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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.

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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 $H^{-(i+1)}$.

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  • $\begingroup$ 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, 2017 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, 2017 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, 2017 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, 2018 at 13:13

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