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Sharp function estimates for fractional integrals and related operators

Part of: Real functions Integral transforms, operational calculus Harmonic analysis in several variables

Published online by Cambridge University Press:  09 April 2009

Douglas S. Kurtz
Douglas S. Kurtz
Affiliation:
New Mexico State UniversityLas Cruces, New Mexico 88003-0001, U.S.A.
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Abstract

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This paper considers analogs of results on integral operators studied by Hörmander. Using the sharp function introduced by Fefferman and Stein, we prove weighted norm inequalities on kernel operators which map an Lp space into an Lq space, with q not equal to p. The techniques recover known results about fractional integral operators and apply to multiplier operators which satisfy a generalization of the Hörmander multiplier condition.

MSC classification

Secondary: 26A33: Fractional derivatives and integrals 42B15: Multipliers 44A35: Convolution
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 1 , August 1990 , pp. 129 - 137
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Cordoba, A. and Fefferman, C., ‘A weighted norm inequality for singular integrals’, Studia Math. 57 (1976), 97–101.CrossRefGoogle Scholar
[2]Fefferman, C. and Stein, E. M., ‘Hp spaces of several variables’, Acta Math. 129 (1972), 137–193.CrossRefGoogle Scholar
[3]Garcia-Cuerva, J. and de Francia, J. L. Rubio, Weighted norm inequalities and related topics, (North-Holland, Amsterdam, 1985).Google Scholar
[4]Heinig, H. P. and Johnson, R., ‘Weighted norm inequalities for Lr-valued integral operators and applications’, Math. Nachr. 107 (1982), 161–174.CrossRefGoogle Scholar
[5]Hörmander, L., ‘Estimates for translation invariant operators in Lp spaces’, Acta Math. 104 (1960), 93–139.CrossRefGoogle Scholar
[6]Kurtz, D. S. and Wheeden, R. L., ‘Results on weighted norm inequalities for multipliers’, Trans. Amer. Math. Soc. 225 (1979), 343–362.CrossRefGoogle Scholar
[7]Muckenhoupt, B. and Wheeden, R. L., ‘Weighted norm inequalities for fractional integrals’, Trans. Amer. Math. Soc. 192 (1974), 261–274.CrossRefGoogle Scholar
[8]Stein, E. M., Singular integrals and differentiability properties of functions, (Princeton Univ. Press, Princeton, N. J., 1970).Google Scholar
[9]Stein, E. M. and Weiss, G., ‘Interpolation of operators with changes of measures’, Trans. Amer. Math. Soc. 87 (1958), 159–172.CrossRefGoogle Scholar
[10]de Francia, J. L. Rubio, Ruiz, F. J., and Torrea, J. L., ‘Calderón-Zygmund theory for operator-valued kernels’, Adv. in Math. 62 (1986), 7–48.CrossRefGoogle Scholar
[11]Welland, G. V., ‘Weighted norm inequalities for fractional integrals’, Proc. Amer. Math. Soc. 51 (1975), 143–148.CrossRefGoogle Scholar