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!
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Do you know the definition of an $L^q$ function? Verifying these properties should come down to checking that some integrals are finite. – Anthony Carapetis Jul 12 '18 at 12:59
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Yes, I do know the definitions but I don't see why there is no blow up of the integral in a neighbourhood of 0. – raffaelluca Jul 12 '18 at 13:01
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1Use polar coordinates. – Jochen Jul 12 '18 at 13:08
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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.
Lorenzo Q
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so in this case by $u$ you mean the integral of $f$ on the sphere? – raffaelluca Jul 12 '18 at 13:27
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No, $u$ is a non-negative one variable function which is obtained from the expression of $f$ by replacing $x$ with a non-negative real number $\rho= |x|$. For instance if $f(x)=\log |x|$ then $|f(x)|=|\log |x||\Rightarrow u(\rho)=|\log \rho|$. Such a function only exists because $f$ is radially symmetric, i.e. $f$ only depends from $|x|$. – Lorenzo Q Jul 12 '18 at 14:16