Weak type estimates for cone multipliers on $H^p$ spaces, $p < 1$
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Abstract:
We consider operators $T^{\delta }$ associated with the Fourier multipliers \begin{equation*}{\bigg (}1- \frac {|{\xi }’|^{2}}{{\xi }_{n+1}^{2}} {\bigg )}_{+}^{\delta },\quad \quad ({\xi }’,\xi _{n+1}) \in {\mathbb R}^{n} \times {\mathbb R},\end{equation*} and show that $T^{\delta }$ is of weak type $(p,p)$ on $H^{p}({\mathbb R}^{n+1})$, $0 < p < 1$, for the critical value $\delta = n(\frac {1}{p}-\frac {1}{2}) - \frac {1}{2}$.References
- J. Bourgain, Estimates for cone multipliers, Geometric aspects of functional analysis (Israel, 1992–1994) Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 41–60. MR 1353448
- Max Jodeit Jr., A note on Fourier multipliers, Proc. Amer. Math. Soc. 27 (1971), 423–424. MR 270072, DOI 10.1090/S0002-9939-1971-0270072-8
- Gerd Mockenhaupt, A note on the cone multiplier, Proc. Amer. Math. Soc. 117 (1993), no. 1, 145–152. MR 1098404, DOI 10.1090/S0002-9939-1993-1098404-6
- Gerd Mockenhaupt, Andreas Seeger, and Christopher D. Sogge, Wave front sets, local smoothing and Bourgain’s circular maximal theorem, Ann. of Math. (2) 136 (1992), no. 1, 207–218. MR 1173929, DOI 10.2307/2946549
- Detlef Müller and Andreas Seeger, Inequalities for spherically symmetric solutions of the wave equation, Math. Z. 218 (1995), no. 3, 417–426. MR 1324536, DOI 10.1007/BF02571912
- L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
- Elias M. Stein, Mitchell H. Taibleson, and Guido Weiss, Weak type estimates for maximal operators on certain $H^{p}$ classes, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1981, pp. 81–97. MR 639468
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Additional Information
- Sunggeum Hong
- Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
- Address at time of publication: Department of Mathematics, Seoul National University, Seoul, 151-742, Korea
- MR Author ID: 648474
- Email: hong@math.wisc.edu, shong@math.snu.ac.kr
- Received by editor(s): September 24, 1998
- Received by editor(s) in revised form: January 28, 1999
- Published electronically: May 11, 2000
- Communicated by: Christopher D. Sogge
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 3529-3539
- MSC (2000): Primary 42B15, 42B30
- DOI: https://doi.org/10.1090/S0002-9939-00-05455-1
- MathSciNet review: 1690992