Suppose $\Phi:\mathbb R\to\mathbb R$ is a positive function such that $\lim_{p\to\infty}\Phi(p)=\infty$. I want to construct a not essentially bounded function $f:(0,1)\to\mathbb R$ such that if $$g(p)=\int_0^1 |f(x)|^p\,dx,$$ then $g(p)\to\infty$ as $p\to\infty$, but slower than $\Phi$.
My initial thought was to consider known functions which have finite integral on $(0,1)$, like $x^{\alpha}$ for $0<\alpha<1$. The problem here is that $p$ will always get large enough to make $g(p)$ infinite. My next thought was $f(x)=-\log(x)$, since this approaches $\infty$ as $x\to 0$ slower than $x^{1/n}$ for all $n\in\mathbb N$. This leads to $g(p)=\Gamma(p+1)$. This is very fast growing in $p$, but at least it is always finite. So now I am thinking about $f(x)=\log(-\log(x/e))$. This function will approach $\infty$ as $x\to 0$ even slower than $\log(x)$. Unfortunately, I cannot find a general form for the integral $$g(p)=\int_0^1\log(-\log(x/e))^p\,dx,$$ so I do not know how fast it grows. Though a numerical integrator seems to suggest it grows like $(p/e)^p$, still very fast. My questions are these:
- How can I characterize the growth of $g(p)=\int_0^1\log(-\log(x/e))^p\,dx$?
- By continuing to iterate logarithms inside the integral (in a clever way), can I generate a function $g(p)$ that grows arbitrarily slowly?
- If I'm thinking way too hard about this and there is a simpler way to construct a function $f$ so that the corresponding $g(p)$ grows at an arbitrarily slow rate, then I would love a hint.
This is motivated by an exercise in Rudin, 3.9, if that helps.
[some baloney was here previously]
As to the main problem, I remember wrestling with this beast. I don't remember exactly how I solved but, but I do know that I looked at a function of the form $$f(x) = \sum_{k=1}^\infty a_k \chi_{B_k}$$ with $B_i \cap B_j = \emptyset$. I think I also have to split up the sum as something like $$\sum_{k=1}^p a_k \chi_{B_k} + \sum_{k=p+1}^\infty a_k \chi_{B_k},$$ which is a good analysis trick to learn at any rate, in order to get the inequalities I wanted. The type of trick shows up in integrals: $$\int_1^p f(x,p) dx + \int_p^\infty f(x,p) dx,$$ for instance, which sometimes makes getting upper and lower bounds a lot easier.
Good luck.