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Functional calculus and harmonic analysis in geometry

  • Special Section: An Homage to Manfredo P. do Carmo
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Abstract

In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these methods as well as their interplay. This is a succinct survey that hopes to inspire geometers and analysts alike to study these methods so that they can be further developed to be potentially applied to a broader range of questions.

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Abbreviations

\(\mathrm{C}^{k}_{\mathrm{c}}({\mathcal {M}}; {\mathcal {E}})\) :

Compactly supported smooth sections \({\mathcal {M}}\rightarrow {\mathcal {E}}\) up to the boundary (if it exists)

\(\mathrm{C}^{k}_{\mathrm{cc}}({\mathcal {M}}; {\mathcal {E}})\) :

Compactly supported smooth sections \({\mathcal {M}}\rightarrow {\mathcal {E}}\) on the interior

\(\mathrm{C}^{k}_{\mathrm{}}({\mathcal {M}}; {\mathcal {E}})\) :

Smooth sections \({\mathcal {M}}\rightarrow {\mathcal {E}}\)

\(\mathrm{T}^*{\mathcal {M}}\) :

Cotangent bundle of \({\mathcal {M}}\)

\(\mathrm{T}^*_x {\mathcal {M}}\) :

Cotangent space of \({\mathcal {M}}\) at x

\(\mathring{{\mathcal {M}}}\) :

Interior of \({\mathcal {M}}\)

\(\mathrm {SymMat}({\mathbb {R}}^n)\) :

Symmetric matrices on \({\mathbb {R}}^n\)

\(\uprho _{{\mathcal {M}}}(\mathrm {g},\mathrm {h})\) :

Extended distance metric measuring the bounded distance between two metric tensors \(\mathrm {g}\) and \(\mathrm {h}\) on \({\mathcal {M}}\)

\({\mathbb {R}}_+\) :

The set \([0, \infty )\)

\(\mathrm{H}^\mathrm{k}_{\mathrm{}}({\mathcal {E}})\) :

Sobolev space of k-th order in \(\mathrm{L}^{2}_{\mathrm{}}\) on a space \({\mathcal {E}}\)

\(\mathrm {spec}(T)\) :

Spectrum of an operator T

\(\upsigma _{\mathrm{D} }(x,\xi )\) :

Principal symbol of \(\mathrm{D} \) in the co-direction \(\xi \) at x

\(\mathrm{T}{\mathcal {M}}\) :

Tangent bundle of \({\mathcal {M}}\)

\(\mathrm{T}_x {\mathcal {M}}\) :

Tangent space of \({\mathcal {M}}\) at x

References

  1. Albrecht, D., Duong, X., McIntosh, A.: Operator theory and harmonic analysis. In: Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), Proceedings of the Centre for Mathematics and Its Applications, ANU, vol. 34, pp. 77–136. Australian National University, Canberra (1996)

  2. Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. Bull. Lond. Math. Soc. 5, 229–234 (1973)

    Article  MathSciNet  Google Scholar 

  3. Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Camb. Philos. Soc. 77, 43–69 (1975)

    Article  MathSciNet  Google Scholar 

  4. Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. II. Math. Proc. Camb. Philos. Soc. 78(3), 405–432 (1975)

    Article  MathSciNet  Google Scholar 

  5. Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. III. Math. Proc. Camb. Philos. Soc. 79(1), 71–99 (1976)

    Article  MathSciNet  Google Scholar 

  6. Atiyah, M.F., Singer, I.M.: Index theory for skew-adjoint Fredholm operators. Inst. Hautes Études Sci. Publ. Math. 37, 5–26 (1969)

    Article  MathSciNet  Google Scholar 

  7. Auscher, P.: Lectures on the Kato square root problem. In: Surveys in Analysis and Operator Theory (Canberra, 2001), Proceedings of the Centre for Mathematics and Its Applications, ANU, vol. 40, pp. 1–18. Australian National University, Canberra (2002)

  8. Auscher, P., Hofmann, S., Lacey, M., McIntosh, A., Tchamitchian, P.: The solution of the Kato square root problem for second order elliptic operators on \({\mathbb{R}}^n\). Ann. Math. (2) 156(2), 633–654 (2002)

    Article  MathSciNet  Google Scholar 

  9. Auscher, P., McIntosh, A., Nahmod, A.: Holomorphic functional calculi of operators, quadratic estimates and interpolation. Indiana Univ. Math. J. 46(2), 375–403 (1997)

    Article  MathSciNet  Google Scholar 

  10. Axelsson, A., Keith, S., McIntosh, A.: Quadratic estimates and functional calculi of perturbed Dirac operators. Invent. Math. 163(3), 455–497 (2006)

    Article  MathSciNet  Google Scholar 

  11. Bandara, L.: Rough metrics on manifolds and quadratic estimates. Math. Z. 283(3–4), 1245–1281 (2016)

    Article  MathSciNet  Google Scholar 

  12. Bandara, L.: Continuity of solutions to space-varying pointwise linear elliptic equations. Publ. Mat. 61(1), 239–258 (2017)

    Article  MathSciNet  Google Scholar 

  13. Bandara, L., McIntosh, A.: The Kato square root problem on vector bundles with generalised bounded geometry. J. Geom. Anal. 26(1), 428–462 (2016)

    Article  MathSciNet  Google Scholar 

  14. Bandara, L., McIntosh, A., Rosén, A.: Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of the metric. Math. Ann. 370(1–2), 863–915 (2018)

    Article  MathSciNet  Google Scholar 

  15. Bandara, L., Rosén, A.: Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of local boundary conditions. Commun. Partial Differ. Equ. 44(12), 1253–1284 (2019)

    Article  MathSciNet  Google Scholar 

  16. Bär, C., Ballmann, W.: Boundary value problems for elliptic differential operators of first order. In: Surveys in Differential Geometry, vol. 17, pp. 1–78. International Press, Boston, MA (2012)

  17. Bär, C., Bandara, L.: Boundary value problems for general first-order elliptic differential operators. arXiv e-prints (2019). arXiv:1906.08581

  18. Carbonaro, A., McIntosh, A., Morris, A.J.: Local Hardy spaces of differential forms on Riemannian manifolds. J. Geom. Anal. 23(1), 106–169 (2013)

    Article  MathSciNet  Google Scholar 

  19. Christ, M.: Lectures on singular integral operators. In: CBMS Regional Conference Series in Mathematics, vol. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1990)

  20. Christ, M.: A \(T(b)\) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61(2), 601–628 (1990)

    Article  MathSciNet  Google Scholar 

  21. Coifman, R.R., McIntosh, A., Meyer, Y.: L’intégrale de Cauchy définit un opérateur borné sur \(L^{2}\) pour les courbes lipschitziennes. Ann. Math. (2) 116(2), 361–387 (1982)

    Article  MathSciNet  Google Scholar 

  22. Cowling, M., Doust, I., McIntosh, A., Yagi, A.: Banach space operators with a bounded \(H^\infty \) functional calculus. J. Austral. Math. Soc. Ser. A 60(1), 51–89 (1996)

    Article  MathSciNet  Google Scholar 

  23. Gigli, N., Mantegazza, C.: A flow tangent to the Ricci flow via heat kernels and mass transport. Adv. Math. 250, 74–104 (2014)

    Article  MathSciNet  Google Scholar 

  24. Grubb, G.: The sectorial projection defined from logarithms. Math. Scand. 111(1), 118–126 (2012)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  26. Hofmann, S., McIntosh, A.: Boundedness and applications of singular integrals and square functions: a survey. Bull. Math. Sci. 1(2), 201–244 (2011)

    Article  MathSciNet  Google Scholar 

  27. Kato, T.: Fractional powers of dissipative operators. J. Math. Soc. Jpn. 13, 246–274 (1961)

    Article  MathSciNet  Google Scholar 

  28. Kato, T.: Perturbation Theory for Linear Operators. Grundlehren der Mathematischen Wissenschaften, Band 132, 2nd edn. Springer, Berlin (1976)

    Google Scholar 

  29. Lions, J.-L.: Espaces d’interpolation et domaines de puissances fractionnaires d’opérateurs. J. Math. Soc. Jpn. 14, 233–241 (1962)

    Article  Google Scholar 

  30. Lott, J.: Optimal transport and Ricci curvature for metric-measure spaces. In: Surveys in Differential Geometry, vol. 11, pp. 229–257. International Press, Somerville, MA (2007)

  31. McIntosh, A.: On the comparability of \(A^{1/2}\) and \(A^{\ast 1/2}\). Proc. Am. Math. Soc. 32, 430–434 (1972)

    MATH  Google Scholar 

  32. McIntosh, A.: Operators which have an \(H_\infty \) functional calculus. In: Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986), Proceedings of the Centre for Mathematics and Its Applications, ANU, vol. 14, pp. 210–231. Australian National University, Canberra (1986)

  33. Morris, A.J.: Local hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds. Ph.D. thesis, Australian National University (2010)

  34. Morris, A.J.: The Kato square root problem on submanifolds. J. Lond. Math. Soc. (2) 86(3), 879–910 (2012)

    Article  MathSciNet  Google Scholar 

  35. Stein, E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton, NJ (1993)

    Google Scholar 

  36. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, NJ (1970)

    Google Scholar 

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Acknowledgements

The author was supported by SPP2026 from the German Research Foundation (DFG). David Rule deserves mention as this paper was borne out of a Colloquium talk given at Linköping University upon his invitation. Andreas Rosén needs to acknowledged for useful feedback on an earlier version of this article. The anonymous referee also deserves a mention for their helpful suggestions. Furthermore, the author acknowledges the gracious hospitality of his auntie Harshya Perera in Sri Lanka who hosted him during the time in which this paper was written.

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  1. Institut für Mathematik, Universität Potsdam, 14476, Potsdam OT Golm, Germany

    Lashi Bandara

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Bandara, L. Functional calculus and harmonic analysis in geometry. São Paulo J. Math. Sci. 15, 20–53 (2021). https://doi.org/10.1007/s40863-019-00149-0

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