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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.
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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

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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