Abstract
In this article, we define the Coifman–Meyer–Stein tent spaces T p,q,α(X) associated with an arbitrary metric measure space (X, d,μ) under minimal geometric assumptions. While gradually strengthening our geometric assumptions, we prove duality, interpolation, and change of aperture theorems for the tent spaces. Because of the inherent technicalities in dealing with abstract metric measure spaces, most proofs are presented in full detail.
Mathematics Subject Classification (2010). 42B35.
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References
F. Albiac and N.J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006.
P. Auscher, A. McIntosh, and E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008), 192–248.
J. Bergh and J. Löfström, Interpolation spaces, Grundlehren der mathematischen Wissenschaften, vol. 223, Springer-Verlag, Berlin, 1976.
A. Bernal, Some results on complex interpolation of \( T_{q}^{p} \) spaces, Interpolation spaces and related topics (Ramat-Gan), Israel Mathematical Conference Proceedings, vol. 5, 1992, pp. 1–10.
V.I. Bogachev, Measure theory, vol. 2, Springer-Verlag, 2007.
W.S. Cohn and I.E. Verbitsky, Factorization of tent spaces and Hankel operators, J. Funct. Anal. 175 (2000), 308–329.
R.R. Coifman, Y. Meyer, and E.M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304–335.
R.R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogenes, Lecture Notes in Mathematics, vol. 242, Springer-Verlag, Berlin, 1971.
J. Diestel and J.J. Uhl, Jr., Vector measures, Mathematical Surveys, vol. 15, American Mathematical Society, Providence, 1977.
L. Forzani, R. Scotto, P. Sjögren, and W. Urbina, On the L p -boundedness of the non-centred Gaussian Hardy–Littlewood maximal function, Proc. Amer. Math. Soc. 130 (2002), no. 1, 73–79.
K. Gowrisankaran, Measurability of functions in product spaces, Proc. Amer. Math. Soc. 31 (1972), no. 2, 485–488.
E. Harboure, J. Torrea, and B. Viviani, A vector-valued approach to tent spaces, J. Anal. Math. 56 (1991), 125–140.
S. Janson, P. Nilsson, and J. Peetre, Notes on Wolff’s note on interpolation spaces, Proc. London Math. Soc. s3-48 (1984), no. 2, 283–299, with appendix by Misha Zafran.
M. Kemppainen, The vector-valued tent spaces T 1 and T ∞, J. Austral. Math. Soc. (to appear), arxiv:1105.0261.
J. Maas, J. van Neerven, and P. Portal, Conical square functions and non-tangential maximal functions with respect to the Gaussian measure, Publ. Mat. 55 (2011), 313–341.
L. Mattner, Product measurability, parameter integrals, and a Fubini counterexample, Enseign. Math. 45 (1999), 271–279.
E. Russ, The atomic decomposition for tent spaces on spaces of homogeneous type, CMA/AMSI research symposium “Asymptotic geometric analysis, harmonic analysis and related topics” (Canberra) (A. McIntosh and P. Portal, eds.), Proceedings of the Centre for Mathematics and its Applications, vol. 42, 2007, pp. 125–135.
P. Sjögren, A remark on the maximal function for measures in \( \mathbb{R}^{n} \), Amer. J. Math. 105 (1983), no. 5, 1231–1233.
H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Company, Amsterdam, 1978.
T.H. Wolff, A note on interpolation spaces, Harmonic Analysis (Minneapolis, Minn., 1981), Lecture Notes in Mathematics, vol. 908, Springer, Berlin, 1982, pp. 199–204.
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Amenta, A. (2014). Tent Spaces over Metric Measure Spaces under Doubling and Related Assumptions. In: Ball, J., Dritschel, M., ter Elst, A., Portal, P., Potapov, D. (eds) Operator Theory in Harmonic and Non-commutative Analysis. Operator Theory: Advances and Applications, vol 240. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-06266-2_1
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DOI: https://doi.org/10.1007/978-3-319-06266-2_1
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