5-17

Question17 : If f \ge 0 and \omega(\alpha)\leq c(1+\alpha)^{-p} for all \alpha > 0, show that f \in L^r, 0 < r 
Proof : From question 16 (please refer to previous page), we obtained
\int_{0}^{\infty} \alpha^{r-1} \omega(\alpha) d \alpha \leq c \int_{0}^{\infty} \frac{\alpha^{r-1}}{(1+\alpha)^{p}} d \alpha<+\infty, where r \in (0,p).
The integral is improper only when it goes to infinity, then convergence follows as below
\frac{\alpha^{r-1}}{(1+\alpha)^{ p }}<\frac{\alpha^{r-1}}{\alpha^{p}}=\frac{1}{\alpha^{p-r+1}}
In the fact that r < p, we have p - r +1 \ge 1, so the integral converges