WEIGHTED ESTIMATES FOR CERTAIN ROUGH SINGULAR INTEGRALS

Title & Authors
WEIGHTED ESTIMATES FOR CERTAIN ROUGH SINGULAR INTEGRALS
Zhang, Chunjie;

Abstract
In this paper we shall prove some weighted norm inequalities of the form $\small{{\int}_{R^n}\;|Tf(x)|^pu(x)dx\;{\leq}\;C_p\;{\int}_{R^n}\;|f(x)|^pNu(x)dx}$ for certain rough singular integral T and maximal singular integral $\small{T^*}$. Here u is a nonnegative measurable function on $\small{R^n}$ and N denotes some maximal operator. As a consequence, some vector valued inequalities for both T and $\small{T^*}$ are obtained. We shall also get a boundedness result of T on the Triebel-Lizorkin spaces.
Keywords
singular integral;weighted norm inequality;vector valued inequality;
Language
English
Cited by
References
1.
J. Bergh and J. Lofstrom, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976

2.
A. P. Calderon and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289-309

3.
J. Chen, D. Fan, and Y. Ying, Singular integral operators on function spaces, J. Math. Anal. Appl. 276 (2002), no. 2, 691-708

4.
J. Chen, Certain operators with rough singular kernels, Canad. J. Math. 55 (2003), no. 3, 504-532

5.
J. Chen and C. Zhang, Boundedness of rough singular integral operators on the Triebel-Lizorkin spaces, J. Math. Anal. Appl. 337 (2008), 1048-1052

6.
Q. Chen and Z. Zhang, Boundedness of a class of super singular integral operators and the associated commutators, Sci. China Ser. A 47 (2004), no. 6, 842-853

7.
R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), no. 2, 249-254

8.
A. Cordoba and C. Fefferman, A weighted norm inequality for singular integrals, Studia Math. 57 (1976), no. 1, 97-101

9.
J. Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), no. 2, 869-880

10.
J. Duoandikoetxea and J. L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), no. 3, 541-561

11.
C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115

12.
M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series in Mathematics, 79. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991

13.
J. Garcıa-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, 116. Notas de Matematica [Mathematical Notes], 104. North-Holland Publishing Co., Amsterdam, 1985

14.
L. Grafakos and A. Stefanov, $L^{p}$ bounds for singular integrals and maximal singular integrals with rough kernels, Indiana Univ. Math. J. 47 (1998), no. 2, 455-469

15.
S. Hofmann, Weighted norm inequalities and vector valued inequalities for certain rough operators, Indiana Univ. Math. J. 42 (1993), no. 1, 1-14

16.
J. L. Rubio de Francia, F. J. Ruiz, and J. L. Torrea, Calderon-Zygmund theory for operator-valued kernels, Adv. in Math. 62 (1986), no. 1, 7-48

17.
E. M. Stein and G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc. 87 (1958), 159-172

18.
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North- Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978

19.
A. M. Vargas, Weighted weak type (1, 1) bounds for rough operators, J. London Math. Soc. 54 (1996), no. 2, 297-310

20.
D. K. Watson, Vector-valued inequalities, factorization, and extrapolation for a family of rough operators, J. Funct. Anal. 121 (1994), no. 2, 389-415