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Real Analysis HW 04-26 (II)

Real Analysis HW 04-26 (I)

psistwu posted @ Wed, 19 May 2010 21:52:46 +0800 in Math with tags math real_analysis , 2610 readers
From Folland p. 289
Problem 1.
A distribution [tex]F[/tex] on [tex]\mathbb{R}^n[/tex] is called homogeneous of degree [tex]\lambda[/tex] if [tex]F\circ S_r=r^\lambda F[/tex] for all [tex]r>0[/tex], where [tex]S_r(x)=rx[/tex].
  1. [tex]\delta[/tex] is homogeneous of degree [tex]-n[/tex].
  2. If [tex]F[/tex] is homogeneous of degree [tex]\lambda[/tex], then [tex]\partial^\alpha F[/tex] is homogeneous of degree [tex]\lambda-\left\lvert\alpha\right\rvert[/tex]
  3. The distribution [tex](d/dx)[\chi_{(0,\infty)}(x)\log x][/tex] discussed in the text is not homogeneous, although it agree on [tex]\mathbb{R}\backslash\{0\}[/tex] with a function that is homogeneous of degree [tex]-1[/tex].
Sol:
  1. Recall the fact that if [tex]T:\mathbb{R}^n\rightarrow\mathbb{R}^n[/tex] is an invertible linear transform, then
    [tex]\begin{align*} \int f(x)\ dx &= \int f\circ T(x)\ dT(x) \\ &= \det(T)\int f\circ T(x)\ dx. \end{align*}[/tex]
    This simple example helps us understand how composition of a distribution with an invertible linear transform behaves.
  2. For a, consider that given any [tex]\phi\in C_c^\infty(\mathbb{R}^n)[/tex],
    [tex]\begin{align*} \left<\delta\circ S_r,\phi\right>&=\det(S_r)^{-1} \left<\delta,\phi\circ {S_r}^{-1}\right> \\ &=r^{-n}\left<\delta,\phi\right>. \end{align*}[/tex]
  3. For b, consider that given an [tex]F\in\mathcal{D}'[/tex] which is homogeneous of degree [tex]\lambda[/tex], then for any [tex]\phi\in C_c^\infty(\mathbb{R}^n)[/tex] we have
    [tex]\begin{align*} \left<\partial^\alpha F\circ S_r,\phi\right>&= \det(S_r)^{-1}\left<\partial^\alpha F,\phi\circ {S_r}^{-1}\right>\\ &=r^{-(n+\left\lvert\alpha\right\rvert)}\left<F, (-1)^{\left\lvert\alpha\right\rvert}\partial^\alpha\phi\circ {S_r}^{-1}\right>\\ &=r^{-\left\lvert\alpha\right\rvert}\left<F\circ S_r,(-1)^{\left\lvert\alpha\right\rvert}\partial^\alpha\phi\right>\\ &=r^{\lambda-\left\lvert\alpha\right\rvert}\left<F,(-1)^{\left\lvert\alpha\right\rvert}\partial^\alpha\phi\right>\\ &=r^{\lambda-\left\lvert\alpha\right\rvert}\left<\partial^\alpha F,\phi\right> .\end{align*}[/tex]
  4. For c, as same as text, we define
    [tex]\begin{align*} L_\epsilon(x)=\chi_{(\epsilon,\infty)}(x)\log x \end{align*}[/tex]
    for some [tex]\epsilon>0[/tex].
    Given any [tex]\phi\in C_c^\infty(\mathbb{R})[/tex], we have
    [tex]\begin{align*} \left<{L_\epsilon}'\circ S_r,\phi\right>&=r^{-2}\left<L_\epsilon,-\phi'\circ {S_r}^{-1}\right>\\ &=r^{-1}\left<L_\epsilon\circ S_r,\phi'\right>\\ &=r^{-1}\left<(\log r+\log x)\chi_{(\epsilon/r,\infty)},-\phi'\right> \end{align*}[/tex]
    If we let [tex]\epsilon\rightarrow 0[/tex], then the equation above becomes
    [tex]\begin{align*} \left<L'\circ S_r,\phi\right>&=r^{-1}\left<\log r\ \delta+L',\phi\right> \end{align*}[/tex]
    Therefore, [tex]L'[/tex] is not homogeneous.
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