Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-04T21:24:02.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Convolution estimates related to surfaces of half the ambient dimension

Published online by Cambridge University Press:  24 October 2008

S. W. Drury  and
Kanghui Guo
S. W. Drury
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, McGill University, Montreal H3A 2K6, Canada
Kanghui Guo
Affiliation:
Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65809, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

Let ƒ be a smooth function of compact support defined in the plane and consider the integral

The estimate

is well-known, see for instance the work of Littman[4]. The operator T amounts to convolution with the measure σ carried by the parabola t → (t, ½t2) and given by dσ = dt. Usually one proves (1) by embedding σ in an analytic family of distributions σz in ℝ2 given by

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 1 , July 1991 , pp. 151 - 159
Copyright
Copyright © Cambridge Philosophical Society 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Christ, M.. On the restriction of the Fourier transform to curves: endpoint results and the degenerate case. Trans. Amer. Math. Soc. 287 (1985), 223–238.CrossRefGoogle Scholar
[2]Guo, K.. On the spectral synthesis property and its application to P.D.E. Ph.D. thesis, McGill University (1989).Google Scholar
[3]Hunt, R. A.. On L(p, q) spaces. Enseign. Math. 12 (1966), 249–275.Google Scholar
[4]Littman, W.. L p – L q-estimates for singular integral operators arising from hyperbolic equations. Proc. Sympos. Pure Math. 23 (1973), 479–481.CrossRefGoogle Scholar
[5]Oberlin, D. M.. Convolution Estimates for some measures on curves. Proc. Amer. Math. Soc. 99 (1987), 56–60.CrossRefGoogle Scholar