# Real Analysis HW 04-26 (II)

psistwu posted @ Fri, 21 May 2010 06:31:41 +0800 in Math with tags math real_analysis , 2940 readers
From Folland p.289
Problem 2.
Let $$f$$ be a continuous function on $$\mathbb{R}^n\backslash\{0\}$$ that is homogeneous of degree $$-n$$ (i.e., $$f(rx)=r^{-n}f(x)$$) and has mean zero on the unit sphere (i.e., $$\int f\ d\sigma=0$$ where $$\sigma$$ is surface measure on the sphere). Then $$f$$ is not locally integrable near the origin (unless $$f=0$$), but the formula
\begin{align*} \left<\mathrm{PV}(f),\phi\right>=\lim_{\epsilon\rightarrow 0}\int_{\left\lvert x\right\rvert>\epsilon}{f(x)\phi(x)\ dx}\qquad (\phi\in C_c^\infty) \end{align*}
defines a distribution $$\mathrm{PV}(f)$$ — “PV” stands for “principal value” — that agrees with $$f$$ on $$\mathbb{R}^n\backslash\{0\}$$ and is homogeneous of degree $$-n$$ in the sense of Exercise 9.
(Hint: For any $$a>0$$, the indicated limit equals
\begin{align*} \int_{\left\lvert x\right\rvert\leq a}{f(x)[\phi(x)-\phi(0)]\ dx} +\int_{\left\lvert x\right\rvert> a}{f(x)\phi(x)\ dx}, \end{align*}
and these integrals converge absolutely.)
Sol:
1. I'll show first that $$f$$ is not locally integrable near the origin. Given any $$r>0$$, consider
\begin{align*} \int_{\left\lvert x\right\rvert\leq r}{\left\lvert f(x)\right\rvert\ dx}&= \int_{s=0}^r\int_{\left\lvert x\right\rvert=s}{\left\lvert f(x)\right\rvert\ d\sigma ds}\\&=\int_{s=0}^r s^{-1}\ ds\ \int_{\left\lvert x\right\rvert=1}{f(x)\ d\sigma}. \end{align*}
2. Next I'll show that $$\mathrm{PV}(f)\in \mathcal{D}'$$. Given $$\phi\in C_c^\infty(\mathbb{R}^n)$$ and $$\epsilon>0$$, then
\begin{align*} &\int_{\left\lvert x\right\rvert>\epsilon}{\left\lvert f(x)\phi(x)\right\rvert\ dx}\\\leq\quad&\int_{\epsilon<\left\lvert x\right\rvert\leq 1}{\left\lvert f(x)\right\rvert\left\lvert\phi(x)-\phi(0)\right\rvert\ dx}+\int_{\left\lvert x\right\rvert>1}{\left\lvert f(x)\right\rvert\left\lvert\phi(x)\right\rvert\ dx}\\=\quad&\mathrm{I}+\mathrm{II} \end{align*}
For part I,
\begin{align*} \mathrm{I}&=\int_{\epsilon<\left\lvert x\right\rvert\leq 1}{\left\lvert f(x)\right\rvert\left\lvert x\cdot \nabla\phi(\xi(x))\right\rvert\ dx}\\&\leq\sup\left\lvert\nabla\phi\right\rvert\ \int_{\epsilon<\left\lvert x\right\rvert\leq 1}{\left\lvert x\right\rvert\left\lvert f(x)\right\rvert\ dx}\\&=\sup\left\lvert\nabla\phi\right\rvert\ \int_{s=\epsilon}^1 ds\ \int_{\left\lvert x\right\rvert=1}{\left\lvert f(x)\right\rvert\ d\sigma}\\&\leq\sup\left\lvert\nabla\phi\right\rvert\ \int_{\left\lvert x\right\rvert=1}{\left\lvert f(x)\right\rvert\ d\sigma}. \end{align*}
On the other hand, for part II,
\begin{align*} \mathrm{II} \leq\left\lvert\mathrm{supp}\ \phi\right\rvert\sup_{x\in\mathbb{R}^n}{\left\lvert\phi(x)\right\rvert}\ \sup_{\left\lvert x\right\rvert\geq 1}{\left\lvert f(x)\right\rvert}. \end{align*}
3. From the estimate above, by dominated convergence theorem, $$\mathrm{PV}(f)$$ is indeed a distribution.
The remaining is obvious.
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