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.