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The term "Euclidean harmonic analysis" means the studying of the classical Fourier transform for functions on $\mathbb{R}^n$ or $\mathbb{T}^n$. This include the basic properties of the Fourier transform such as Plancherel's theorem for $L^2$ functions as well as other more sophisticated techniques and results, for example, the Fourier transform for $L^p$ functions, Hardy-Littlewood maximal function, Littlewood-Paley functionals, etc.

My question is if there is any application for these theories to geometry? What I know about applications of the Fourier transform to geometry is that it can be used to define Sobolev spaces for $L^2$ functions on manifolds and then pseudodifferential operators. Combining these with the topological $K$-theory, one can prove the Atiyah-Singer index theorem which gives a lot of interesting results for compact manifolds. A standard reference for this is Lawson's book, Spin Geometry.

But to do this we don't need anything that is essentially more delicate then the Plancherel's theorem or formulae such as $\widehat{\partial_x^\alpha u}(\xi)=\xi^\alpha \widehat{u}(\xi)$. So do all those $L^p$ results or Hardy-Littlewood maximal functions, etc. help us understand more about geometry? If so, does anyone has some reference for this kind of results?

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Analysis results involving $L^p$ spaces for $p \neq 2$ show up naturally when you are studying non-linear partial differential equations. This occurs for example in the theory of $J$-holomorphic curves in symplectic geometry which gives you a powerful tool to study symplectic invariants of your manifold.

The basic idea is that given a symplectic manifold $(M,\omega)$, one can choose an almost complex structure $J$ that is compatible with $\omega$ and study the solutions inside $M$ of the non-linear Cauchy-Riemann equation $du \circ j + J \circ du = 0$ where $u \colon C \rightarrow M$ and $(C,j)$ is a Riemann surface. Such maps are called pseudoholomorphic or $J$-holomorphic curves (inside $M$). In local coordinates, this equation is a non-linear version of the regular Cauchy-Riemann equation and one wants to show various properties such as regularity and removal of singularities. Since $C$ is a (real) two-dimensional surface, the natural arena on which the equation is defined is on a Sobolev space $W^{1,p}$ with $p > 2$ (by the Sobolev embedding theorem, this is the minimal Sobolev space in which functions are guaranteed to be continuous and there are various problems in even making sense of the equation if $p \leq 2$). By applying standard techniques, one is lead to use results from the theory of elliptic regularity in $L^p$ spaces for $p \neq 2$. Such results are in turn based on more advanced analysis involving the theory of singular integrals and the Calderon-Zygmund decomposition.

A standard reference for what I described above is the book "J-Holomorphic Curves and Symplectic Topology" by McDuff and Salamon in which there is a whole appendix dedicated to proving and using hard analysis to study the properties of $J$-holomorphic curves.

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  • $\begingroup$ Today I mentioned this to some other guy and he told me that in addition to $J$-holomorphic curves, we also need this kind of analysis to establish transversality and compactness for the moduli spaces of the Seiberg-Witten invariants of $4$-manifolds. A standard reference is the lecture notes "Spin Geometry and Seiberg Witten invariants" by Salamon and Z\"{u}rich. Also, for J-Holomorphic curves, McDuff and Salamon have a shorter notes, "J-holomorphic Curves and Quantum Cohomology". $\endgroup$ – Chris Kuo Dec 2 '16 at 5:24

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