Consider $$f(x)=\frac{1}{x^{\alpha}(\log x)^{\beta}},$$ $\alpha,\beta\in\mathbb{R}$. For which $\alpha,\beta \in \mathbb{R}$ is $u \in L^p((1,\infty))$, $1\le p\le \infty$?
For $p=1$ I know how to do that (Hölder). $p=\infty$ is no problem too.
My problem is $1<p\le \infty$ .
For 1. I have to find out for which $\alpha, \beta \in \mathbb{R}$ the integral $$\int_1^\infty \left \vert \frac{1}{x^{\alpha}(\log x)^{\beta}}\right\vert^p\,dx$$is finite. I don't know how to estimate, could you give me a hint?
Edit: With the hint I obtain for $1<p<\infty$: $$\int_0^\infty ( \frac{e^{t(1-\alpha)}}{t^{\beta}})^p\,dt=\int_0^1 \frac{e^{t(1-\alpha)p}}{t^{\beta p}}\,dt+\int_1^\infty \frac{e^{t(1-\alpha)p}}{t^{\beta p}}\,dt.$$ How to continue?
Hint: Substitute (also the limits) in your integral $x\mapsto e^x$. Then you can split the integral in a part around the origin and the rest.