Is the distribution $f\mapsto \int_{S} \frac{\partial^i }{\partial \nu^i}f\,\mathrm{dvol}$ in a Bessel potential space?

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.

Answer

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

Attribution
Source : Link , Question Author : Simon Willerton , Answer Author : Michael Renardy

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