Littlewood-Paley-Stein theory for semigroups in UMD spaces. (English) Zbl 1213.42012

Summary: The Littlewood-Paley theory for a symmetric diffusion semigroup \(T^t\), as developed by Stein, is here generalized to deal with the tensor extensions of these operators on the Bochner spaces \(L^p(\mu,X)\), where \(X\) is a Banach space. The \(g\)-functions in this situation are formulated as expectations of vector-valued stochastic integrals with respect to a Brownian motion. A two-sided \(g\)-function estimate is then shown to be equivalent to the UMD property of \(X\). As in the classical context, such estimates are used to prove the boundedness of various operators derived from the semigroup \(T^t\), such as the imaginary powers of the generator.


42A61 Probabilistic methods for one variable harmonic analysis
42B25 Maximal functions, Littlewood-Paley theory
46B09 Probabilistic methods in Banach space theory
46B20 Geometry and structure of normed linear spaces
Full Text: DOI Euclid EuDML


[1] Berkson, E. and Gillespie, T. A.: Spectral decompositions and harmonic analysis on UMD spaces. Studia Math. 112 (1994), no. 1, 13-49. · Zbl 0823.42004
[2] Berkson, E. and Gillespie, T. A.: An \(\mathfrakM_q(\mathbbT)\)-functional calculus for power-bounded operators on certain UMD spaces. Studia Math. 167 (2005), no. 3, 245-257. · Zbl 1076.47010
[3] Bourgain, J.: Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat. 21 (1983), no. 2, 163-168. · Zbl 0533.46008
[4] Bourgain, J.: Vector-valued singular integrals and the \(H^1\)-\(\BMO\) duality. In Probability theory and harmonic analysis (Cleveland, Ohio, 1983) , 1-19. Monogr. Textbooks Pure Appl. Math. 98 . Marcel Dekker, New York, 1986. · Zbl 0602.42015
[5] Clément, P., de Pagter, B., Sukochev, F. A. and Witvliet, H.: Schauder decomposition and multiplier theorems. Studia Math. 138 (2000), no. 2, 135-163. · Zbl 0955.46004
[6] Clément, P. and Prüss, J.: Some remarks on maximal regularity of parabolic problems. In Evolution equations: applications to physics, industry, life sciences and economics (Levico Terme, 2000) , 101-111. Progr. Nonlinear Differential Equations Appl. 55 . Birkhäuser, Basel, 2003. · Zbl 1036.35115
[7] Cowling, M., Doust, I., McIntosh, A. and Yagi, A.: Banach space operators with a bounded \(H^\infty\) functional calculus. J. Austral. Math. Soc. Ser. A 60 (1996), no. 1, 51-89. · Zbl 0853.47010
[8] Diestel, J., Jarchow, H. and Tonge, A.: Absolutely summing operators . Cambridge Studies in Advanced Math. 43 . Cambridge Univ. Press, Cambridge, 1995. · Zbl 0855.47016
[9] Dodds, P. G., Dodds, T. K. and de Pagter, B.: Fully symmetric operator spaces. Integral Equations Operator Theory 15 (1992), no. 6, 942-972. · Zbl 0807.46028
[10] Doob, J. L.: A ratio operator limit theorem. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1 (1962/1963), 288-294. · Zbl 0122.36302
[11] Guerre-Delabrière, S.: Some remarks on complex powers of \((-\Delta)\) and UMD spaces. Illinois J. Math. 35 (1991), no. 3, 401-407. · Zbl 0703.47024
[12] Kalton, N. and Weis, L.: The \(H^\infty\)-functional calculus and square function estimates. Manuscript in preparation. · Zbl 0992.47005
[13] Kunstmann, P. C. and Weis, L.: Maximal \(L_p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty\)-functional calculus. In Functional analytic methods for evolution equations , 65-311. Lecture Notes in Math. 1855 . Springer, Berlin, 2004. · Zbl 1097.47041
[14] Martínez, T., Torrea, J. L. and Xu, Q.: Vector-valued Littlewood-Paley-Stein theory for semigroups. Adv. Math. 203 (2006), no. 2, 430-475. · Zbl 1111.46008
[15] van Neerven, J. M. A. M., Veraar, M. C. and Weis, L.: Stochastic integration in UMD Banach spaces. Ann. Probab. 35 (2007), no. 4, 1438-1478. · Zbl 1121.60060
[16] van Neerven, J. M. A. M. and Weis, L.: Stochastic integration of functions with values in a Banach space. Studia Math. 166 (2005), no. 2, 133-170. · Zbl 1073.60059
[17] Rosiński, J. and Suchanecki, Z.: On the space of vector-valued functions integrable with respect to the white noise. Colloq. Math. 43 (1980), no. 1, 183-201. · Zbl 0478.60017
[18] Rota, G.-C.: An \?\? alternierende Verfahren\?\? for general positive operators. Bull. Amer. Math. Soc. 68 (1962), 95-102. · Zbl 0116.10403
[19] Rubio de Francia, J. L.: Martingale and integral transforms of Banach space valued functions. In Probability in Banach spaces (Zaragoza, 1985) , 195–222. Lecture Notes in Math. 1221 . Springer, Berlin, 1986. · Zbl 0615.60041
[20] Stein, E. M.: Interpolation of linear operators. Trans. Amer. Math. Soc. 83 (1956), 482-492. JSTOR: · Zbl 0072.32402
[21] Stein, E. M.: Topics in harmonic analysis related to Littlewood-Paley theory . Annals of Mathematics Studies 63 . Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. · Zbl 0193.10502
[22] Weis, L.: Operator-valued Fourier multiplier theorems and maximal \(L_ p\)-regularity. Math. Ann. 319 (2001), no. 4, 735-758. · Zbl 0989.47025
[23] Xu, Q.: Littlewood-Paley theory for functions with values in uniformly convex spaces. J. Reine Angew. Math. 504 (1998), 195-226. · Zbl 0904.42016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.