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Lp-Lq estimates off the line of duality

Part of: Harmonic analysis in several variables Partial differential equations Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

J.-G. Bak ,
D. McMichael  and
D. Oberlin
J.-G. Bak
Affiliation:
Department of Mathematics, Florida State Unviersity, Tallahassee, FL 32306-3027, USA, email address: back@math.fsu.edu, email address: mcmichael@math.fsu.edu, email address: oberlin@math.fsu.edu
D. McMichael
Affiliation:
Department of Mathematics, Florida State Unviersity, Tallahassee, FL 32306-3027, USA, email address: back@math.fsu.edu, email address: mcmichael@math.fsu.edu, email address: oberlin@math.fsu.edu
D. Oberlin
Affiliation:
Department of Mathematics, Florida State Unviersity, Tallahassee, FL 32306-3027, USA, email address: back@math.fsu.edu, email address: mcmichael@math.fsu.edu, email address: oberlin@math.fsu.edu
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Abstract

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Theorems 1 and 2 are known results concerning Lp–Lq estimates for certain operators wherein the point (1/p, 1/q) lies on the line of duality 1/p + 1/q = 1. In Theorems 1′ and 2′ we show that with mild additional hypotheses it is possible to prove Lp-Lq estimates for indices (1/p, 1/q) off the line of duality. Applications to Bochner-Riesz means of negative order and uniform Sobolev inequalities are given.

MSC classification

Secondary: 46B70: Interpolation between normed linear spaces 42B15: Multipliers 35B45: A priori estimates
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 58 , Issue 2 , April 1995 , pp. 154 - 166
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Börjeson, L., ‘Estimates for the Bochner-Riesz operator with negative index’, Indiana Univ. Math. J. 35 (1986), 225–233.CrossRefGoogle Scholar
[2]Carbery, A. and Soria, F., ‘Almost-every where convergence of Fourier integrals for functions in Sobolev spaces, and an L2-localisation principle’, Rev. Mat. Iberoamericana 4 (1988), 319–337.CrossRefGoogle Scholar
[3]Drury, S., ‘Degenerate curves and harmonic analysis’, Math. Proc. Cambridge Philos. Soc. 108 (1990), 89–96.CrossRefGoogle Scholar
[4]Fefferman, C., ‘Inequalities for strongly singular convolution operators’, Acta Math. 124 (1970), 9–36.CrossRefGoogle Scholar
[5]Harmse, J., ‘On Lebesgue space estimates for the wave equation’, Indiana Univ. Math. J. 39 (1990), 229–248.CrossRefGoogle Scholar
[6]Journé, J. L., Soffer, A. and Sogge, C., ‘Lp - Lq′ estimates for time-dependent Schrödinger operators’, Bull. Amer. Math. Soc. 23 (1990), 519–524.CrossRefGoogle Scholar
[7]Kenig, C., Ruiz, A. and Sogge, C., ‘Uniform Sobolev inequalities and unique continuation theorems for second order constant coefficient differential operators’, Duke Math. J. 55 (1987), 329–347.CrossRefGoogle Scholar
[8]Keig, C. and Sogge, C., ‘A note on unique continuation for Schrödinger's operator’, Proc. Amer. Math. Soc. 103 (1988), 543–546.Google Scholar
[9]Littman, W., Lp - Lq estimates for singular integral operators, volume 23 of Proc. Sympos. Pure Math. (Amer. Math. Soc., providence, 1973).Google Scholar
[10]Oberlin, D., ‘Convolution estimates for some measures on curves’, Proc. Amer. Math. Soc. 99 (1987), 56–60.CrossRefGoogle Scholar
[11]Oberlin, D., ‘Convolution estimates for some distributions with singularities on the light cone’, Duke Math. J. 59 (1989), 747–757.CrossRefGoogle Scholar
[12]Ricci, F. and Stein, E. M., ‘Harmonic analysis on nilpotent groups and singular integrals III: fractional integration along manifolds’, J. Funct. Anal. 86 (1989), 360–389.CrossRefGoogle Scholar
[13]Sogge, C., ‘Oscillatory integrals and spherical harmonics’, Duke Math. J. 53 (1986), 43–65.CrossRefGoogle Scholar
[14]Stein, E. M., ‘Interpolation of linear operators’, Trans. Amer. Math. Soc. 83 (1956), 482–492.CrossRefGoogle Scholar
[15]Strichartz, R., ‘Convolutions with kernels having singularities on a sphere’, Trans. Amer. Math. Soc. 148 (1970), 461–471.CrossRefGoogle Scholar
[16]Strichartz, R., ‘A priori estimates for the wave equation and some applications’, J. Funct. Anal. 5 (1970), 218–235.CrossRefGoogle Scholar
[17]Strichartz, R., ‘Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations’, Duke Math. J. 44 (1977), 705–714.CrossRefGoogle Scholar
[18]Tomas, P., ‘A restriction theorem for the Fourier transform’, Bull. Amer. Math. Soc. 81 (1975), 477–478.CrossRefGoogle Scholar
[19 ]Zygmund, A., ‘On Fourier coefficients and transforms of two variables’, Studia Math. 50 (1974), 189–201.CrossRefGoogle Scholar