Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A convolution estimate
for a measure on a curve in ${\mathbb {R}}^{4}$


Author: Daniel M. Oberlin
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. M. Christ, A convolution inequality concerning Cantor-Lebesgue measures, Revista Mat. Iberoamericana 1 (1985), 79-83. MR 87k:42011
  • 2. S.W. Drury, Degenerate curves and harmonic analysis, Math. Proc. Cam. Phil. Soc. 108 (1990), 89-96. MR 91h:42021
  • 3. I.M. Gelfand & G.E. Shilov, Generalized functions I, Academic Press, New York, 1964. MR 29:3869
  • 4. J.D. McMichael, Damping oscillatory integrals with polynomial phase, Math. Scand. 73 (1993), 215-228. MR 95f:42020
  • 5. D. Oberlin, Convolution estimates for some measures on curves, Proc. Amer. Math. Soc. 99 (1987), 56-60. MR 88f:42033
  • 6. D. Oberlin, Oscillatory integrals with polynomial phase, Math. Scand. 69 (1991), 45-56. MR 93c:41048
  • 7. D. Oberlin, Estimates for oscillatory integrals with polynomial phase, Trans. Amer. Math. Soc. (to appear).
  • 8. Y. Pan, Convolution estimates for some degenerate curves, Math. Proc. Cam. Phil. Soc. 116 (1994), 143-146. MR 95h:42026
  • 9. Y. Pan, $L^{p}$-improving properties for some measures supported on curves, Math. Scand. (to appear).
  • 10. Y. Pan, A remark on convolution with measures supported on curves, Can. Math. Bull. 36 (1993), 245-250. MR 94f:42022

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42B15, 42B20

Retrieve articles in all journals with MSC (1991): 42B15, 42B20


Additional Information

Daniel M. Oberlin
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306-3027
Email: oberlin@math.fsu.edu

DOI: https://doi.org/10.1090/S0002-9939-97-03716-7
Received by editor(s): July 18, 1995
Received by editor(s) in revised form: October 31, 1995
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society