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Hello everyone !

In Fourier Analysis, Self-Adjointness by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2), Theorem IX.14 tells us that (I'll take dimension 1 for simplicity) :

if $T$ is a tempered distribution on $\mathbb R$ such that :

  • $\hat T$ has an analytic continuation to $|\Im z| < a$ for some $a > 0$
  • on each slice $\mathbb R + i\eta$ with $|\eta| < a$, $\hat T$ is integrable
  • for each $0 < b < a$, the supremum of the integrals on the slices $\mathbb R + i\eta$ with $|\eta| < b$ is finite

then $T$ is a bounded continuous function and for any $0 < b < a$, there exists $C_b \geq 0$ s.t. $$|T(x)| \leq > C_b e^{-b|x|}$$

I was wondering : what can effectively make $C_b$ explode as $b \to a$ ? For instance, if we look at $\hat T(\xi) = \frac{1}{1+\xi^2}$, we have exactly the three hypotheses with $a=1$, and nothing more, but $T(x) = e^{-|x|}$ has exponential decay with exponent exactly $1$, i.e. we can take $b=a=1$.


I've tried several other computations and can never get a $C_b$ exploding as $b \to a$.

I'm not really an expert in the proofs of such result because I'm not familiar enough with Paley-Wiener theory, so excuse me if the answer is obvious.

The motivation, in a very unclear nutshell : I'm solving some linear elliptic PDE in a strip (by partial Fourier transform) with some strange boundary condition on the upper boundary part, and I want to show that exponential decay of the datum on this boundary yields the same exponential decay for the solution. I have other ways to prove it so this is not really important, but I'm interested in digging this way of seeing it.

But interestingly, in my other proof I need some assumption on the $0$-order coefficient of my operator : if I take $-\Delta + A$, for instance, I need $A > r^2$ where $r$ is the exponent in the exponential decay of the datum. Since none of this appear in the theorem above, I guess that either my proof is non optimal, or there are indeed ways s.t. $C_b$ explodes as $b\to a$. And I tend to trust more Reed & Simon than myself, so...

Thank you very much,

Laurent


EDIT : Oh ! I was wrong when I wrote that we had nothing more. Indeed, in my example, the supremum of the integrals on the slices is finite for all $|\eta| < 1$, since it is $\pi$ independantly from $\eta$. So no need to put a $b$, and I guess this is the reason... I also found a $L^2$ version of the theorem (which is the same as theorem IX.13 in Reed & Simon) in Krantz, Parks, A primer in real analytic functions that has no $b$ in their assumption and in their result, so the proof might be adapted...

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