Abstract
Suppose that {T t : t ≥ 0} is a symmetric diffusion semigroup on L 2(X) and denote by \({\{\widetilde{T}_t:t\geq0\}}\) its tensor product extension to the Bochner space \({L^p(X,\,\mathcal{B})}\), where \({\mathcal{B}}\) belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf–Dunford–Schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of \({\{\widetilde{T}_t\,:\,t\,\geq\,0\}}\) on \({L^p(X,\,\mathcal{B})}\). As an application, we show that such continuations exhibit pointwise convergence.
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References
Bourgain, J.: Vector-valued singular integrals and the H 1-BMO duality. In: Probability Theory and Harmonic Analysis (Cleveland, OH, 1983), Monogr. Textbooks Pure Appl. Math., vol. 98, pp. 1–19. Dekker, New York (1986)
Burkholder, D.L.: Martingale transforms and the geometry of Banach spaces. In: Probability in Banach Spaces, III (Medford, MA, 1980), Lecture Notes in Math., vol. 860, pp. 35–50. Springer, Berlin (1981)
Burkholder, D.L.: A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. In: Conference on Harmonic Analysis in Honor of Antoni Zygmund, vol. I, II (Chicago, IL, 1981), Wadsworth Math. Ser., pp. 270–286. Wadsworth, Belmont, CA (1983)
Coifman, R.R., Rochberg, R., Weiss, G.: Applications of transference: the L p version of von Neumann’s inequality and the Littlewood–Paley–Stein theory. In: Linear Spaces and Approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1977), Internat. Ser. Numer. Math., vol. 40, pp. 53–67. Birkhäuser, Basel (1978)
Cowling, M.: Harmonic analysis on semigroups (2). Ann. Math. 117(2), 267–283 (1983)
Cowling, M., Doust, I., McIntosh, A., Yagi, A.: Banach space operators with a bounded H ∞ functional calculus. J. Aust. Math. Soc. Ser. A 60(1), 51–89 (1996)
Dodds, P.G., Dodds, T.K., de Pagter, B.: Fully symmetric operator spaces. Integral Equ. Oper. Theory 15(6), 942–972 (1992)
Dore, G., Venni, A.: On the closedness of the sum of two closed operators. Math. Z. 196(2), 189–201 (1987)
Dunford, N., Schwartz, J.T.: Linear Operators. Part I. Interscience Publishers, New York (1958)
Fendler, G.: Dilations of one parameter semigroups of positive contractions on L p spaces. Can. J. Math. 49(4), 736–748 (1997)
Guerre-Delabrière, S.: Some remarks on complex powers of ( − Δ) and UMD spaces. Ill. J. Math. 35(3), 401–407 (1991)
Hieber, M., Prüss, J.: Functional calculi for linear operators in vector-valued L p-spaces via the transference principle. Adv. Differ. Equ. 3(6), 847–872 (1998)
Hytönen, T.: Littlewood–Paley–Stein theory for semigroups in UMD spaces. Preprint
Kipnis, C.: Majoration des semi-groupes de contractions de L 1 et applications. Ann. Inst. H. Poincaré Sect. B (N.S.) 10(1974), 369–384 (1975)
Krengel, U.: Ergodic Theorems, de Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter & Co., Berlin (1985)
Kubokawa, Y.: Ergodic theorems for contraction semi-groups. J. Math. Soc. Jpn 27, 184–193 (1975)
Kunstmann, P.C., Weis, L.: Maximal L p -regularity for parabolic equations, Fourier multiplier theorems and H ∞-functional calculus. In: Functional Analytic Methods for Evolution Equations, Lecture Notes in Math., vol. 1855, pp. 65–311. Springer, Berlin (2004)
Martínez, T., Torrea, J.L., Xu, Q.: Vector-valued Littlewood–Paley–Stein theory for semigroups. Adv. Math. 203(2), 430–475 (2006)
Peller, V.: An analogue of an inequality of J. von Neumann, isometric dilation of contractions, and approximation by isometries in spaces of measurable functions. A translation of Trudy Mat. Inst. Steklov. 155(1981), pp. 103–150. Proc. Steklov Inst. Math. 1983, vol. 1. American Mathematical Society, Providence, RI (1983)
Pisier, G.: Complex interpolation and regular operators between Banach lattices. Arch. Math. 62, 261–269 (1994)
Prüss, J., Sohr, H.: On operators with bounded imaginary powers in Banach spaces. Math. Z. 203(3), 429–452 (1990)
Rubio de Francia, J.L.: Martingale and integral transforms of Banach space valued functions. In: Probability and Banach Spaces (Zaragoza, 1985), Lecture Notes in Math., vol. 1221, pp. 195–222. Springer, Berlin (1986)
Stein, E.M.: Topics in harmonic analysis related to the Littlewood–Paley theory. Ann. Math. Stud., vol. 63. Princeton University Press, Princeton (1970)
Taggart, R.J.: Evolution equations and vector-valued L p spaces. Ph.D. dissertation, The University of New South Wales, Sydney (2008)
Titchmarsh, E.C.: The Theory of Functions. Oxford University Press, Oxford (1978)
Weis, L.: Operator-valued Fourier multiplier theorems and maximal L p -regularity. Math. Ann. 319(4), 735–758 (2001)
Xu, Q.: Littlewood–Paley theory for functions with values in uniformly convex spaces. J. Reine Angew. Math. 504, 195–226 (1998)
Zimmermann, F.: On vector-valued Fourier multiplier theorems. Stud. Math. 93(3), 201–222 (1989)
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Taggart, R.J. Pointwise convergence for semigroups in vector-valued L p spaces. Math. Z. 261, 933–949 (2009). https://doi.org/10.1007/s00209-008-0360-3
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DOI: https://doi.org/10.1007/s00209-008-0360-3