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Problem set 12, Harmonic Analysis

Due May 13

Problem 1

Let $latex X$ be a closed subspace of the Hilbert space $latex \mathscr H$.
  1. $latex X^\perp = \{x\in\mathscr H: x\perp X \}$ is a closed subspace of $latex \mathscr H$.
  2. $latex \mathscr H \cong X\oplus X^\perp$

Problem 2

If $latex g$ is the weak derivative of $latex f\in L^2(\mathbb R^d)$ with respect to $latex x_j$, then
$latex \hat g(\xi) = 2\pi i \xi_j \hat f(\xi).$

Problem 3

Let $latex \mathscr H^1(\Omega)$ be the set of equivalence classes in $latex H^1(\Omega)$ under the relation $latex f\sim g$ if and only if $latex f-g$ is a constant.
  1. $latex \mathscr H^1(\Omega)$ is a vector space.
  2. The bilinear form $latex \mathcal E$ is an inner product on $latex \mathscr H^1(\Omega)$.
  3. $latex \mathscr H^1(\Omega)$ is a Hilbert space with respect to $latex \mathcal E$.
  4. $latex H_0^1(\Omega)$ is a closed subspace of $latex \mathscr H^1(\Omega)$.

Problem 4

Let $latex \Omega$ be a bounded $latex C^1$-domain in $latex \mathbb R^d$. Then there exists a constant $latex A>0$ such that, for every $latex f\in H^1(\Omega)$,
$latex \displaystyle\int_{\partial\Omega} |f|^2 d\sigma \le A ||f||^2_{H^1},$
where $latex ||\cdot||^2_{H^1}$ is the norm in $latex H^1(\Omega)$ induced by the inner product
$latex \displaystyle \langle f, g \rangle_1 = \int_\Omega f \cdot \bar g + \nabla f \cdot \overline{\nabla g}$.
Hence, the restriction $latex f\mapsto f|_{\partial\Omega}$ is a bounded operator from $latex H^1(\Omega)$ into $latex L^2(\partial\Omega,d\sigma)$. (Hint: Extend the normal field $latex \nu$ from $latex \partial\Omega$ into $latex \Omega$ --$latex \Omega$ is a $latex C^1$-domain-- and apply the divergence theorem to the field $latex F = |f|^2 \nu$.)

Problem 5

Consider, in the disk $latex \mathbb D$, the functions $latex u_k(r,\theta) = r \cos k\theta$, $latex k\in\mathbb N$, in polar coordinates.
  1. The $latex u_k$ are orthogonal in $latex H^1(\mathbb D)$, with respect to $latex \langle \cdot, \cdot \rangle_1$ of the previous problem. (Hint: Use integration in polar coordinates.)
  2. $latex ||u_k||_{H^1} \sim k$, and thus $latex f = \sum_k a_k u_k \in H^1(\mathbb D)$ if and only if $latex \sum_k |k a_k|^2 < \infty$.
  3. Construct a function $latex f\in L^2(\mathbb S^1)$ that is not the restriction of an $latex H^1(\mathbb D)$ function to the circle $latex \mathbb S^1.$
Thus, the restriction operator $latex f\mapsto f|_{\partial\Omega}$ is not onto $latex L^2(\partial\Omega,d\sigma)$.