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    Jeong, Eunhee Kwon, Yehyun and Lee, Sanghyuk 2016. Uniform Sobolev inequalities for second order non-elliptic differential operators. Advances in Mathematics, Vol. 302, p. 323.


    Cho, Yonggeun Kim, Youngcheol Lee, Sanghyuk and Shim, Yongsun 2005. Sharp <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math>-<mml:math altimg="si2.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msup></mml:math> estimates for Bochner–Riesz operators of negative index in <mml:math altimg="si3.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math>, <mml:math altimg="si4.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>n</mml:mi><mml:mo>⩾</mml:mo><mml:mn>3</mml:mn></mml:math>. Journal of Functional Analysis, Vol. 218, Issue. 1, p. 150.


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  • Currently known as: Journal of the Australian Mathematical Society Title history
    Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, Volume 58, Issue 2
  • April 1995, pp. 154-166

Lp-Lq estimates off the line of duality

  • J.-G. Bak (a1), D. McMichael (a1) and D. Oberlin (a1)
  • DOI: http://dx.doi.org/10.1017/S1446788700038209
  • Published online: 01 April 2009
Abstract
Abstract

Theorems 1 and 2 are known results concerning LpLq 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.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]L. Börjeson , ‘Estimates for the Bochner-Riesz operator with negative index’, Indiana Univ. Math. J. 35 (1986), 225233.

[4]C. Fefferman , ‘Inequalities for strongly singular convolution operators’, Acta Math. 124 (1970), 936.

[5]J. Harmse , ‘On Lebesgue space estimates for the wave equation’, Indiana Univ. Math. J. 39 (1990), 229248.

[6]J. L. Journé , A. Soffer and C. Sogge , ‘Lp - Lq′ estimates for time-dependent Schrödinger operators’, Bull. Amer. Math. Soc. 23 (1990), 519524.

[7]C. Kenig , A. Ruiz and C. Sogge , ‘Uniform Sobolev inequalities and unique continuation theorems for second order constant coefficient differential operators’, Duke Math. J. 55 (1987), 329347.

[10]D. Oberlin , ‘Convolution estimates for some measures on curves’, Proc. Amer. Math. Soc. 99 (1987), 5660.

[11]D. Oberlin , ‘Convolution estimates for some distributions with singularities on the light cone’, Duke Math. J. 59 (1989), 747757.

[12]F. Ricci and E. M. Stein , ‘Harmonic analysis on nilpotent groups and singular integrals III: fractional integration along manifolds’, J. Funct. Anal. 86 (1989), 360389.

[13]C. Sogge , ‘Oscillatory integrals and spherical harmonics’, Duke Math. J. 53 (1986), 4365.

[14]E. M. Stein , ‘Interpolation of linear operators’, Trans. Amer. Math. Soc. 83 (1956), 482492.

[15]R. Strichartz , ‘Convolutions with kernels having singularities on a sphere’, Trans. Amer. Math. Soc. 148 (1970), 461471.

[16]R. Strichartz , ‘A priori estimates for the wave equation and some applications’, J. Funct. Anal. 5 (1970), 218235.

[17]R. Strichartz , ‘Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations’, Duke Math. J. 44 (1977), 705714.

[18]P. Tomas , ‘A restriction theorem for the Fourier transform’, Bull. Amer. Math. Soc. 81 (1975), 477478.

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Journal of the Australian Mathematical Society
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  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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