$\log|x|\in L^q(B_1(0)) ~~\forall q<\infty$ where $B_1(0)\subseteq \mathbb{R}^n$

Question

Unfortunately I don't really see why the following two properties hold. I need to understand them for a proof in Functional Analysis:

i) $\log|x|\in L^q(B_1(0))~ \forall q<\infty$

ii) $\frac{x}{|x|^2} \in L^p(B_1(0))~ \forall p<n$

Thanks a lot!

Answer 1

In the situations, you should use the coarea formula. Specifically, if you need to check whether a function $f$ such that $|f|$ is radially symmetric, i.e. $|f(x)|=u(|x|)$ for some $u:[0,\infty)\to [0,\infty)$ (which both $f(x)=\log |x|$ and $f(x)=\frac{x}{|x|^2}$ satisfy), is such that $f\in L^p(B_1(0))$, you are allowed to first integrate alongside the sphere of radius $\rho$ and then integrate with respect to $\rho$:

$$\int_{B_1(0)}|f(x)|^pdx=\int_{0}^{1}\int_{|x|=\rho}|f(x)|^pdx\,d\rho=\alpha_1\int_0^1\rho^{n-1}u(\rho)^pd\rho $$ Where for all $\rho\geq 0$ we set $\alpha_\rho:=|\left\{x\in \mathbb{R}^n:|x|=\rho\right\}|$ the measure of the sphere of radius $\rho$ in $\mathbb{R}^n$, and we have $\alpha_{\rho}=\rho^{n-1}\alpha_1$.

Your problem is now reduced to one-dimensional calculus.