Lebesgues spaces

Question

we consider the function $f(x)= \ln(|x|)$ defined in $\mathbb{R}^2 \setminus \{0\}$. I have difficlties for these two questions:

1. How we prouve that $f \in L^p_\text{loc}(\mathbb{R}^2)$ for $p < +\infty$?

2. How we prouve that $\partial_i f \in L^p_\text{loc}(\mathbb{R}^2), i=1,2$ for $p < 2$?

I know that $f$ is continuous then $f \in L^1_\text{loc}(\mathbb{R})$, but i have difficulties with two questions.

Thank you in advance to the help.

Answer 1

Local integrability is equivalent to integrabilty over some ball around each point. Considering the fact that f and its partials are continuous on $\mathbb R^{2} \setminus \{0\}$ we only have to consider integrals over $\{x:|x|<\delta\}$for some $\delta >0$. This is very easy using polar coordinates: $\int_{\{x:|x|<\delta\}} ln|x| dx = \int_0^{2\pi}\int_0^{\delta} r \ln (r) dr d\theta$ which is finite because $r \ln (r) \to 0$ as $r \to 0$. For the partial write $f(x)=\frac 1 2 \ln (x^{2}+y^{2})$. If you differentiate and change to polar coordinates again you will get integrals of $\frac {\cos (\theta)} r$ and $\frac {\sin (\theta)} r$ with respect to $rdrd\theta$ which are finite.

Answer 2

To prove that $g \in L^p_{\text{loc}}(\mathbb R^2)$, you need to prove that $$\int_K \lvert g(x) \rvert^p dx < \infty $$ for any compact $K \subseteq \mathbb R^2$. Here $f$ is smooth on any compact set not containing the origin (thus $\lvert f \rvert^p$ and $\lvert \partial_i f \rvert^p$ are continuous there) so if $K \subseteq \mathbb R^2$ is compact and doesn't contain $0$, it is clear that $$\int_{K} \lvert f(x) \rvert^p dx, \,\, \int_K \lvert \partial_i f \rvert^p dx < \infty.$$ Thus to prove $f, \partial_i f \in L^p_{\text{loc}}(\mathbb R^2)$, it suffices to show that $$\int_{B_1(0)} \lvert f(x) \rvert^p dx, \,\, \int_{B_1(0)} \lvert \partial_i f \rvert^p dx < \infty.$$ [Do you see why we need only consider the unit ball centered at $0$?] To this end, consider, by integration in polar coordinates $$\int_{B_1(0)} \lvert f(x) \rvert^p dx =\int_{B_1(0)} \lvert\log(\lvert x \rvert)\rvert^p dx = 2\pi \int_0^1 \lvert\log(r)\rvert^p r \,dr = 2\pi \int_1^\infty \frac{\lvert \log(t) \rvert^p}{t^3} dt < \infty$$ where I've used $r = 1/t$ and the latter converges since, for example, $\frac{\lvert \log(t) \rvert^p}{t^3} \le \frac{1}{t^2}$ for $t$ large enough. This shows that $f$ is in $L^p_{\text{loc}}(\mathbb R^2)$ for any $p < \infty$.

Next, consider $$\partial_if(x) = \frac{x_i}{\lvert x \rvert^2} \,\,\,\, \text{ for } x \in \mathbb R^2 \setminus \{ 0 \}.$$ Thus $$\int_{B_1(0)} \lvert \partial_i f(x) \rvert^p dx = \int_{B_1(0)}\frac{\lvert x_i \rvert^p}{\lvert x \rvert^{2p}} dx \le \int_{B_1(0)}\frac{dx}{\lvert x \rvert^p} = 2\pi \int^1_0 \frac{r \, dr}{r^p} = 2\pi \int^1_0 r^{1-p} dr$$ which converges when $1-p > -1$ or $p < 2$.