Proceedings of the American Mathematical Society

Author(s): Leslie C. Cheng
Journal: Proc. Amer. Math. Soc. 135 (2007), 3241-3247.
MSC (2000): Primary 42B25
Posted: June 20, 2007
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Abstract: We prove that, for a compactly supported

function $\Phi$ with vanishing integral on $\mathbf{R}^n$ , the corresponding square function operator $S_\Phi$ is bounded on

for $\vert 1/p - 1/2\vert < \min\{(q-1)/2, 1/2\}$ .

1.: Al-Salman, A., Al-Qassem, H., Cheng, L. C., Pan, Y., Bounds for the Function of Marcinkiewicz, Math. Res. Lett. 9 (2002), 697-700. MR 1906071 (2004d:42020)
2.: Benedek, A., Calderón, A. and Panzone, R., Convolution operators on Banach space valued functions, Proc. Natl. Acad. Sci. USA 48 (1962), 356-365. MR 0133653 (24:A3479)
3.: Bergh, J. and Löfström, J., Interpolation Spaces, Springer-Verlag, 1976.
4.: Coifman, R. and Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. MR 0358205 (50:10670)
5.: Coifman, R. and Meyer, Y., Au delà des opérateurs pseudo-différentiels, Astérisque 57 (1978). MR 518170 (81b:47061)
6.: Ding, Y., On Marcinkiewicz integral, Proc. of the conference ``Singular Integrals and Related Topics, III'', Osaka, Japan (2001), 28-38.
7.: Ding, Y., Fan, D.. and Pan, Y., On Littlewood-Paley functions and singular integrals, Hokkaido Math. J. 29 (2000), 537-552. MR 1795491 (2001m:42038)
8.: Duoandikoetxea, J. and Rubio de Francia, J. L., Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541-561. MR 837527 (87f:42046)
9.: Duoandikoetxea, J. and Seijo, E., Weighted inequalities for rough square functions through extrapolation, Studia Math. 149 (2002), 239-252. MR 1890732 (2003a:42022)
10.: Journé, J.-L., Calderón-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calderón, Lecture Notes in Math. 994 (1983), Springer-Verlag. MR 706075 (85i:42021)
11.: Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal functions, Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 0293384 (45:2461)
12.: Sato, S., Remarks on square functions in the Littlewood-Paley theory, Bull. Austral. Math. Soc. 58 (1998), 199-221. MR 1642027 (99g:42024)
13.: Stein, E.M., On the functions of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430-466. MR 0112932 (22:3778)
14.: Stein, E.M., Harmonic Analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993. MR 1232192 (95c:42002)
15.: Stein, E.M., Singular integrals and differentiability properties of functions, Princeton University Press, 1970. MR 0290095 (44:7280)
16.: Walsh, T., On the function of Marcinkiewicz, Studia Math. 44 (1972), 203-217. MR 0315368 (47:3917)

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