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Outer measures and weak type estimates of Hardy-Littlewood maximal operators
Journal of Inequalities and Applications volume 2006, Article number: 15063 (2006)
Abstract
We will introduce the times modified centered and uncentered Hardy-Littlewood maximal operators on nonhomogeneous spaces for. We will prove that the times modified centered Hardy-Littlewood maximal operator is weak type bounded with constant when if the Radon measure of the space has "continuity" in some sense. In the proof, we will use the outer measure associated with the Radon measure. We will also prove other results of Hardy-Littlewood maximal operators on homogeneous spaces and on the real line by using outer measures.
References
Bernal A: A note on the one-dimensional maximal function. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1989,111(3–4):325–328. 10.1017/S030821050001859X
Calderón A-P: Inequalities for the maximal function relative to a metric. Studia Mathematica 1976,57(3):297–306.
Carlsson H: A new proof of the Hardy-Littlewood maximal theorem. The Bulletin of the London Mathematical Society 1984,16(6):595–596. 10.1112/blms/16.6.595
Coifman RR, Weiss G: Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics. Volume 242. Springer, Berlin; 1971:v+160.
Evans LC, Gariepy RF: Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Florida; 1992:viii+268.
Hardy GH, Littlewood JE: A maximal theorem with function-theoretic applications. Acta Mathematica 1930, 54: 81–116. 10.1007/BF02547518
Melas AD: The best constant for the centered Hardy-Littlewood maximal inequality. Annals of Mathematics. Second Series 2003,157(2):647–688. 10.4007/annals.2003.157.647
Muckenhoupt B, Stein EM: Classical expansions and their relation to conjugate harmonic functions. Transactions of the American Mathematical Society 1965, 118: 17–92.
Nazarov F, Treil S, Volberg A: Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces. International Mathematics Research Notices 1998,1998(9):463–487. 10.1155/S1073792898000312
Sawano Y: Sharp estimates of the modified Hardy-Littlewood maximal operator on the nonhomogeneous space via covering lemmas. Hokkaido Mathematical Journal 2005, 34: 435–458.
Sierpinski W: Un lemme métrique. Fundamenta Mathematicae 1923, 4: 201–203.
Termini D, Vitanza C: Weighted estimates for the Hardy-Littlewood maximal operator and Dirac deltas. The Bulletin of the London Mathematical Society 1990,22(4):367–374. 10.1112/blms/22.4.367
Trinidad Menarguez M, Soria F: Weak typeinequalities of maximal convolution operators. Rendiconti del Circolo Matematico di Palermo. Serie II 1992,41(3):342–352. 10.1007/BF02848939
Wiener N: The ergodic theorem. Duke Mathematical Journal 1939, 5: 1–18. 10.1215/S0012-7094-39-00501-6
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Terasawa, Y. Outer measures and weak type estimates of Hardy-Littlewood maximal operators. J Inequal Appl 2006, 15063 (2006). https://doi.org/10.1155/JIA/2006/15063
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DOI: https://doi.org/10.1155/JIA/2006/15063