The endpoint Fefferman-Stein inequality for the strong maximal function with respect to nondoubling measure
Volume 11, Issue 11, pp 1271--1281
http://dx.doi.org/10.22436/jnsa.011.11.07
Publication Date: September 05, 2018
Submission Date: December 17, 2017
Revision Date: July 21, 2018
Accteptance Date: August 19, 2018
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Authors
Wei Ding
- School of Sciences, Nantong University, Nantong 226007, P. R. China.
Abstract
Let \(d\mu(x_1, \ldots, x_n)=d\mu_1(x_1)\cdots d\mu_n(x_n)\) be a
product measure which is not necessarily doubling in
\(\mathbb{R}^n\) (only assuming \(d\mu_i\) is doubling on \(\mathbb{R}\)
for \(i=2, \ldots, n\)), and \(M_{d\mu}^n\) be the strong maximal function defined by
\[ M_{d\mu}^n f(x)=\sup_{x\in R\in \mathcal{R}}\frac{1}{\mu(R)}\int_{R}|f(y)|d\mu(y),\]
where \(\mathcal{R}\) is the collection of rectangles with sides
parallel to the coordinate axes in \(\mathbb{R}^n\), and \(\omega,\nu\) are two nonnegative functions. We
give a sufficient condition on \(\omega,\nu\) for which the operator \(M_{d\mu}^n\) is bounded from \(L(1+(\log^{+})^{n-1})(\nu d\mu)\) to \(L^{1,\infty}(\omega d\mu)\). By interpolation,
\(M^{n}_{d\mu}\) is bounded from \(L^{p}(\nu d\mu)\) to \(L^{p}(\omega d\mu)\), \(1<p<\infty\).
Share and Cite
ISRP Style
Wei Ding, The endpoint Fefferman-Stein inequality for the strong maximal function with respect to nondoubling measure, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 11, 1271--1281
AMA Style
Ding Wei, The endpoint Fefferman-Stein inequality for the strong maximal function with respect to nondoubling measure. J. Nonlinear Sci. Appl. (2018); 11(11):1271--1281
Chicago/Turabian Style
Ding, Wei. "The endpoint Fefferman-Stein inequality for the strong maximal function with respect to nondoubling measure." Journal of Nonlinear Sciences and Applications, 11, no. 11 (2018): 1271--1281
Keywords
- Fefferman-Stein inequality
- strong maximal function
- nondoubling measure
- \(A^\infty\) weights
- reverse Holder's inequality
MSC
References
-
[1]
R. J. Bagby , Maximal functions and rearrangements: some new proofs, Indiana Univ. Math. J., 32 (1983), 879–891.
-
[2]
A. Chang, R. Fefferman , Some recent developments in Fourier Analysis and \(H^p\) theory on product domains, Bull. Amer. Math. Soc., 12 (1985), 1–43.
-
[3]
A. Cordoba, Maximal functions, covering lemmas and Fourier multipliers, In: Harmonic Analysis in Euclidean Spaces (Proc. Sympos. Pure Math.), 1979 (1979), 29–50.
-
[4]
A. Cordoba, R. Fefferman , A geometric proof of the strong maximal theorem, Ann. Math. Second Series, 102 (1975), 95–100.
-
[5]
A. Cordoba, C. Fefferman, A weighted norm inequality for singular integrals, Studia Math., 57 (1976), 97–101.
-
[6]
D. Cruz-Uribe, J. M. Martell, C. Pérez, Sharp two weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture, Adv. Math., 216 (2007), 647–676.
-
[7]
W. Ding, L. X. Jiang, Y. P. Zhu, Boundedness of strong maximal functions with respect to non-doubling measures, J. Inequal. Appl., 2016 (2016), 14 pages.
-
[8]
R. Fefferman, Some weighted norm inequalities for Cordoba’s maximal function, Amer. J. Math., 106 (1984), 1261–1264.
-
[9]
R. Fefferman, Strong differentiation with respect to measures, Amer. J. Math., 103 (1981), 33–40.
-
[10]
R. Fefferman, J. Pipher, Multiparameter operators and sharp weighted inequalites, Amer. J. Math., 119 (1997), 337–369.
-
[11]
C. Fefferman, E. Stein, Some maximal inequalities, Amer. J. Math., 93 (1971), 107–115.
-
[12]
J. Garcia-Cuerva, J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Publishing Co., Amsterdam (1985)
-
[13]
L. Grafakos, Classical Fourier Analysis , Springer-Verlag, New York (2008)
-
[14]
Y. Han, M. Y. Lee, C. C. Lin, Y. C. Lin, Calderón-Zygmund operators on product Hardy spaces, J. Fourier Anal., 258 (2010), 2834–2861.
-
[15]
Y. S. Han, C. C. Lin, G. Z. Lu, Z. P. Ruan, E. T. Sawyer, Hardy spaces associated with different homogeneities and boundedness of composition operators, Rev. Mat. Iberoam., 29 (2013), 1127–1157.
-
[16]
B. Jawerth, A. Torchinsky, The strong maximal function with respect to measures, Studia Math., 80 (1984), 261–285.
-
[17]
J. L. Journé, Calderman-Zygmund operators on product spaces, Rev. Mat. Iberoam, 1 (1985), 55–92.
-
[18]
R. L. Long, Z. W. Shen, A note on a covering lemma of A. Cordoba and R. Fefferman, Chinese Ann. Math. Ser. B, 9 (1988), 283–291.
-
[19]
T. Luque, I. Parissis, The endpoint Fefferman-Stein inequality for the strong maximal function, J. Fourier Anal., 266 (2014), 199–212.
-
[20]
J. Mateu, P. Mattila, A. Nicolau, J. Orobitg, BMO for nondoubling measures, Duke Math. J., 102 (2000), 533–565.
-
[21]
T. Mitsis, The weighted weak type inequality for the strong maximal function, J. Fourier Anal. Appl., 12 (2006), 645–652.
-
[22]
B. Muckenhoupt, E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc., 118 (1965), 17–92.
-
[23]
F. Nazarov, S. Treil, A. Volberg, Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices, 15 (1997), 703–726.
-
[24]
F. Nazarov, S. Treil, A. Volberg, The Bellman function and two weight inequalities for Haar multipliers, J. Amer. Math. Soc., 12 (1999), 909–928.
-
[25]
J. Orobitg, C. Pérez, \(A_p\) weights for nondoubling measures in \(R^n\) and applications, Trans. Amer. Math. Soc., 354 (2002), 2013–2033.
-
[26]
C. Perez, A remark on weighted inequalities for general maximal operators, Proc. Amer. Math. Soc., 119 (1993), 1121– 1126.
-
[27]
C. Perez, Weighted norm inequalities for general maximal operators, Publ. Mat., 35 (1991), 169–186.
-
[28]
E. T. Sawyer, A characterization of a two weight norm inequality for maximal operators, Studia Math., 75 (1982), 1–11.
-
[29]
P. Sjögren, A remark on maximal function for measures in \(R^n\), Amer. J. Math., 105 (1983), 1231–1233.
-
[30]
E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton (1993)
-
[31]
X. Tolsa , BMO, \(H^1\) and Calderón-Zygmund operators for non doubling measures, Math. Ann., 319 (2001), 89–149.
