Almost analytic continuation

Question

Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost analytic continuation of f if it satisfies

(1) $\tilde{f}$ is a smooth function on $\mathbb{C}$ and $\tilde{f}=f$ for $x\in \mathbb{R}$;

(2) For any $N\geq 0$ $$|\partial_{\bar{z}}\tilde{f}(z)|\leq C_{N}\langle z \rangle^{\alpha-1-N}|\Im z|^{N},\qquad z\in \mathbb{C}$$ where $\partial_{\bar{z}}\tilde{f}(x+iy)=(\partial_{x}+i\partial_{y})\tilde{f}(x+iy)$

Obviously,this is a weaker assumption than the analytic continuation of a function(a somehow similar spirit of this is the definition of almost orthogonal operators).One of its application is in functional calculus,if A is a selfadjoint operator in a hilbert space,and $f\in S^{-\epsilon}$,$\epsilon>0$,then we have $$f(A)=\frac{1}{2\pi i}\int_{C}(\partial_{\bar{z}}\tilde{f}(z))(A-z)^{-1}dzd\bar{z}$$ This formular is sometimes useful when various types of $L^{p}-L^{q}$ estimates are concerned in wave or schrodinger equation.

My first question is that is there an explicit way to construct almost analytic continuation for functions in $ S^{-\epsilon}$,$\epsilon>0$, it seems non-trivial even when $f\in C_{0}^{\infty}(\mathbb{R})$.

Another question I'm interested is about application.In some cases,the analyticity may be too strong to obtain so that we could only hope for the weak analyticity or almost analyticity instead.so are there exactly any other use of this in mathematics(such as complex analysis or function theory)?

Answer 1

In Chapter 8 of Sj\"{o}strand's book Spectral asymptotics in the semiclassical limit, two constructions are given (at least for $f\in C_0^\infty$, it should work for $S^\alpha$ with certain modifications).

The first is H\"{o}rmander's approach, based on Borel's construction.

$$\tilde{f}(x+iy)=\sum_{k=0}^\infty\frac{f^{(k)}(y)}{k!}(iy)^k\chi(\lambda_k y)$$

where $\chi\in C_0^\infty(\mathbb{R})$ equal to 1 near 0 and $\lambda_k$ tending to $\infty$ sufficiently fast.

The second construction is introduced by Mather, Jensen and Nakamura based on Fourier inversion formula: If $\psi(x)\in C_0^\infty(\mathbb{R})$ equal to 1 in a neighborhood of the support of $f$. $\chi$ as above.

$$\tilde{f}(x+iy)=\int_\mathbb{R}e^{i(x+iy)\xi}\chi(y\xi)\hat{f}(\xi)d\xi$$

where $\hat{f}$ is the Fourier transform of $f$.

Sj\"{o}strand uses the almost analytic continuation to discuss functional calculus of the pseudo-differential operators and gives a trace formula in this book. There are also references listed there.

Zworski's new book Semiclassical Analysis also contains some discussion on the almost analytic continuation and its applications.