For which $p$ do the following functions converge in $L^p([0,\infty),leb)$?

Question

Consider the functions $f_n(x):= n^{1/2}\exp\left\{ (-n^2x^2)/(x+1)\right\}$. We want to figure out for which $p$ the functions converge in $L^p([0,\infty),leb)$.

I note that $L^p$ is a Banach space, so in the affirmative case you can show that the sequence is Cauchy. In the negative case, I used a graphing calculator to look for possible lower bounds and found that the sequence is not Cauchy using those lower bounds. Specifically, I found it is not Cauchy for $p\geq 2.$

However, I still don't know how to rigorously show that these functions are $L^p$ for, say, $1\leq p<2.$ Any hints would be great.

Answer 1

For $0<x<1$, $\frac{-1}{x+1}< -\frac{1}{2}$, therefore $$ ∫_0^1|f_n|^p = ∫_0^1n^{p/2} e^{-p\,n^2\,x^2/(x+1)}\,\mathrm{d}x \leq ∫_0^1n^{p/2} e^{-p\,n^2\,x^2/2}\,\mathrm{d}x $$ Moreover, with the change of variable $y=n\, x$ we have $$ ∫_0^1 n^{p/2} e^{-p\,n^2\,x^2}\,\mathrm{d}x = ∫_0^{\sqrt n} n^{p/2-1} e^{-p\,y^2}\,\mathrm{d}y \sim n^{p/2-1} C_p \underset{n\to\infty}{\longrightarrow} 0 $$ On the other hand, for $x>1$, $\frac{-1}{x+1} < -\frac{1}{2x}$, $$ ∫_1^\infty |f_n|^p \leq ∫_1^∞ n^{p/2} e^{-p\,n^2\,x/2}\,\mathrm{d}x = C \,n^{p/2-2} \underset{n\to\infty}{\longrightarrow} 0 $$ Therefore $$ f_n \underset{n\to\infty}{\longrightarrow} 0 $$ in any $L^p$ when $p<2$.

Answer 2

It's clear that $f_n(x)\to 0$ pointwise everywhere in $(0,\infty).$ So if $f_n\to f$ in $L^p,$ then $f=0$ in $L^p,$ since some subsequence $f_{n_k}\to f$ pointwise a.e.

Let's look at the case $1\le p<2$ since you've already handled $p\ge 2.$ Make the substitution $x=y/n$ to see

$$ \int_0^\infty |f_n(x)|^p\,dx = n^{p/2-1}\int_0^\infty \exp (-py^2/(y/n+1))\,dy.$$

On $[0,1]$ the last integrand is $\le \exp (-py^2/2).$ While on $[1,\infty),$ that integrand is bounded above by

$$\exp (-py^2/(2y/n)) = \exp (-pyn/2)\le \exp (-py/2).$$ Both $\exp (-py^2/2)$ and $\exp (-py/2)$ are integrable on $[0,\infty).$ Since $n^{p/2-1} \to 0$ for $1\le p<2,$ we see $f_n\to 0$ in $L^p$ for this range of $p.$