4
$\begingroup$

In order to finish a paper on 'metric space magnitude' I need to prove that a certain distribution on $\mathbb{R}^{2p+1}$ is in Mark Meckes' weighting space (see Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces). My question requires no knowledge of that background, however: for what I want to do, it suffices to show that certain simple distributions are in a specific Bessel potential space.

Let $w_i$ be the distribution on the odd-dimensional space $\mathbb{R}^{2p+1}$ which is defined by integrating the $i$th normal derivative of a function over the radius $R$ sphere $S^{2p}_R$: $$ \langle w_i, f\rangle := \int_{x\in S_R^{2p}} \frac{\partial^i }{\partial \nu^i}f(x) \,\mathrm{dvol}(x), $$ 'normal' meaning normal to the sphere, of course.

Define the Bessel potential space $H^{-(p+1)}(\mathbb{R}^{2p+1})$ by $$ H^{-(p+1)}(\mathbb{R}^{2p+1}) := \left\{\phi\in \mathcal{S}'(\mathbb{R}^{2p+1})\mid (1+{\left\| \cdot \right\|}^2)^{-(p+1)/2}\widehat\phi\in L^2(\mathbb{R}^{2p+1})\right\}, $$ where $\mathcal{S}'(\mathbb{R}^{2p+1})$ is the space of tempered distributions and $\widehat\phi$ is the Fourier transform of the distribution $\phi$.

Question: Is it true that for $0\le i\le p$ the distribution $w_i$ is in the Bessel potential space $H^{-(p+1)}(\mathbb{R}^{2p+1})$?

I should add that my background is very far from analysis, so I'm a bit sketchy in this area, to say the least.

$\endgroup$
2
$\begingroup$

Yes, this is true. Your distribution is defined as an integral over the sphere of a derivative of order i. By using the divergence theorem, you can convert this to an integral over the ball of a derivative of order i+1. Hence the distribution is an element of $H^{-(i+1)}$.

$\endgroup$
  • $\begingroup$ Sorry, I has taken me a while to think about this! As this isn't my area it I want to make sure I've understood what you're saying. Are you saying the following? $H^{-(p+1)}$ is dual to $H^{p+1}$ and so it suffices to prove that for functions with derivatives up to $i+1$th in $L^2(\mathbf{R}^{2p+1})$, we have $|\langle w_i, f\rangle\rangle |< \infty$. As such derivatives are in $L^2(\mathbf{R}^{2p+1})$ they're also in $L^1(B^{2p+1})$... (ctd) $\endgroup$ – Simon Willerton Apr 4 '17 at 8:49
  • $\begingroup$ ... Now we can extend the normal vector field $\nu$ on the sphere to a compactly supported vector field $\bar \nu$ on the whole of $\mathbf{R}^{2p+1}$ (using the Tubular Neighbourhood Theorem if we had a more complicated boundary than the sphere). Then we can use the Divergence Theorem to convert this to an integral over the ball of a derivative of f of order $i+1$ involving the Laplacian and the vector field $\bar \nu$. As such a thing is in $L^1(B^{2p+1})$ we can conclude that the integral is finite, as required. $\endgroup$ – Simon Willerton Apr 4 '17 at 8:50
  • $\begingroup$ In the first comment that should be "with derivatives up to the $(p+1)$th" not "up to $i+1t$h", but I can't edit it now. $\endgroup$ – Simon Willerton Apr 4 '17 at 10:13
  • $\begingroup$ This appears as Lemma 9 in my paper The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials. $\endgroup$ – Simon Willerton Jul 30 '18 at 13:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.