Two estimates for curves in the plane

Oberlin, Daniel

doi:10.1090/S0002-9939-04-07610-5

Two estimates for curves in the plane
HTML articles powered by AMS MathViewer

by Daniel M. Oberlin

Proc. Amer. Math. Soc. 132 (2004), 3195-3201

DOI: https://doi.org/10.1090/S0002-9939-04-07610-5

Published electronically: June 16, 2004

PDF | Request permission

Abstract:

We obtain a Fourier transform estimate and an $L^{3/2}({\mathbb {R}}^{2}) -L^{3}({\mathbb {R}}^{2})$ convolution estimate for certain measures on a class of convex curves in the plane.

References

Youngwoo Choi, Convolution operators with the affine arclength measure on plane curves, J. Korean Math. Soc. 36 (1999), no. 1, 193–207. MR 1669101
Michael Christ, On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, Trans. Amer. Math. Soc. 287 (1985), no. 1, 223–238. MR 766216, DOI 10.1090/S0002-9947-1985-0766216-6
S. W. Drury, Degenerate curves and harmonic analysis, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 1, 89–96. MR 1049762, DOI 10.1017/S0305004100068973
S. W. Drury and Kang Hui Guo, Convolution estimates related to surfaces of half the ambient dimension, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 1, 151–159. MR 1104610, DOI 10.1017/S0305004100070201
Walter Littman, $L^{p}-L^{q}$-estimates for singular integral operators arising from hyperbolic equations, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 479–481. MR 0358443
Daniel M. Oberlin, Oscillatory integrals with polynomial phase, Math. Scand. 69 (1991), no. 1, 45–56. MR 1143473, DOI 10.7146/math.scand.a-12368
Daniel M. Oberlin, Convolution with affine arclength measures in the plane, Proc. Amer. Math. Soc. 127 (1999), no. 12, 3591–3592. MR 1690999, DOI 10.1090/S0002-9939-99-05462-3
Daniel M. Oberlin, Fourier restriction for affine arclength measures in the plane, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3303–3305. MR 1845006, DOI 10.1090/S0002-9939-01-06012-9
Per Sjölin, Fourier multipliers and estimates of the Fourier transform of measures carried by smooth curves in $R^{2}$, Studia Math. 51 (1974), 169–182. MR 385437, DOI 10.4064/sm-51-2-169-182