Abstract
Let (T t ) t ≥ 0 be a bounded analytic semigroup on L p(Ω), with 1 < p < ∞. Let −A denote its infinitesimal generator. It is known that if A and A * both satisfy square function estimates \({\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}\) and \({\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}\) for \({x\in L^p(\Omega)}\) and \({y\in L^{p^\prime}(\Omega)}\) , then A admits a bounded \({H^{\infty}(\Sigma_\theta)}\) functional calculus for any \({\theta>\frac{\pi}{2}}\) . We show that this actually holds true for some \({\theta<\frac{\pi}{2}}\) .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berkson E., Gillespie T. A.: Spectral decompositions and harmonic analysis on UMD Banach spaces. Studia Math. 112, 13–49 (1994)
Clément P., de Pagter B., Sukochev F. A., Witvliet H.: Schauder decompositions and multiplier theorems. Studia Math. 138, 135–163 (2000)
Cowling M., Doust I., McIntosh A., Yagi A.: Banach space operators with a bounded H ∞ functional calculus. J. Aust. Math. Soc., Ser. A 60, 51–89 (1996)
Dore G., Venni A.: On the closedness of the sum of two closed operators. Math. Z. 196, 189–201 (1987)
J. Garcia-Cuerva, and J. L. Rubio de Francia, Weighted norm inequalities. North-Holland Mathematics Studies 116, 1985, pp. x+604
M. Haase, The functional calculus for sectorial operators, Operator Theory: Advances and Applications, 169, Birkhuser Verlag, Basel, 2006, pp. xiv+392
M. Junge, C. Le Merdy, and Q. Xu, H ∞ functional calculus and square functions on noncommutative L p-spaces, Soc. Math. France, Ast erisque 305, 2006.
Kalton N.: A remark on sectorial operators with an H ∞-calculus. Contemp. Math. 321, 91–99 (2003)
Kalton N., Kunstmann P., Weis L.: Perturbation and interpolation theorems for the H ∞-calculus with applications to differential operators. Math. Ann. 336, 747–801 (2006)
Kalton N. J., Weis L.: The H ∞-calculus and sums of closed operators. Math. Ann. 321, 319–345 (2001)
N. J. Kalton, and L. Weis, The H ∞-functional calculus and square function estimates, Unpublished manuscript (2004).
P. C. Kunstmann, and L. Weis, Maximal L p -regularity for parabolic equations, Fourier multiplier theorems and H ∞-functional calculus, pp. 65–311 in “Functional analytic methods for evolution equations”, Lect. Notes in Math. 1855, Springer, 2004.
Le Merdy C.: H ∞-functional calculus and applications to maximal regularity. Publ. Math. Besançon 16, 41–77 (1998)
Le Merdy C.: On square functions associated to sectorial operators. Bull. Soc. Math. France 132, 137–156 (2004)
Le Merdy C.: Square functions, bounded analytic semigroups, and applications. Banach Center Publ. 75, 191–220 (2007)
C. Le Merdy, H ∞ functional calculus and square function estimates for Ritt operators, Preprint 2011, arXiv:1202.0768.
Lindenstrauss J., Tzafriri L.: Classical Banach spaces II. Springer-Verlag, Berlin (1979)
McIntosh A.: Operators which have an H ∞ functional calculus. Proc. CMA Canberra 14, 210–231 (1986)
Weis L.: Operator valued Fourier multiplier theorems and maximal regularity. Math. Annalen 319, 735–758 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author is supported by the research program ANR 2011 BS01 008 01.
Rights and permissions
About this article
Cite this article
Le Merdy, C. A sharp equivalence between H ∞ functional calculus and square function estimates. J. Evol. Equ. 12, 789–800 (2012). https://doi.org/10.1007/s00028-012-0154-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-012-0154-2