L^p-spaces in R

Question

Let $\Omega = (0, \frac{1}{2})$ and $f: \Omega \to \mathbb R: x \to (x(\log(x))^2)^{-\frac{1}{p}}$

In the lecture, we had this statement: $f \in L^p (\Omega)$ but $f \not \in L^{p+ \epsilon} (\Omega)$ for all $\epsilon > 0$.

Why does this hold true?

Answer 1

It is just computation. \begin{align} \int_\Omega|f(x)|^p\,dx&=\int_0^{1/2}\frac{dx}{x\,(\log x)^2}\\ &=\lim_{t\to0^+}\int_t^{1/2}\frac{dx}{x\,(\log x)^2}\\ &=\frac{1}{\log 2}\\ &<\infty. \end{align} This shows $f\in L^p$. Now let $q=(p+\epsilon)/p>1$. $$ \int_\Omega|f(x)|^{p+\epsilon}\,dx=\int_0^{1/2}\frac{dx}{x^q\,(\log x)^{2q}}. $$ Comparison with $1/x$ shows that the integral diverges.