Skip to main content
SpringerLink
Log in

\(H^{\infty }\) calculus for submarkovian semigroups on weighted \(L^2\) spaces

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Let \((T_t)_{t \geqslant 0}\) be a markovian (resp. submarkovian) semigroup on some \(\sigma \)-finite measure space \((\Omega ,\mu )\). We prove that its negative generator A has a bounded \(H^{\infty }(\Sigma _\theta )\) calculus on the weighted space \(L^2(\Omega ,wd\mu )\) as long as the weight \(w : \Omega \rightarrow (0,\infty )\) has finite characteristic defined by \(Q^A_2(w) = \sup _{t > 0} \left\| T_t(w) T_t \left( w^{-1} \right) \right\| _{L^\infty (\Omega )}\) (resp. by a variant for submarkovian semigroups). Some additional technical conditions on the semigroup have to be imposed and their validity in examples is discussed. Any angle \(\theta > \frac{\pi }{2}\) is admissible in the above \(H^{\infty }\) calculus, and for some semigroups also certain \(\theta = \theta _w < \frac{\pi }{2}\) depending on the size of \(Q^A_2(w)\). The norm of the \(H^{\infty }(\Sigma _\theta )\) calculus is linear in the \(Q^A_2\) characteristic for \(\theta > \frac{\pi }{2}\). We also discuss negative results on angles \(\theta < \frac{\pi }{2}\). Namely we show that there is a markovian semigroup on a probability space and a \(Q^A_2\) weight w without Hörmander functional calculus on \(L^2(\Omega ,w d\mu )\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1. We do not need any continuity assumption of \(t \mapsto S_t\).

References

  1. Arhancet, C., Fackler, S., Le Merdy, C.: Isometric dilations and \(H^{\infty }\) calculus for bounded analytic semigroups and Ritt operators. Trans. Am. Math. Soc. 369(10), 6899–6933 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Auscher, P., Hofmann, S., Lacey, M., Lewis, J., McIntosh, A., Tchamitchian, P.: The solution of Kato’s conjectures. C. R. Acad. Sci. Paris Sér. I Math. 332(7), 601–606 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Auscher, P., Tchamitchian, P.: Square root problem for divergence operators and related topics. Astérisque 249, viii+172 (1998)

    MathSciNet  MATH  Google Scholar 

  4. Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223, p. x+207. Springer, Berlin (1976)

    Book  Google Scholar 

  5. Blunck, S., Kunstmann, P.: Calderón–Zygmund theory for non-integral operators and the \(H^{\infty }\) functional calculus. Rev. Mat. Iberoamericana 19(3), 919–942 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Böttcher, B., Schilling, R., Wang, J.: Lévy Matters. III. Chapter 1. Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Lecture Notes in Mathematics, 2099. Lévy Matters, p. xviii+199. Springer, Cham (2013)

    MATH  Google Scholar 

  7. Carbonaro, A., Dragičević, O.: Functional calculus for generators of symmetric contraction semigroups. Duke Math. J. 166(5), 937–974 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Coifman, R., Weiss, G.: Some Examples of Transference Methods in Harmonic Analysis. Symposia Mathematica, Vol. XXII (Convegno sull’Analisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti, INDAM, Rome, 1976), pp. 33–45. Academic Press, London (1977)

    Google Scholar 

  9. Cowling, M.: Harmonic analysis on semigroups. Ann. Math. (2) 117(2), 267–283 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cowling, M., Doust, I., McIntosh, A., Yagi, A.: Banach space operators with a bounded \(H^\infty \) functional calculus. J. Austral. Math. Soc. 60, 51–89 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dahmani, K.: Sharp dimension free bound for the Bakry–Riesz vector. arXiv:1611.07696 (preprint on)

  12. Denk, R., Dore, G., Hieber, M., Prüss, J., Venni, A.: New thoughts on old results of R. T. Seeley. Math. Ann. 328(4), 545–583 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. de Simon, L.: Un’applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine. Rend. Sem. Mat. Univ. Padova 34, 205–223 (1964)

    MathSciNet  MATH  Google Scholar 

  14. Domelevo, K., Petermichl, S.: Differential Subordination under change of law. arXiv:1604.01606 (preprint on)

  15. Duong, X. T.: \(H^{\infty }\) functional calculus of second order elliptic partial differential operators on \(L^p\) spaces. Miniconference on Operators in Analysis (Sydney, 1989), Proc. Centre Math. Anal. Austral. Nat. Univ., 24, Austral. Nat. Univ., Canberra, pp 91–102(1990)

  16. Duong, X.T., Robinson, D.: Semigroup kernels, Poisson bounds, and holomorphic functional calculus. J. Funct. Anal. 142(1), 89–128 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Duong, X.T., Ouhabaz, E.M., Sikora, A.: Plancherel-type estimates and sharp spectral multipliers. J. Funct. Anal. 196(2), 443–485 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Duong, X.T., Sikora, A., Yan, L.: Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers. J. Funct. Anal. 260, 1106–1131 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fendler, G.: On dilations and transference for continuous one-parameter semigroups of positive contractions on \({\cal{L}}^p\)-spaces. Ann. Univ. Sarav. Ser. Math. 9(1), iv+97 (1998)

    MathSciNet  MATH  Google Scholar 

  20. García-Cuerva, J., Mauceri, G., Meda, S., Sjögren, P., Torrea, J.L.: Functional calculus for the Ornstein–Uhlenbeck operator. J. Funct. Anal. 183(2), 413–450 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. García-Cuerva, J., Rubio de Francia, J. L.: Weighted norm inequalities and related topics. North-Holland Mathematics Studies, 116. Notas de Matemática [Mathematical Notes], 104. North-Holland Publishing Co., Amsterdam, p x+604 (1985)

  22. Giga, Y.: Domains of fractional powers of the Stokes operator in \(L_r\) spaces. Arch. Rational Mech. Anal. 89(3), 251–265 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gong, R., Yan, L.: Littlewood-Paley and spectral multipliers on weighted \(L^p\) spaces. J. Geom. Anal. 24(2), 873–900 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Grigor’yan, A.: Gaussian Upper Bounds for the Heat Kernel and for Its Derivatives on a Riemannian Manifold. Classical and modern potential theory and applications (Chateau de Bonas, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 430, pp. 237–252. Kluwer Academic Publisher, Dordrecht (1994)

    Google Scholar 

  25. Grigor’yan, A., Telcs, A.: Two-sided estimates of heat kernels on metric measure spaces. Ann. Probab. 40(3), 1212–1284 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Haak, B., Le Merdy, C.: \(\alpha \)-admissibility of observation and control operators. Houston J. Math. 31(4), 1153–1167 (2005)

    MathSciNet  MATH  Google Scholar 

  27. Haak, B., Ouhabaz, E.M.: Exact observability, square functions and spectral theory. J. Funct. Anal. 262(6), 2903–2927 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Haase, M.: The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, vol. 169, p. xiv+392. Birkhäuser Verlag, Basel (2006)

    Book  MATH  Google Scholar 

  29. 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)

    MATH  Google Scholar 

  30. Hytönen, T., van Neerven, J., Portal, P.: Conical square function estimates in UMD Banach spaces and applications to \(H^{\infty }\)-functional calculi. J. Anal. Math. 106, 317–351 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Junge, M., Le Merdy, C., Xu, Q.: \(H^{\infty }\) functional calculus and square functions on noncommutative \(L^p\)-spaces. Astérisque 305, vi+138 (2006)

    MATH  Google Scholar 

  32. Kalton, N., Weis, L.: The \(H^{\infty }\)-calculus and sums of closed operators. Math. Ann. 321(2), 319–345 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kalton, N., Weis, L.: The \(H^{\infty }\)-functional calculus and square function estimates. Preprint on arXiv:1411.0472v2

  34. Kalton, N., Kunstmann, P., Weis, L.: Perturbation and interpolation theorems for the \(H^{\infty }\)-calculus with applications to differential operators. Math. Ann. 336(4), 747–801 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kriegler, C.: Analyticity angle for non-commutative diffusion semigroups. J. Lond. Math. Soc. (2) 83(1), 168–186 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kriegler, C.: Spectral multipliers, \(R\)-bounded homomorphisms and analytic diffusion semigroups. https://hal.archives-ouvertes.fr/tel-00461310v1

  37. Kriegler, C., Weis, L.: Spectral multiplier theorems via \(H^{\infty }\) calculus and \(R\)-bounds. To appear in Math. Z

  38. Kunstmann, P., Weis, L.: Maximal \(L_p\)-Regularity for Parabolic Equations, Fourier Multiplier Theorems and \(H^{\infty }\)-Functional Calculus. Functional Analytic Methods for Evolution Equations, Lecture Notes in Mathematics, vol. 1855, pp. 65–311. Springer, Berlin (2004)

    MATH  Google Scholar 

  39. Lancien, F., Lancien, G., Le Merdy, C.: A joint functional calculus for sectorial operators with commuting resolvents. Proc. Lond. Math. Soc. (3) 77(2), 387–414 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lancien, G., Le Merdy, C.: A generalized \(H^{\infty }\) functional calculus for operators on subspaces of \(L^p\) and application to maximal regularity. Ill. J. Math. 42(3), 470–480 (1998)

    MATH  Google Scholar 

  41. Lebedev, N.N.: Special Functions and Their Applications. Revised English Edition. Translated and Edited by Richard A. Silverman, p. xii+308. Prentice-Hall Inc, Englewood Cliffs (1965)

    Google Scholar 

  42. Le Merdy, C.: \(H^{\infty }\)-functional calculus and applications to maximal regularity. Publ. Math. UFR Sci. Tech. Besançon 16, 41–77 (1998)

    MathSciNet  MATH  Google Scholar 

  43. Le Merdy, C.: The Weiss conjecture for bounded analytic semigroups. J. Lond. Math. Soc. (2) 67(3), 715–738 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  44. Meda, S.: A general multiplier theorem. Proc. Am. Math. Soc. 110(3), 639–647 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ouhabaz, E.M.: Analysis of Heat Equations on Domains. London Mathematical Society Monographs Series, vol. 31, p. xiv+284. Princeton University Press, Princeton (2005)

    Google Scholar 

  46. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44, p. viii+279. Springer, New York (1983)

    Google Scholar 

  47. Prüss, J., Simonett, G.: \(H^{\infty }\)-calculus for the sum of non-commuting operators. Trans. Am. Math. Soc. 359(8), 3549–3565 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  48. Stein, E.M.: Interpolation of linear operators. Trans. Am. Math. Soc. 83, 482–492 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  49. Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Annals of Mathematics Studies, vol. 63, p. viii+146. Princeton, Princeton University Press (1970)

    Google Scholar 

  50. van Neerven, J., Portal, P.: The Weyl calculus for group generators satisfying the canonical commutation relations. arXiv:1806.00980 (Preprint on)

  51. Weis, L.: A New Approach to Maximal \(L_p\)-Regularity. Evolution Equations and Their Applications in Physical and Life Sciences (Bad Herrenalb, 1998). Lecture Notes in Pure and Applications and Mathematics, vol. 215, pp. 195–214. Dekker, New York (2001)

    Google Scholar 

  52. Xu, Q.: \(H^{\infty }\) functional calculus and maximal inequalities for semigroups of contractions on vector-valued \(L_p\)-spaces. Int. Math. Res. Not. 14, 5715–5732 (2015)

    Article  MATH  Google Scholar 

  53. Yagi, A.: Coïncidence entre des espaces d’interpolation et des domaines de puissances fractionnaires d’opérateurs. C. R. Acad. Sci. Paris Sér. I Math. 299(6), 173–176 (1984)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The second author acknowledges support by the grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front) and by the grant ANR-18-CE40-0021 (project HASCON). All authors acknowledge support by the European Research Council grant DLV-862402 (CHRiSHarMa). The third author acknowledges support by the Alexander von Humboldt Stiftung.

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Würzburg, Emil-Fischer-Straße 40, 97074, Würzburg, Germany

    Komla Domelevo & Stefanie Petermichl

  2. Université Clermont Auvergne, CNRS, LMBP, 63000, Clermont-Ferrand, France

    Christoph Kriegler

Authors
  1. Komla Domelevo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christoph Kriegler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefanie Petermichl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christoph Kriegler.

Additional information

Communicated by Loukas Grafakos.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Domelevo, K., Kriegler, C. & Petermichl, S. \(H^{\infty }\) calculus for submarkovian semigroups on weighted \(L^2\) spaces. Math. Ann. 381, 1137–1195 (2021). https://doi.org/10.1007/s00208-021-02175-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-021-02175-w

Mathematics Subject Classification

Search

Navigation