When and how is the map $ z\mapsto \int |x|^{-z}\phi(x)\ dx $ analytic?

Question

In Wolff's Lectures on Harmonic Analysis (see also the AMS published version here), the author claims without proofs in details the following:

for $\phi\in\mathcal{S}(\mathbb{R}^n)$, where $\mathcal{S}$ denotes the Schwartz space, the map $$ z\mapsto \int |x|^{-z}\phi(x)\ dx $$ is analytic in the "indicated regime", which may be done by using the dominated convergence theorem to justify complex differentiation under the integral sign.

I have the following questions:

• The "indicated regime" is not indicated in Wolff's notes. Where should it be?
• in order to apply the dominated convergence theorem, it seems (doesn't it?) that one needs to estimate the integral $$ \int \bigg||x|^{-z}\bigg|\cdot \bigg|\log |x|\bigg|\cdot |\phi(x)|\ dx. $$ What dominated function should one use?

Answer 1

The basic theorem is that $g(z) = \int_{\mathbb R^n} f(x,z)\; dx$ is analytic in $z \in U \subseteq \mathbb C$ if

1. $f(x,z)$ is analytic in $U$ for every $x\in \mathbb R^n$,
2. $f(x,z)$ is jointly measurable in $x$ and $z$,
3. $\int_{\mathbb R^n} |f(x,z)|\; dx$ is uniformly bounded on every compact subset of $U$.

Then we can use Fubini's theorem to say if $\Gamma$ is any closed triangle in $U$, $$ \oint_\Gamma g(z)\; dz = \int_{\mathbb R^n} \oint_\Gamma f(x,z)\; dz\; dz = 0$$ and then Morera's theorem says $g(z)$ is analytic in $U$.

In your case, the only problem is the singularity of $|x|^{-z}$ at the origin: we have $$\left| |x|^{-z} \right| = |x|^{-\text{Re}\; z}$$ and we need $n-1 - \text{Re}\; z > -1$, i.e. $\text{Re}\; z < n$.