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On inequalities for integral operators

Published online by Cambridge University Press:  18 May 2009

G. O. Okikiolu
G. O. Okikiolu
Affiliation:
University of East Anglia
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In two papers [3] and [4], the author has extended the inequality of Schur (Theorem 319 of [2]) to cases involving kernels which satisfy identities of the form

The purpose of this paper is to prove a general inequality, which includes the above and also the inequality of Young (Theorem 281 of [2]) as special cases. We shall give the results a general setting by considering functions defined on abstract measure spaces. From this we shall deduce an extension to n dimensions of the results given in [3], which also generalises a similar extension of the Schur inequality given by Stein and Weiss. In fact some cases of the other results given in [5] will follow directly from our theorem.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 11 , Issue 2 , July 1970 , pp. 126 - 133
Copyright
Copyright © Glasgow Mathematical Journal Trust 1970

References

REFERENCES

1.Cotlar, M. and Ortiz, E. L., On some inequalities for potential operators, Univ. Nac. La Plata Publ. Fac. Ci. Fisicomat. Serie Segunda Rev. 8 (1962), No. 1, 16–34.Google Scholar
2.Hardy, G. H., Littlewood, J. E. and Polya, G., Inequalities (Cambridge, 1934).Google Scholar
3.Okikiolu, G. O., Bounded linear transformations in Lp space, J. London Math. Soc, 41 (1966), 407–414.Google Scholar
4.Okikiolu, G. O., On certain bounded linear transformations in Lp′, Proc. London Math. Soc. (3) 17 (1967), 700–714.Google Scholar
5.Stein, E. M. and Weiss, G., Fractional integrals on n-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514.Google Scholar