Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Sharp estimates for the Bochner-Riesz operator of negative order in $ \mathbf {R}^{2}$


Author: Jong-Guk Bak
Journal: Proc. Amer. Math. Soc. 125 (1997), 1977-1986
MSC (1991): Primary 42B15
DOI: https://doi.org/10.1090/S0002-9939-97-03723-4
MathSciNet review: 1371114
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Bochner-Riesz operator $T^{\alpha }$ on $\mathbf {R}^{n}$ of order $\alpha $ is defined by

\begin{equation*}(T^{\alpha } f)\;\widehat {}\;(\xi ) = {\frac {(1-|\xi |^{2})_{+}^{\alpha } }{\Gamma (\alpha +1)}} \hat {f}(\xi ) \end{equation*}

where $\;\widehat {}\;$ denotes the Fourier transform and $r_{+}^{\alpha } = r^{\alpha }$ if $r>0$, and $r_{+}^{\alpha }=0$ if $r\leq 0$. We determine all pairs $(p,q)$ such that $T^{\alpha }$ on $\mathbf {R}^{2}$ of negative order is bounded from $L^{p}(\mathbf {R}^{2})$ to $L^{q}(\mathbf {R}^{2})$. To be more precise, we prove that for $0<\delta < 3/2$ the estimate $\Vert T^{-\delta }f \Vert _{L^{q}(\mathbf {R}^{2})} \leq C \Vert f \Vert _{L^{p}(\mathbf {R}^{2})}$ holds if and only if $(1/p,1/q) \in \Delta ^{-\delta }$, where

\begin{equation*}\Delta ^{-\delta }=\bigg \{ \bigg ({\frac {1}{p}},{\frac {1}{q}} \bigg )\in [0,1]\times [0,1]\colon \;\; {\frac {1}{p}}-{\frac {1}{q}} \geq {\frac {2\delta }{3}}, \;\; {\frac {1}{p}}> {\frac {1}{4}} + {\frac {\delta }{2}} , \;\; {\frac {1}{q}} < {\frac {3}{4}} - {\frac {\delta }{2}} \bigg \} .\end{equation*}

We also obtain some weak-type results for $T^{\alpha }$.


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

  • [B] J.-G. Bak, Sharp convolution estimates for measures on flat surfaces, J. Math. Anal. Appl. 193 (1995), 756-771. CMP 95:15
  • [BMO] J.-G. Bak, D. McMichael, and D. Oberlin, $L^{p}$-$L^{q}$ estimates off the line of duality, J. Austral. Math. Soc. (Series A) 58 (1995), 154-166. MR 96j:42004
  • [BR] C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissertationes Math. 175 (1980), 1-67.
  • [Bo] L. Börjeson, Estimates for the Bochner-Riesz operator with negative index, Indiana U. Math. J. 35 (1986), 225-233. MR 87f:42036
  • [CS] A. Carbery and F. Soria, Almost-everywhere convergence of Fourier integrals for functions in Sobolev spaces, and an $L^{2}$-localisation principle, Rev. Mat. Iberoamericana 4 (1988), 319-337. MR 91d:42015
  • [CaS] L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287-299. MR 50:14052
  • [F] C. Fefferman, The multiplier problem for the ball, Ann. of Math. 94 (1971), 330-336. MR 45:5661
  • [H] L. Hörmander, Oscillatory integrals and multipliers on $FL^{p}$, Ark. f. Mat. 11 (1973), 1-11. MR 49:5674
  • [Hu] R. Hunt, On $L(p,q)$ spaces, L'Ens. Math. 12 (1966), 249-275. MR 36:6921
  • [Se] A. Seeger, Endpoint inequalities for Bochner-Riesz multipliers in the plane, Pacific J. Math. 174 (1996), 543-553.
  • [So] C. D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), 43-65. MR 87g:42026
  • [S] E. M. Stein, Harmonic analysis: Real-variable 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), 477-478. MR 50:10681

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42B15

Retrieve articles in all journals with MSC (1991): 42B15


Additional Information

Jong-Guk 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 790-784, Korea
Email: bak@euclid.postech.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-97-03723-4
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

American Mathematical Society