Open Access
2014 Sharp Maximal Function Estimates for Parameterized Littlewood-Paley Operators and Area Integrals
Y. Lin, C.H. Song, X. Xuan
Commun. Math. Anal. 16(1): 66-83 (2014).
Abstract

In this paper, we establish sharp maximal function estimates for parameterized Littlewood-Paley operators and area integrals with kernel satisfying the logarithmic type Lipschitz condition. As applications, the weighted $L^p$ bounds and Morrey boundedness of these operators can be obtained.

References

1.

Y. Ding, D. Fan and Y. Pan, Weighted boundedness for a class of rough Macinkiewicz integrals. J. Indiana Univ. Math. 48 (1999), pp 1037-1055.  MR1736970 10.1512/iumj.1999.48.1696 Y. Ding, D. Fan and Y. Pan, Weighted boundedness for a class of rough Macinkiewicz integrals. J. Indiana Univ. Math. 48 (1999), pp 1037-1055.  MR1736970 10.1512/iumj.1999.48.1696

2.

Y. Ding, S. Z. Lu and K. Yabuta, A problem on rough parametric Marcinkiewicz functions. J. Austral. Math. Soc. 72 (2002), pp 13-21.  MR1868701 10.1017/S1446788700003542 Y. Ding, S. Z. Lu and K. Yabuta, A problem on rough parametric Marcinkiewicz functions. J. Austral. Math. Soc. 72 (2002), pp 13-21.  MR1868701 10.1017/S1446788700003542

3.

J. Duoandikoetxea and E. Seijo, Weighted inequalities for rough square functions through extrapolation. Studia Math. 149 (2002), pp 239-252.  MR1890732 10.4064/sm149-3-2 J. Duoandikoetxea and E. Seijo, Weighted inequalities for rough square functions through extrapolation. Studia Math. 149 (2002), pp 239-252.  MR1890732 10.4064/sm149-3-2

4.

L. Hömander, Estimates for translation invariant operators in $L^p$ spaces. Acta Math. 104 (1960), pp 93-140.  MR121655 10.1007/BF02547187 L. Hömander, Estimates for translation invariant operators in $L^p$ spaces. Acta Math. 104 (1960), pp 93-140.  MR121655 10.1007/BF02547187

5.

J. Lee and K. S. Rim, Estimates of Marcinkiewicz integral with bounded homogeneous kernel of degree zero. Integral Equations and Operator Theory. 48 (2004), pp 213-223.  MR2030528 10.1007/s00020-001-1193-1 J. Lee and K. S. Rim, Estimates of Marcinkiewicz integral with bounded homogeneous kernel of degree zero. Integral Equations and Operator Theory. 48 (2004), pp 213-223.  MR2030528 10.1007/s00020-001-1193-1

6.

Y. Lin, Z. G. Liu and F. L. Gao, Endpoint estimates for parametrized Littlewood-Paley operator. Commun. Math. Anal. 12 (2012), pp 11-25.  MR2905128 euclid.cma/1331929868 Y. Lin, Z. G. Liu and F. L. Gao, Endpoint estimates for parametrized Littlewood-Paley operator. Commun. Math. Anal. 12 (2012), pp 11-25.  MR2905128 euclid.cma/1331929868

7.

Y. Lin, Z. G. Liu, D. L. Mao and Z. K. Sun, Parametrized area integrals on Hardy spaces and weak Hardy spaces. Acta Math. Sinica 29 (2013), pp 1857-1870.  MR3096549 10.1007/s10114-013-2440-5 Y. Lin, Z. G. Liu, D. L. Mao and Z. K. Sun, Parametrized area integrals on Hardy spaces and weak Hardy spaces. Acta Math. Sinica 29 (2013), pp 1857-1870.  MR3096549 10.1007/s10114-013-2440-5

8.

L. Z. Liu, Interior estimates in Morrey spaces for solutions of elliptic equations and weighted boundedness for commutators of singular integral operators. Acta Math Scientia. 46 (2005), pp 89-94.  MR2119340 L. Z. Liu, Interior estimates in Morrey spaces for solutions of elliptic equations and weighted boundedness for commutators of singular integral operators. Acta Math Scientia. 46 (2005), pp 89-94.  MR2119340

9.

M. Sakamoto and K. Yabuta, Boundedness of Marcinkiewicz functions. Studia Math. 135 (1999), pp 103-142.  MR1690749 M. Sakamoto and K. Yabuta, Boundedness of Marcinkiewicz functions. Studia Math. 135 (1999), pp 103-142.  MR1690749

10.

A. Torchinsky and S. Wang, A note on the Marcinkiewicz integral. Colloq. Math. 60/61 (1990), pp 235-243.  MR1096373 A. Torchinsky and S. Wang, A note on the Marcinkiewicz integral. Colloq. Math. 60/61 (1990), pp 235-243.  MR1096373

11.

Q. Y. Xue, Parametrized Littlewood-Pley operators. Doctor Degree Dissertation, Beijing Normal University. (2004). Q. Y. Xue, Parametrized Littlewood-Pley operators. Doctor Degree Dissertation, Beijing Normal University. (2004).
Copyright © 2014 Mathematical Research Publishers
Y. Lin, C.H. Song, and X. Xuan "Sharp Maximal Function Estimates for Parameterized Littlewood-Paley Operators and Area Integrals," Communications in Mathematical Analysis 16(1), 66-83, (2014). https://doi.org/
Published: 2014
Vol.16 • No. 1 • 2014
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