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Real Analysis HW 04-26 (I)

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 [tex]f[/tex] be a continuous function on [tex]\mathbb{R}^n\backslash\{0\}[/tex] that is homogeneous of degree [tex]-n[/tex] (i.e., [tex]f(rx)=r^{-n}f(x)[/tex]) and has mean zero on the unit sphere (i.e., [tex]$\int f\ d\sigma=0$[/tex] where [tex]\sigma[/tex] is surface measure on the sphere). Then [tex]f[/tex] is not locally integrable near the origin (unless [tex]f=0[/tex]), but the formula
[tex] \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*} [/tex]
defines a distribution [tex]\mathrm{PV}(f)[/tex] — “PV” stands for “principal value” — that agrees with [tex]f[/tex] on [tex]\mathbb{R}^n\backslash\{0\}[/tex] and is homogeneous of degree [tex]-n[/tex] in the sense of Exercise 9.
(Hint: For any [tex]a>0[/tex], the indicated limit equals
[tex] \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*} [/tex]
and these integrals converge absolutely.)
Sol:
  1. I'll show first that [tex]f[/tex] is not locally integrable near the origin. Given any [tex]r>0[/tex], consider
    [tex] \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*} [/tex]
  2. Next I'll show that [tex]\mathrm{PV}(f)\in \mathcal{D}'[/tex]. Given [tex]\phi\in C_c^\infty(\mathbb{R}^n)[/tex] and [tex]\epsilon>0[/tex], then
    [tex] \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*} [/tex]
    For part I,
    [tex] \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*} [/tex]
    On the other hand, for part II,
    [tex] \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*}[/tex]
  3. From the estimate above, by dominated convergence theorem, [tex]\mathrm{PV}(f)[/tex] is indeed a distribution.
    The remaining is obvious.
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