Volume 8, Issue 3, June 2019, Page: 47-53
Two Weight Characterization of New Maximal Operators
Hu Yunpeng, Department of Mathematics and Systems Sciences, Xinjiang University, Urumqi, China
Cao Yonghui, Department of Mathematics and Systems Sciences, Xinjiang University, Urumqi, China
Received: Jun. 23, 2019;       Accepted: Jul. 19, 2019;       Published: Aug. 5, 2019
DOI: 10.11648/j.pamj.20190803.11      View  30      Downloads  10
Abstract
For the last twenty years, there has been a great deal of interest in the theory of two weight. In the present paper, we investigate the two weight norm inequalities for fractional new maximal operator on the Lebesgue space. More specifically, we obtain that the sufficient and necessary conditions for strong and weak type two weight norm inequalities for a new fractional maximal operators by introducing a class of new two weight functions. In the discussion of strong type two weight norm inequalities, we make full use of the properties of dyadic cubes and truncation operators, and utilize the space decomposition technique which space is decomposed into disjoint unions. In contrast, weak type two weight norm inequalities are more complex. We have the aid of some good properties of Ap weight functions and ingeniously use the characteristic function. What should be stressed is that the new two weight functions we introduced contains the classical two weights and our results generalize known results before. In this paper, it is worth noting that w(x)dx may not be a doubling measure if our new weight functions ω∈Ap (φ). Since φ(|Q|)≥1, our new weight functions are including the classical Muckenhoupt weights.
Keywords
Two Weight, Maximal Operator, Lebesgue Space
To cite this article
Hu Yunpeng, Cao Yonghui, Two Weight Characterization of New Maximal Operators, Pure and Applied Mathematics Journal. Vol. 8, No. 3, 2019, pp. 47-53. doi: 10.11648/j.pamj.20190803.11
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
B. Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc.. 165, 207-226, 1972.
[2]
B. Muckenhoupt, R. L. Wheeden. Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform. Studia Math.. 55 (3): 279-294, 1976.
[3]
E. T. Sawyer. A characterization of a two-weight norm inequality for maximal operators. Studia Math.. 75 (1), 1-11, 1982.
[4]
D. Cruz-Uribe. New proofs of two-weight norm inequalities for the maximal operator. Georgian Math. J.. 7 (1), 33-42, 2000.
[5]
E. T. Sawyer. Weighted norm inequalities for fractional maximal operators. Amer. Math. Soc., 1, 283-309, 1981.
[6]
C. Perez. Two weighted inequalities for potential and fractional type maximal operators. Indiana Math. J., 43, 663-683, 1994.
[7]
R. Wheeden. A characterization of some weighted norm inequalities for the fractional maximal function. Studia Math., 107, 257-272, 1993.
[8]
A. Gogatishvili, V. Kokilashvili. Criteria of strong type two-weighted inequalities for fractional maximal functions. Georgian Math. J., 3, 423-446, 1996.
[9]
L. Tang. Weighted norm inequalities for pseudo-differential oprators with smooth symbols and their commutators. J. Funct. Anal., 262 (4): 1603-1629, 2012.
[10]
J. García-Cuerva, J. Rubio de Francia. Weighted norm inequalities and related topics. Amsterdam-New York, North-Holland, 387-395, 1985.
[11]
A. Laptev. Spectral asympotics of a class of Fourier integral operators (Russian). Trudy Moskov. Mat. Obshch., 43, 92-115, 1981.
[12]
J. Alverez, J. Hounie. Estimates for the kernel and continuity properties of pseudo difffferential operators. Ark. Mat. 28, 1-22, 1990.
[13]
P. Auscher, M. Taylor. Paradifffferential operators and commutators estimates. Comm. Partial Difffferential Equations 20, 1743-1775, 1995.
[14]
R. Coifman, C. Fefffferman. Weighted norm inequalities for maximal functions and singular integrals, Studia. Math. 51, 241-250, 1974.
[15]
S. Chanillo, A. Torchinsky. Sharp fuction and weighted L estimates for a class of pseudo-difffferential operators. Ark. Mat. 28, 1-25, 1985.
[16]
G. X. Pan, L. Tang. New weighted norm inequalities for certain classes of multilinear operators and their iterated commutators. Potential Analysis, 43 (3), 371-398, 2015.
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