A convolution estimate for a measure on a curve in $\mathbb {R}^4$
HTML articles powered by AMS MathViewer
- by Daniel M. Oberlin PDF
- Proc. Amer. Math. Soc. 125 (1997), 1355-1361 Request permission
Abstract:
Let $\gamma (t)=(t,t^{2},t^{3},t^{4})$ and fix an interval $I\subset {\mathbb {R}}$. If $T$ is the operator on ${\mathbb {R}}^{4}$ defined by $Tf(x)=\int \nolimits _{I}f(x-\gamma (t)) dt$, then $T$ maps $L^{\frac {5}{3}}({\mathbb {R}}^{4})$ into $L^{2}({\mathbb {R}}^{4})$.References
- Michael Christ, A convolution inequality concerning Cantor-Lebesgue measures, Rev. Mat. Iberoamericana 1 (1985), no. 4, 79–83. MR 850410, DOI 10.4171/RMI/19
- 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
- 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
- David McMichael, Damping oscillatory integrals with polynomial phase, Math. Scand. 73 (1993), no. 2, 215–228. MR 1269260, DOI 10.7146/math.scand.a-12467
- Daniel M. Oberlin, Convolution estimates for some measures on curves, Proc. Amer. Math. Soc. 99 (1987), no. 1, 56–60. MR 866429, DOI 10.1090/S0002-9939-1987-0866429-6
- 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
- D. Oberlin, Estimates for oscillatory integrals with polynomial phase, Trans. Amer. Math. Soc. (to appear).
- Yibiao Pan, Convolution estimates for some degenerate curves, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 1, 143–146. MR 1274164, DOI 10.1017/S0305004100072431
- Y. Pan, $L^{p}$-improving properties for some measures supported on curves, Math. Scand. (to appear).
- Yibiao Pan, A remark on convolution with measures supported on curves, Canad. Math. Bull. 36 (1993), no. 2, 245–250. MR 1222541, DOI 10.4153/CMB-1993-035-2
Additional Information
- Daniel M. Oberlin
- Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306-3027
- Email: oberlin@math.fsu.edu
- Received by editor(s): July 18, 1995
- Received by editor(s) in revised form: October 31, 1995
- Communicated by: Christopher D. Sogge
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 1355-1361
- MSC (1991): Primary 42B15, 42B20
- DOI: https://doi.org/10.1090/S0002-9939-97-03716-7
- MathSciNet review: 1363436