Sharp estimates for the BochnerRiesz operator of negative order in
Author:
JongGuk Bak
Journal:
Proc. Amer. Math. Soc. 125 (1997), 19771986
MSC (1991):
Primary 42B15
MathSciNet review:
1371114
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Abstract: The BochnerRiesz operator on of order is defined by where denotes the Fourier transform and if , and if . We determine all pairs such that on of negative order is bounded from to . To be more precise, we prove that for the estimate holds if and only if , where We also obtain some weaktype results for .
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Oberlin, 𝐿^{𝑝}𝐿^{𝑞} estimates off
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functions in Sobolev spaces, and an 𝐿²localisation
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(91d:42015), http://dx.doi.org/10.4171/RMI/76
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 [B]
 J.G. Bak, Sharp convolution estimates for measures on flat surfaces, J. Math. Anal. Appl. 193 (1995), 756771. CMP 95:15
 [BMO]
 J.G. Bak, D. McMichael, and D. Oberlin,  estimates off the line of duality, J. Austral. Math. Soc. (Series A) 58 (1995), 154166. MR 96j:42004
 [BR]
 C. Bennett and K. Rudnick, On LorentzZygmund spaces, Dissertationes Math. 175 (1980), 167.
 [Bo]
 L. Börjeson, Estimates for the BochnerRiesz operator with negative index, Indiana U. Math. J. 35 (1986), 225233. MR 87f:42036
 [CS]
 A. Carbery and F. Soria, Almosteverywhere convergence of Fourier integrals for functions in Sobolev spaces, and an localisation principle, Rev. Mat. Iberoamericana 4 (1988), 319337. MR 91d:42015
 [CaS]
 L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287299. MR 50:14052
 [F]
 C. Fefferman, The multiplier problem for the ball, Ann. of Math. 94 (1971), 330336. MR 45:5661
 [H]
 L. Hörmander, Oscillatory integrals and multipliers on , Ark. f. Mat. 11 (1973), 111. MR 49:5674
 [Hu]
 R. Hunt, On spaces, L'Ens. Math. 12 (1966), 249275. MR 36:6921
 [Se]
 A. Seeger, Endpoint inequalities for BochnerRiesz multipliers in the plane, Pacific J. Math. 174 (1996), 543553.
 [So]
 C. D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), 4365. MR 87g:42026
 [S]
 E. M. Stein, Harmonic analysis: Realvariable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton, New Jersey, 1993. MR 95c:42002
 [SW]
 E. M. Stein and G. Weiss, An introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, New Jersey, 1971.
 [T]
 P. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477478. MR 50:10681
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Additional Information
JongGuk Bak
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306–3027
Address at time of publication:
Department of Mathematics, Pohang University of Science and Technology, Pohang 790784, Korea
Email:
bak@euclid.postech.ac.kr
DOI:
http://dx.doi.org/10.1090/S0002993997037234
PII:
S 00029939(97)037234
Received by editor(s):
October 3, 1995
Received by editor(s) in revised form:
December 19, 1995
Additional Notes:
The author’s research was partially supported by a grant from the Pohang University of Science and Technology
Communicated by:
Christopher D. Sogge
Article copyright:
© Copyright 1997 American Mathematical Society
