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A lower bound for the spectrum of a one-dimensional pseudo-differential operator

Published online by Cambridge University Press:  24 October 2008

M. W. Wong
M. W. Wong
Affiliation:
Department of Mathematics, York University, Ontario, Canada
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Extract

In a 1928 paper [1], Hardy and Littlewood published the following inequality which is valid for all non-negative measurable functions ƒ and g on ( − ∞, ∞):

where 1 < p, q < ∞, 0 < λ < 1, l/p + 1/q + λ = 2 and Cλp, q depends only on λ, p, q; ‖ƒ‖p and ‖g‖q are the Lp and Lq norms of ƒ and g respectively. See also p. 288 of the monograph [2] by Hardy, Littlewood and Pólya for a discussion of this inequality and related matters.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 2 , March 1988 , pp. 317 - 320
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Hardy, G. H. and Littlewood, J. E.. Some properties of fractional integrals (1). Math. Z. 27 (1928), 565–606.Google Scholar
[2]Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities, 2nd ed. (Cambridge University Press, 1951).Google Scholar
[3]Lieb, E. H.. Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. 118 (1983), 349–374.CrossRefGoogle Scholar
[4]Schechter, M.. Spectra of partial differential operators (North-Holland, 1971).Google Scholar