Abstract
In this paper we prove that the maximal L p-regularity property on the interval (0,T), T>0, for Cauchy problems associated with the square root of Hermite, Bessel or Laguerre type operators on L 2(Ω,dμ;X), characterizes the UMD property for the Banach space X.
Similar content being viewed by others
References
Abu-Falahah, I., Torrea, J.L.: Hermite function expansions versus Hermite polynomial expansions. Glasg. Math. J. 48, 203–215 (2006)
Abu-Falahah, I., Stinga, P.R., Torrea, J.L.: Square functions associated to Schrödinger operators. Stud. Math. 203, 171–194 (2011)
Amann, H.: Maximal regularity for nonautonomous evolution equations. Adv. Nonlinear Stud. 4, 417–430 (2004)
Arendt, W., Bu, S.: The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Math. Z. 240, 311–343 (2002)
Arendt, W., Bu, S.: Tools for maximal regularity. Math. Proc. Camb. Philos. Soc. 134, 317–336 (2003)
Arendt, W., Bu, S.: Fourier series in Banach spaces and maximal regularity. In: Vector Measures, Integration and Related Topics. Oper. Theory Adv. Appl., vol. 201, pp. 21–39. Birkhäuser, Basel (2010)
Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-valued Laplace transforms and Cauchy problems. Monographs in Mathematics, vol. 96. Birkhäuser, Basel (2001)
Auscher, P., Axelsson, A.: Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I. Invent. Math. 184, 47–115 (2011)
Auscher, P., Monniaux, S., Portal, P.: The maximal regularity operator on tent spaces. Commun. Pure Appl. Anal. 11, 2213–2219 (2012)
Betancor, J.J., Fariña, J.C., Martínez, T., Torrea, J.L.: Riesz transform and g-function associated with Bessel operators and their appropriate Banach spaces. Isr. J. Math. 157, 259–282 (2007)
Betancor, J.J., Fariña, J.C., Rodríguez-Mesa, L., Sanabria, A., Torrea, J.L.: Transference between Laguerre and Hermite settings. J. Funct. Anal. 254, 826–850 (2008)
Betancor, J.J., Chicco Ruiz, A., Fariña, J.C., Rodríguez-Mesa, L.: Odd \({\rm BMO}(\Bbb{R})\) functions and Carleson measures in the Bessel setting. Integral Equ. Oper. Theory 66, 463–494 (2010)
Betancor, J.J., Fariña, J.C., Rodríguez-Mesa, L., Sanabria, A., Torrea, J.L.: Lusin type and cotype for Laguerre g-functions. Isr. J. Math. 182, 1–30 (2011)
Betancor, J.J., Castro, A.J., Rodríguez-Mesa, L.: Characterization of Banach valued BMO functions and UMD Banach spaces by using Bessel convolutions. Positivity (2012). doi:10.1007/s11117-012-0189-1
Betancor, J.J., Crescimbeni, R., Fariña, J.C., Stinga, P.R., Torrea, J.L.: A T1 criterion for Hermite-Calderón-Zygmund operators on the \(\mathrm{BMO}_{H}(\mathbb{R}^{n})\) space and applications. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12, 157–187 (2013)
Bourgain, J.: Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat. 21, 163–168 (1983)
Burkholder, D.L.: Martingales and Fourier analysis in Banach spaces. In: Probability and Analysis, Varenna, 1985. Lecture Notes in Math., vol. 1206, pp. 61–108. Springer, Berlin (1986)
Coulhon, T., Lamberton, D.: Régularité L p pour les équations d’évolution. In: Séminaire d’Analyse Fonctionelle 1984/1985, Paris, 1986. Publ. Math. Univ. Paris VII, vol. 26, pp. 155–165 (1986)
Dore, G.: Maximal regularity in L p spaces for an abstract Cauchy problem. Adv. Differ. Equ. 5, 293–322 (2000)
Dziubański, J., Zienkiewicz, J.: Hardy space H 1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality. Rev. Mat. Iberoam. 15, 279–296 (1999)
Dziubański, J., Garrigós, G., Martínez, T., Torrea, J.L., Zienkiewicz, J.: BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality. Math. Z. 249, 329–356 (2005)
Harboure, E., Torrea, J.L., Viviani, B.: Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup. J. Anal. Math. 91, 1–29 (2003)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1934)
Kalton, N.J., Lancien, G.: A solution to the problem of L p-maximal regularity. Math. Z. 235, 559–568 (2000)
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)
Lebedev, N.N.: Special Functions and Their Applications. Dover, New York (1972)
LeCrone, J.: Elliptic operators and maximal regularity on periodic little-Hölder spaces. J. Evol. Equ. 12, 295–325 (2012)
Muckenhoupt, B.: Hermite conjugate expansions. Trans. Am. Math. Soc. 139, 243–260 (1969)
Muckenhoupt, B.: Poisson integrals for Hermite and Laguerre expansions. Trans. Am. Math. Soc. 139, 231–242 (1969)
Muckenhoupt, B.: Conjugate functions for Laguerre expansions. Trans. Am. Math. Soc. 147, 403–418 (1970)
Muckenhoupt, B., Stein, E.M.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965)
Nowak, A., Stempak, K.: Riesz transforms and conjugacy for Laguerre function expansions of Hermite type. J. Funct. Anal. 244, 399–443 (2007)
Nowak, A., Stempak, K.: Riesz transforms for multi-dimensional Laguerre function expansions. Adv. Math. 215, 642–678 (2007)
Nowak, A., Stempak, K.: On L p-contractivity of Laguerre semigroups. To appear in Illinois J. Math. arXiv:1009.1767v1
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)
Stempak, K., Torrea, J.L.: Poisson integrals and Riesz transforms for Hermite function expansions with weights. J. Funct. Anal. 202, 443–472 (2003)
Stempak, K., Torrea, J.L.: On g-functions for Hermite function expansions. Acta Math. Hung. 109, 99–125 (2005)
Stinga, P.R., Zhang, C.: Harnack’s inequality for fractional nonlocal equations. Discrete Contin. Dyn. Syst., Ser. A 33, 3153–3170 (2013)
Szegő, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975)
Thangavelu, S.: Lectures on Hermite and Laguerre expansions. Mathematical Notes, vol. 42. Princeton University Press, Princeton (1993)
Weis, L.: Operator-valued Fourier multiplier theorems and maximal L p -regularity. Math. Ann. 319, 735–758 (2001)
Acknowledgements
We would like to offer thanks to the referee for his advice which have improved quite a lot this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Markus Haase.
Rights and permissions
About this article
Cite this article
Almeida, V., Betancor, J.J. & Castro, A.J. UMD Banach spaces and the maximal regularity for the square root of several operators. Semigroup Forum 88, 21–51 (2014). https://doi.org/10.1007/s00233-013-9498-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-013-9498-3