Parabolic Littlewood-Paley \(g\) -function with rough kernel. (English) Zbl 1151.42007
Summary: We give the \(L^p (1 < p < \infty)\) boundedness of the parabolic Littlewood-Paley \(g\)-function with rough kernel.
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
References:
| [1] | Calderón, A. P., Torchinsky, A.: Parabolic maximal functions associated with a distribution. Adv. Math., 16, 1–64 (1975) · Zbl 0315.46037 · doi:10.1016/0001-8708(75)90099-7 |
| [2] | Fabes, E. B., Rivière, N. M.: Singular integrals with mixed homogeneity. Studia Math., 27, 19–38 (1966) · Zbl 0161.32403 |
| [3] | Hofmann, S.: A characterization of commutators of parabolic singular integrals, Fourier analysis and partial differential equations (Miraflores de la Siera, 1992), Stud. Adv. Math. CRC, Boca Raton, FL 1995, 195–210 |
| [4] | Hofmann, S.: Parabolic singular integrals of Calderon-type, rough operators, and caloric layer potentials. Duke Math. J., 90(2), 209–259 (1997) · Zbl 0941.42006 · doi:10.1215/S0012-7094-97-09008-6 |
| [5] | Tao, T.: The weak-type (1,1) of Llog L homogeneous convolution operators. Indiana U. Math. J., 48, 1547–1584 (1999) · Zbl 0943.43002 · doi:10.1512/iumj.1999.48.1784 |
| [6] | Stein, E. M.: On the functions of Littlewood-Paley, Lusin and Marcinkiewicz. Trans. Amer. Math. Soc., 88, 430–466 (1958) · Zbl 0105.05104 · doi:10.1090/S0002-9947-1958-0112932-2 |
| [7] | Ding, Y., Fan, D., Pan, Y.: L p-boundedness of Marcinkiewicz integrals with Hardy space function kernel. Acta Mathematica Sinica, English Series, 16(4), 593–600 (2000) · Zbl 0966.42008 · doi:10.1007/s101140000015 |
| [8] | Madych, W. R.: On Littlewood-Paley functions. Studia Math., 50, 43–63 (1974) · Zbl 0247.44005 |
| [9] | Hörmander, L.: Estimates for translation invariant operators in Lp spaces. Acta Math. 104, 93–140 (1960) · Zbl 0093.11402 · doi:10.1007/BF02547187 |
| [10] | Stein, E. M., Wainger, S.: Problems in harmonic analysis related to curvature. Bull. Amer. Math. Soc., 84, 1239–1295 (1978) · Zbl 0393.42010 · doi:10.1090/S0002-9904-1978-14554-6 |
| [11] | Stein, E. M.: Maximal functions: Homogeneous curves. Proc. Natl. Acad. Sci. USA., 73, 2176–2177 (1976) · Zbl 0332.42019 · doi:10.1073/pnas.73.7.2176 |
| [12] | Ding, Y., Lu, S., Yabuta, K.: A problem on rough parametric Marcinkiewicz functions. J. Austral. Math. Soc. 72, 13–21 (2002) · Zbl 0993.42008 · doi:10.1017/S1446788700003542 |
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