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Let $p>2, \frac{1}{p} < \alpha < 1- \frac{1}{p}$ and define $g_\alpha(t) := t^{\alpha - 1} \chi_{[1, \infty]}$. Then $g_\alpha \in L^p(\mathbb R)$. Define $$f(z) := \int_1^\infty g_\alpha(t) \exp(-izt) \, dt.$$ for $z\in \mathbb C, \operatorname{Im }z <0.$ Then:

$a)$ $f$ is holomorphic in the lower half plane,
$b)$ The limit $$f_0(x):= \lim_{\substack{y\to 0 \\ y < 0}} f(x+iy)$$ exists for all $x\neq 0$,
$c)$ $f(z)$ does not satisfy an estimate of the form $$\forall \epsilon >0: \quad \lvert f(z) \rvert \leq C_\epsilon \exp((1+\epsilon)\lvert z \rvert),$$ $d)$ $f_0$ is not $p$-integrable in any neighbourhood of $0$.

I managed to show $a)$ using Lebesgue's theorem on interchanging derivative and integral sign. However, I am unsure with the other assertions, especially $c)$ and $d)$: How does one estimate the integral from below? And how I can I show something about $f_0 $ if I don't explicitly know it? Any help appreciated!

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First we prove that $$\tag{1}f(z) := \int_1^\infty t^{\alpha-1} e^{-izt} \, \mathrm{d} t$$ is convergent (as an improper Riemann integral) for all $z = x-iy$ with $x \ne 0$ and $y >0$. More presioulsy, the convergence is uniform if $|x| \ge x_0$ for fixed $x_0>0$. Thus (1) defines a continuous function and we have for $x \ne 0$ that $$\tag{2}f_0(x) = \int_1^\infty t^{\alpha-1} e^{-ixt} \, \mathrm{d} t$$ Prove: For any $z= x-iy$ with $|x| \ge x_0$ and $y \ne 0$ we have with $1 \le a < b$ that \begin{align} \tag{3}\int_a^b t^{\alpha-1} e^{-izt} dt = \frac{1}{iz} ( a^{\alpha-1} e^{-iza} - b^{\alpha-1} e^{-izb}) - \frac{\alpha-1}{iz} \int_a^b t^{\alpha-2} e^{-izt} dt. \end{align} The last line can be bounded by $$\frac{1}{x_0} (a^{\alpha-1} + b^{\alpha-1}) +\frac{1-\alpha}{x_0} \int_a^b t^{\alpha-2} dt \le \frac{2}{x_0} a^{\alpha-1} $$ and thus (3) is a Cauchy sequence and thus convergent. In fact, it is (uniformly) convergent and thus continuous. (The whole argument is also known as Dirchlet's test, see for example here.)

Prove of (c) and (d): So we have shown (b). With the help of the explicit identity (2) we can also verify (c) and (d). In fact, we have (after coordinate of change) the identity $$f_0(x) = x^{-\alpha} \int_x^\infty s^{\alpha-1} e^{-is} \, \mathrm{d} s.$$ Since the integrand is locally integrable (also in $s=0$) the function $$g(x) := \int_x^\infty s^{\alpha-1} e^{-is} \, \mathrm{d} s$$ is continuous in $x=0$ with $$\tag{4}g(0) = \int_0^\infty s^{\alpha-1} e^{-is} \, \mathrm{d} s.$$ Thus, it remains to show that $g(0) \ne 0$. Here we use the integral representation of (4). In fact, (4) is related to the Gamma function. One representation of the Gamma function (which was proven by Euler by using a contour shift argument) is $$\Gamma(w) = e^{i \pi w /2} \int_0^\infty s^{w-1} e^{-is} \, \mathrm{d} s$$ if $0 < \mathrm{Re}(w) < 1$. Hence $$g(0) = e^{-i \alpha \pi/2} \Gamma(\alpha) \neq 0.$$ All in all, we see that $$\tag{5} f_0(x) \sim_\alpha x^{-\alpha} \quad (\text{for} \ x \rightarrow 0) $$ up to an non-zero constant (depending on $\alpha$).

c) can not hold, because c) would imply that $f_0$ is bounded. (5) implies that $f_0$ is not $p$-integrable (note that $\alpha p >1$) in any neighbourhood of $0$.

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  • $\begingroup$ Thanks for taking your time! What do you mean by "$(3)$ is a Cauchy sequence"? In which way do you interpret this as a sequence? $\endgroup$ – Staki42 Jan 13 at 12:29
  • $\begingroup$ Never mind, I got it. Thanks! $\endgroup$ – Staki42 Jan 13 at 12:40
  • $\begingroup$ More precisely, I should say 'Cauchy-net'. There is a generalization of sequences called nets, see Wikipedia. The bound shows that (3) can be made arbitary small if $a$ is large enough and that is exactly the property to be a Cauchy-net. $\endgroup$ – p4sch Jan 13 at 12:47
  • $\begingroup$ I see. But I also could just take a sequence, right? $\endgroup$ – Staki42 Jan 13 at 13:28
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    $\begingroup$ No, for $y=0$ the integral is not absolute convergent, because $t^{\alpha-1}$ is not in $L^1([1,\infty)$. For $y >0$ it is not a problem, but we would like to determine the integral for $y=0$ and also see that it is continuous in $y$ in order to determine the limes. $\endgroup$ – p4sch Jan 13 at 14:54

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