \subset \mathbf {R}^{n}$ and $\sigma$ be the measure defined by $\langle \sigma , \phi \rangle = \int _{\mathbf {R}^{n-1}}\phi (y, |y|^{k}) dy$. Let $\sigma _{P}$ denote the measure obtained by restricting $\sigma$ to the set $P=[0,\infty )^{n-1}$. We prove estimates on $\sigma _{P}*\sigma _{P}$. As a corollary we obtain results on the restriction to $S_{k} \subset \mathbf {R}^{3}$ of the Fourier transform of functions on $\mathbf {R}^{3}$ for $k\in \mathbf {R}$, $2<k<6$.

References

J.-G. Bak, Sharp convolution estimates for measures on flat surfaces, J. Math. Anal. Appl. 193 (1995), 756–771.
J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. MR 1097257, DOI 10.1007/BF01896376
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. MR 257819, DOI 10.1007/BF02394567
I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Academic Press, New York-London, 1964. Translated by Eugene Saletan. MR 0166596
Lars Hörmander, Oscillatory integrals and multipliers on $FL^{p}$, Ark. Mat. 11 (1973), 1–11. MR 340924, DOI 10.1007/BF02388505
Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
Christopher D. Sogge, A sharp restriction theorem for degenerate curves in $\textbf {R}^2$, Amer. J. Math. 109 (1987), no. 2, 223–228. MR 882421, DOI 10.2307/2374572
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
Peter A. Tomas, Restriction theorems for the Fourier transform, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–114. MR 545245
A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189–201. MR 387950, DOI 10.4064/sm-50-2-189-201