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Functional calculus and harmonic analysis in geometry

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

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


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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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Correspondence to Lashi Bandara.

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Bandara, L. Functional calculus and harmonic analysis in geometry. São Paulo J. Math. Sci. 15, 20–53 (2021).

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