Comments for lecture 7

So in this lecture, I wanted to gather together some preliminary ideas that we will need when we move on to discuss pseudodifferential operators on Thursday.

The Schwarz Kernels Theorem

We defined a smoothing operator (on a compact manifold) as one that maps any Sobolev space to any other Sobolev space, or equivalently as one that maps distributions to smooth functions. A simple calculation shows that such an operator  T can always be written

\[ Tu(x) = \int_M k(x,y) u(y) d\lambda(y), \quad k \in C^\infty(M\times M), \]

where \(\lambda\) is a smooth measure on \(M\).  The smooth function \(k\) is called the kernel of the operator \(T\).

Not all operators can be expressed this way.  For example the identity operator cannot.  However distribution theory allows one to express the identity operator by a formula like that above where \(k\) is a “distributional” kernel \(k(x,y)=\delta(x-y)\).  The kernels theorem says that if we allow “distributional” kernels of this sort then the formula above can represent any continuous linear operator from smooth functions to distributions.  This allows one to reformulate questions about the behavior of differential or pseudodifferential operators on \(M\) as questions about their distributional kernels on \(M\times M\).

The kernels theorem is really a result about duality and topological tensor products,  For a proof using Sobolev spaces, see my functional analysis notes from 2009. (lectures 11 through 13).

Oscillatory Integrals

Consider the Fourier inversion theorem (in one dimension for simplicity),

\[ \hat{u}(\xi) = \frac{1}{2\pi}\int  e^{-iy\xi} u(y) dy, \quad u(x) = \int e^{ix\xi} \hat{u}(\xi) d\xi. \]

Formally combining these one obtains

\[ u(x) = \frac{1}{2\pi} \iint e^{i(x-y)\xi} u(y) d\xi dy, \]

or, equivalently, \(   \int e^{i(x-y)\xi} d\xi = 2\pi \delta (x-y) \).  What might this formal expression mean?

Returning to the Fourier inversion formula, we understand that the inversion integral makes sense because \(\hat{u}\) is of rapid decay (if, say, \(u\) is compactly supported).  To prove that, we make use of integration by parts to show that \(\xi^k \hat{u}(\xi) \) is bounded for any \(k\) (because it is, essentially, the Fourier transform of the \(k\)’th derivative of \(u\)).

We’ll use similar integration-by-parts arguments to interpret the oscillatory integrals of the form

\[ \int e^{i(x-y)\cdot\xi} a(x,y,\xi) d\xi \]

which will arise when we discus pseudodifferential operators.

Asymptotics

Pseudodifferential operator theory makes extensive use of \(C^\infty\) techniques like bump functions.  The corresponding Taylor series may not converge, or if they do converge, may not converge to the original function.    But they are always asymptotic series in the sense that, if \(f\) has Taylor series \(\sum a_n x^n \) about the origin, then for any \(N\),

\[ \left| f(x) – \sum_{n=0}^{N-1} a_n x^n \right| = O(x^N). \]

Moreover, a theorem of Borel states that every sequence of reals is the sequence of Taylor coefficients of some smooth function.  (There is a cute proof of this using functional analysis, see Example 10.2 in my functional analysis lectures, but it is really more enlightening to give an ad hoc construction with bump functions.)  Thus if we are prepared to work “asymptotically”, the relation between series and functions is perfect – a calculus student’s dream.

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