Optimal boundedness of central oscillating multipliers on compact Lie groups

Chen, Jiecheng; Fan, Dashan

Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 123-145.

Fefferman-Stein, Wainger and Sjölin proved optimal $H^p$ boundedness for certain oscillating multipliers on ${\mathbf{R}}^d$. In this article, we prove an analogue of their result on a compact Lie group.

[1] Alexopoulos, G. Oscillating multipliers on Lie groups and Riemannian manifolds, Tohoku Math. J., Volume 46 (1994), pp. 457-468 | DOI | MR | Zbl

[2] Bennett, C.; Sharpley, R. Interpolation of Operators, Pure and Applied Math., Academic Press, Florida, 1988 | MR | Zbl

[3] Blank, B.; Fan, D. $H^p$ spaces on compact Lie groups, Ann. Fac Sci. Toulouse Math., Volume 6 (1997), pp. 429-479 | DOI | Numdam | MR | Zbl

[4] Bloom, W. R.; Xu, Z. Approximation of $H^p$ functions by Bochner-Riesz means on compact Lie groups, Math. Z., Volume 216 (1994), pp. 131-145 | DOI | MR | Zbl

[5] Chen, J.; Fan, D. Central Oscillating Multipliers on Compact Lie Groups, Math. Z., Volume 267 (2011), pp. 235-259 | DOI | MR

[6] Chen, J.; Fan, D.; Sun, L. Hardy Space Estimates on Wave Equations on Compact Lie Groups, J. Funct. Anal., Volume 259 (2010), pp. 3230-3264 | DOI | MR

[7] Chen, J.; Wang, S. Decomposition of BMO functions on normal groups, Acta. Math. Sinica (Ser. A, Volume 32 (1989), pp. 345-357 | MR | Zbl

[8] Clerc, J. L. Bochner-Riesz means of $H^p$ functions ($0<p<1$) on compact Lie groups, Lecture Notes in Math., Volume 1234 (1987), pp. 86-107 | DOI | MR | Zbl

[9] Coifman, R.; Weiss, G. Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., Volume 83 (1977), pp. 569-645 | DOI | MR | Zbl

[10] Colzani, L. Hardy space on sphere, Ph.D. Thesis, Washington University, St Louis, 1982

[11] Cowling, M.; Mantero, A.M.; Ricci, F. Pointwise estimates for some kernels on compact Lie groups, Rend. Circ. Mat. Palerma, Volume 31 (1982), pp. 145-158 | DOI | MR | Zbl

[12] Domar, Y. On the spectral synthesis problem for ($n-1$)-dimensional subset of ${ℝ}^n,n\le 2$, Ark. Math., Volume 9 (1971), pp. 23-37 | DOI | MR | Zbl

[13] Fan, D. Calderón-Zygmund operators on compact Lie groups, Math. Z., Volume 216 (1994), pp. 401-415 | DOI | EuDML | MR | Zbl

[14] Fefferman, C.; Stein, E. M. $H^p$ space of several variables, Acta Math., Volume 129 (1972), pp. 137-193 | DOI | MR | Zbl

[15] Giulini, S.; Meda, S. Oscillating multiplier on noncompact symmetric spaces, J. Reine Angew. Math., Volume 409 (1990), pp. 93-105 | EuDML | MR | Zbl

[16] Latter, R. H. A characterization of $H^p\left({ℝ}^n\right)$ in terms of atoms, Studia Math., Volume 62 (1978), pp. 93-101 | EuDML | MR | Zbl

[17] Littman, W. Fourier transforms of surface-carried measures and differentiability of surface average, Bull. Amer. Math. Soc., Volume 69 (1963), pp. 766-770 | DOI | MR | Zbl

[18] Marias, Michel $L^p$-boundedness of oscillating spectral multipliers on Riemannian manifolds, Ann. Math. Blaise Pascal, Volume 10 (2003) no. 1, pp. 133-160 | DOI | EuDML | Numdam | MR | Zbl

[19] Miyachi, A. On some singular Fourier multipliers, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 28 (1981), pp. 267-315 | MR | Zbl

[20] Sjölin, P. An $H^p$ inequality for strongly singular integrals, Math Z., Volume 165 (1979), pp. 231-238 | DOI | EuDML | MR | Zbl

[21] Stein, E. M. Topics in Harmonic Analysis, Ann. of Math. Studies, Princeton University Press, Princeton, NJ, 1970 | MR | Zbl

[22] Wainger, S. Special trigonometric series in k dimensions, Mem. Amer. Math. Soc., 1965 | MR | Zbl

[23] Watson, G. N. A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922 | JFM | MR