Quantitative estimates for bounded holomorphic semigroups. (English) Zbl 07825705
The authors of the paper under review revisit the theory of one-parameter semigroups of linear operators on Banach spaces in order to prove quantitative bounds for bounded holomorphic semigroups. In particular, they obtain new quantitative versions of two recent results of Xu related to the vector-valued Littlewood-Paley-Stein theory for symmetric diffusion semigroups. Moreover, the authors alsoquantify the dependence of the analyticity bounds for extensions of diffusion semigroups \((T(t))_{t\geq0}\) to uniformly convex spaces. Especially, they provide an explicit dependence of \(\sup_{t>0}\|t\partial T(t)\|\)s on the martingale cotype constant.
Reviewer: Christian Budde (Bloemfontein)
MSC:
47D03 | Groups and semigroups of linear operators |
47D07 | Markov semigroups and applications to diffusion processes |
42B25 | Maximal functions, Littlewood-Paley theory |
Keywords:
bounded holomorphic semigroups; symmetric diffusion semigroups; uniform convexity; martingale cotype; Littlewood-Paley-Stein theoryReferences:
[1] | Arendt, W.; Batty, CJK; Hieber, M.; Neubrander, F., Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 2011, Basel: Birkhäuser, Basel · Zbl 1226.34002 · doi:10.1007/978-3-0348-0087-7 |
[2] | Bergh, J.; Löfström, J., Interpolation Spaces, 1976, Berlin: Springer, Berlin · Zbl 0344.46071 · doi:10.1007/978-3-642-66451-9 |
[3] | Engel, KJ; Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 2000, New York: Springer, New York · Zbl 0952.47036 |
[4] | Fackler, S.: Holomorphic Semigroups and the Geometry of Banach Spaces. Diploma Thesis, University of Ulm. https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/abschlussarbeiten/fackler_diplom.pdf (2011) |
[5] | Hytönen, T.; van Neerven, J.; Veraar, M.; Weis, L., Analysis in Banach Spaces, 2016, Cham: Springer, Cham · Zbl 1366.46001 · doi:10.1007/978-3-319-48520-1 |
[6] | Hytönen, T.; Naor, A., Heat flow and quantitative differentiation, J. Eur. Math. Soc., 21, 11, 3415-3466, 2019 · Zbl 1436.46024 · doi:10.4171/jems/906 |
[7] | Kato, T., A characterization of holomorphic semigroups, Proc. Am. Math. Soc., 25, 3, 495-498, 1970 · Zbl 0199.45604 · doi:10.1090/S0002-9939-1970-0264456-0 |
[8] | Martínez, T.; Torrea, JL; Xu, Q., Vector-valued Littlewood-Paley-Stein theory for semigroups, Adv. Math., 203, 2, 430-475, 2006 · Zbl 1111.46008 · doi:10.1016/j.aim.2005.04.010 |
[9] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 1983, New York: Springer, New York · Zbl 0516.47023 |
[10] | Pisier, G., Martingales with values in uniformly convex spaces, Israel J. Math., 20, 326-350, 1975 · Zbl 0344.46030 · doi:10.1007/BF02760337 |
[11] | Pisier, G., Holomorphic semigroups and the geometry of Banach spaces, Ann. Math., 115, 2, 375-392, 1982 · Zbl 0487.46008 · doi:10.2307/1971396 |
[12] | Pisier, G.: Probabilistic methods in the geometry of Banach spaces. In: Letta, G., Pratelli, M. (eds.), Probability and Analysis (Varenna, 1985). Lecture Notes in Mathematics, vol. 1206, pp. 167-241. Springer, Berlin (1986) · Zbl 0606.60008 |
[13] | Pisier, G., Martingales in Banach Spaces, Cambridge Studies in Advanced Mathematics, 2016, Cambridge: Cambridge University Press, Cambridge · Zbl 1382.46002 · doi:10.1017/CBO9781316480588 |
[14] | Stein, EM, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Mathematics Studies, 1970, Princeton: Princeton University Press, Princeton · Zbl 0193.10502 |
[15] | Xu, Q., Littlewood-Paley theory for functions with values in uniformly convex spaces, J. Reine Angew. Math., 504, 195-226, 1998 · Zbl 0904.42016 · doi:10.1515/crll.1998.107 |
[16] | Xu, Q., Vector-valued Littlewood-Paley-Stein theory for semigroups II, Int. Math. Res. Not., 21, 7769-7791, 2020 · Zbl 1507.47096 · doi:10.1093/imrn/rny200 |
[17] | Xu, Q.: Holomorphic functional calculus and vector-valued Littlewood-Paley-Stein theory for semigroups. Preprint. arXiv:2105.12175 (2021) |
[18] | Xu, Q.: Optimal orders of the best constants in the Littlewood-Paley inequalities. J. Funct. Anal. 283(6), 109570 (2022) · Zbl 1498.42033 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.