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A note on restricted weak-type estimates
for Bochner-Riesz operators
with negative index in $\mathbb{R}^n$, $n\geq 2$


Author: Susana Gutiérrez
Journal: Proc. Amer. Math. Soc. 128 (2000), 495-501
MSC (1991): Primary 42B15
DOI: https://doi.org/10.1090/S0002-9939-99-05144-8
Published electronically: June 24, 1999
MathSciNet review: 1641626
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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the Bochner-Riesz operator on $\mathbb{R}^n$ of negative order $\alpha $ is of restricted weak type in the critical points $(p_0,q_0)$ and $(q_0', p_0')$, where $1/q_0=3/4+\alpha /2$, $q_0=p_0'/3$ for $-3/2<\alpha<0$ in the two-dimensional case and $1/q_0=(n+1+2\alpha)/2n$, $q_0=(n-1)p_0'/(n+1),\,$ for $-(n+ 1)/2<\alpha<-1/2$ if $n\geq 3$.


References [Enhancements On Off] (What's this?)

  • [BSh] C. Bennet & R. Sharpley, Interpolation of operators, Academic Press, Inc., Orlando, Florida, (1988).
  • [BeJ] J. Bergh & J. Löfström, Interpolation of spaces, An introduction, Springer-Verlag, Berlin-Heidelberg-New York, (1976).
  • [Bo] J. Bourgain, Besicovitch-type maximal operators and applications to Fourier analysis, Geom. and Funct. Anal., 1, (1991), no.2, 147-187. MR 92g:42010
  • [Bk] J.-G Bak, Sharp estimates for the Bochner-Riesz operator of negative order in $\mathbb{R}^2$, Proc. Amer. Math. Soc., 125, (1997), no. 7, 1977-1986. MR 97i:42011
  • [BMO] J.-G. Bak, D. McMichael & D. Oberlin, $L^p$-$L^q$ Estimates off the line of duality, J. Austral. Math. Soc., 58, (1995), 154-166.
  • [Br] L. Börjeson, Estimates for the Bochner-Riesz operator with negative index, Indiana U. Math. J., 35, (1986), no.2, 225-233. MR 87f:42036
  • [CSr] A. Carbery & F. Soria, Almost-everywhere convergence of Fourier integrals for functions in Sobolev spaces, and an $L^2$-localisation principle, Rev. Mat. Iberoamericana, 4, (1988), no.2, 319-337. MR 91d:42015
  • [Sg] C. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J., 53, (1986), no.1, 43-65. MR 91d:35037
  • [St] E.M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton U. Press, Princeton, New Jersey, (1993). MR 95c:42002
  • [StWe] E.M. Stein & G.Weiss, An introduction to Fourier analysis on Euclidean spaces, Princeton U. Press, Princeton, New Jersey, 1975.
  • [T1] P. Tomas, A restriction theorem for Fourier transform, Bull. Amer. Math. Soc., 81, (1975), no.2, 477-478.MR 50:10681
  • [T2] P.A. Tomas, Restriction theorems for the Fourier transform, Harmonic Analysis in Euclidean Spaces (2 volumes), G. Weiss and W.Wainger (eds.), Proc. Symp. Pure Math. # 35, American Math. Society, (1979), part 1, 111-114. MR 81d:42029
  • [W] G.W. Watson, Theory of Bessel functions, Cambridge University Press, Cambridge, 1944. MR 6:64a

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Additional Information

Susana Gutiérrez
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco (UPV-EHU), Aptdo 644, 48080 Bilbao, Spain
Email: mtbgugrs@lg.ehu.es

DOI: https://doi.org/10.1090/S0002-9939-99-05144-8
Received by editor(s): March 29, 1998
Published electronically: June 24, 1999
Additional Notes: Supported by a grant from Spanish Ministry of Education and Sciences.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1999 American Mathematical Society

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