# Applications of Microlocal Analysis?

What examples are there of striking applications of the ideas of Microlocal Analysis?

Ideally i'm looking for specific results in any of the relevant fields (PDE, algebraic/differential geometry/topology, representation theory etc.) which follow as natural consequences of Microlocal Analysis and are not known to be obtained easily without it.

Microlocal analysis is used in computed tomography and other tomographic imaging techniques e.g. in medicine .

Specifically, it is used to describe which wavefront sets (here: boundaries of objects, e.g. of organs in the human body) can or can not be detected by a specific tomographic measurement setup and also helps to understand reconstruction artifacts and develop strategies to overcome these. Details can be found for example in

There are striking applications in dynamical systems, due to Dyatlov and Zworski, where dynamical zeta functions were analysed using techniques from microlocal analysis. These zeta functions have a long history in dynamical systems and microlocal analysis is giving a range of new tools.

Although microlocal analysis was developed originally exclusively for linear problems, it has played an increasingly important role in nonlinear PDE via what's known as paradifferential calculus. There are some nice write-ups about this. Here are a few:

http://www.math.ucla.edu/~tao/247b.1.07w/notes6.pdf

http://www.ams.org/notices/201007/rtx100700858p.pdf

http://www.unc.edu/math/Faculty/met/micro2.pdf

Here are some examples:

1. Helgason conjecture: Generically, the spherical principal series representations of a connected real semisimple Lie group $G$ are isomorphic via the Poisson transform to translation representations of $G$ on the spaces of joint eigenfunctions on the symmetric space $X=G/K$. The conjecture was proved in Kashiwara et al., Ann. Math. 107 (1978) using the boundary value theory in Kashiwara, Oshima, Ann. Math. 106 (1977). The inverse of the Poisson transform is shown to be a boundary value map defined by transforming, microlocally at the conormal bundle of a corner (Weyl chamber at infinity), the differential operators of the eigensystem on $X$ to simple normal form.
2. Second term in Weyl asymptotics: The number $N(\lambda)$ of eigenvalues $\leq\lambda$ for the Dirichlet problem in $\Omega\subset\mathbb{R}^3$ of the Laplacian has the asymptotics $N(\lambda)=c_1\lambda^{3/2}+c_2\lambda+o(\lambda)$ if the closed geodesics cover only a set of measure zero in the unit sphere bundle. Up to constant factors $c_1$ and $c_2$ are the volumes of $\Omega$ and of its boundary $\partial\Omega$. This was proved by Ivrii (1980); see Corollary 29.3.4 in volume 4 of Hörmander's treatise. The existence of a two-term asymptotics under certain assumptions had been a long-standing conjecture.
3. Distribution of scattering poles: The Lax--Phillips scattering matrix for acoustic scattering by a non-trapping obstacle has no poles in a logarithmic neighbourhood of the real axis. This was conjectured by Lax and Phillips in their 1967 book (Conjecture (a) on page 155) and proved by Melrose and Sjöstrand, CPAM 35 (1982), as a corollary of their results on propagation of wavefront sets in boundary problems.
4. Controllability in control theory: The solutions of a wave equation are controllable in time $T>0$ from a subset $\Gamma$ in the space-time boundary iff every geometric optics ray of time length $T$ intersects $\Gamma$. This was shown in Bardos, Lebeau, Rauch, SIAM J. Control Opt. 30 (1992); the proof uses the results of Melrose and Sjöstrand mentioned above.

The list can be continued with trace formulae, applications to inverse problems, and more. Many results have been developed further in semiclassical analysis, a modern variant of microlocal analysis, which works under weaker assumptions, and is applicable to Schrödinger's equation, for example.

(1) The older and more widely known applications are to regularity and solvability of PDEs of any order that are not necessarily elliptic.

(2) The phenomenon of the propagation of singularities along certain submanifolds of the cotangent space is also another striking application.

These refined notions of hypoellipticity and solvability as well as the propagation of singularities involve many fantastic results which are possible because of microlocal analysis. Even the statements of many of these results require microlocal analysis.

(3) There are also several applications to several complex variables and a very active subfield called CR (for Cauchy Riemann) manifolds.

Some of the standard references on these topics are the two volumes of F. Treves on Introduction to Pseudodifferential and Fourier Integral Operators and the L. Hormander's treatise (Volumes I, II, III, etc).

Search for papers by Maarten de Hoop (Colorado School of Mines, then Purdue, now a Simons chair at Rice University), who has used microlocal analysis very extensively to study problems in global and exploration seismology and imaging.

Broadly speaking, microlocal analysis helps one in 'geometrization' of certain results on the singularities of distributions. In this direction, one striking application of microlocal analysis is the Hormander condition for the existence of product of distributions. Study of nonlinear pdes and the renormalization problem in Quantum Field Theory are two major applications of product of distributions.

An important tool in this context is the wavefront set of a distribution. Roughly speaking, the wavefront set locates the singularities of a distribution in space and also helps in the analysis of the 'direction' of singularities. For $$u\in \mathcal{D}'(\mathbb{R}^n)$$ the wavefront set $$WF(u)\subset T^{*}(\mathbb{R}^n)$$. If $$(x,\xi)\in WF(u)$$ then $$\xi\in \mathbb{R}^{n}-\{0\}$$ is the direction of 'propagation of singularity' that is located at $$x\in\mathbb{R}^{n}$$.

Hormander condition for existence of product: Let $$u,v\in \mathcal{D}'({\mathbb{R}^n})$$. If for every $$(x,\xi)\in WF(u)$$ we have $$(x,-\xi)\notin WF(v)$$ then the product $$uv$$ exists and the Leibniz rule $$\partial(uv)=\partial(u)v+u\partial(v)$$ holds. Moreover, $$WF(uv)\subseteq WF(u)\cup WF(v)\cup \{(x,\xi+\eta):(x,\xi)\in WF(u), (x,\eta)\in WF(v)\}$$.

We can note the following from the above condition:

1. If the point of singularity is same for both distributions then for the product to exist the 'directions' of the singularity should not cancel each other.
2. We can also estimate the wavefront set of the product.
3. The condition is only a sufficiency condition. The condition only provides information on the product of distributions that are relevant for pdes - those that satisfy the Lebiniz rule. For example, the Heaviside function can be multiplied with itself but the Lebniz rule does not hold for the square of Heaviside function. So, the Hormander condition rules out square of Heaviside function.