Regularity of the law of a random variable

Question

I'm trying to understand the following lemma, it's proof, and the subsequent remark (found here):

My questions:

1. Why does the assumption (4.1) imply every distributional derivative of $\mu$ belongs to $H^{-s}$? Is this simply a definition or is there more to it than that?
2. Along the same lines, why does the fact that every distributional derivative of $\mu$ belongs to $H^{-s}$ imply $\mu$ belongs to $H^\ell$ for every $\ell$?
3. (Question of greatest interest) Extending on Remark 4.2, is it true that if (4.1) only holds for all $1\leq k\leq m$ for some $m$ finite, then the law of $X$ has a $C^{m-1}$ (or $C^m$?) density with respect to Lebesgue?

I know it says "By duality and the density of $\mathcal{C}^\infty_0$ in $H^s$" but the above assertions and my Question 3 still aren't clear to me. I feel like I'm missing something obvious.

Answer 1

In case it's of interest to someone else, here is a slightly more explicit statement and argument. It combines Sobolev embedding with similar proofs found in Chapter 1 of Degenerate Stochastic Differential Equations and Hypoellipticity by Denis Bell and Chapter 8 of Diffusions and Elliptic Operators by Richard Bass. Chapter 9 of Gerald Folland's Real Analysis is also a nice reference.

Claim. Let $\mu$ be a probability measure on $\mathbb{R}^d$ and for $\xi\in\mathbb{R}^d$ define $\phi_\xi(x)=e^{i\langle \xi,x\rangle}$. If \begin{align}\label{eq:Hypothesis} \left\vert\int\partial^\alpha\phi_\xi\ d\mu\right\vert &\leq C \end{align} for all $\lvert\alpha\rvert\leq k+d+1$ and some constant $C$ (depending on $k$ and $d$), then $\mu$ has a $C^k_0$ density.

A few remarks before the proof.

Remark 1. I'll use the following characterization of the Sobolev space $H^s$: \begin{align} H^s = \left\{u\in \mathscr{S}' : \big[(1+\lvert\xi\rvert^2)^{s/2}\hat{u}(\xi)\big]\check\ \in L^2(\mathbb{R}^d)\right\} \end{align} where $\mathscr{S}'$ is the space of Schwartz distributions and $\hat{u}$ and $\check{u}$ are the Fourier and inverse Fourier transforms of $u$, respectively. Note all Borel probability measures belong to $\mathscr{S}'$ and that the inverse Fourier transform is applied to the entire expression in brackets above.

Remark 2. $u\in H^s$ if and only if $(1+\lVert\xi\rVert^2)^{s/2}\hat{u}(\xi)\in L^2(\mathbb{R}^d)$.

Remark 3. (Sobolev embedding) $H^s\subset C^k$ whenever $s>k+\frac{1}{2}d$. In particular, if a probability measure $\mu$ belongs to $H^s$ for $s>k+\frac{1}{2}d$ then $\mu(dx)=f(x)dx$ for some function $f\in C^k(\mathbb{R}^d)$.

Proof of claim. Set $m=k+d+1$ and $\partial_j=\partial/\partial x_j$. By assumption \begin{align} C &\geq \left\lvert\int \partial_j^m e^{i\langle\xi,x\rangle}\mu(dx)\right\rvert =\lvert \xi_j\rvert^m\lvert\hat{\mu}(\xi)\rvert. \end{align} And by Jensen's inequality, \begin{align} \lVert \xi\rVert^m &= \bigg(\sum_{j=1}^d \xi_j^2\bigg)^{m/2} = \bigg(\frac{d}{d}\sum_{j=1}^d \xi_j^2\bigg)^{m/2} \leq d^{\frac{m-2}{2}}\sum_{j=1}^d\lvert\xi_j\rvert^m. \end{align} Combining the above inequalities gives \begin{align}\label{eq:Inequality} \lVert\xi\rVert^m\lvert\hat{\mu}(\xi)\rvert &\leq d^{\frac{m-2}{2}}\sum_{j=1}^d\lvert\xi_j\rvert^m\lvert\hat{\mu}(\xi)\rvert \leq d^{m/2}C = C_1. \end{align} Set $s=k+\tfrac{1}{2}(d+1)$. Note $s>k+\tfrac{1}{2}d$ so $H^s\subset C^k_0$ by Sobolev embedding. And by the above inequality, \begin{align*} \big(1+\lVert\xi\rVert^2\big)^s\lvert\hat{\mu}(\xi)\rvert^2 &\leq \frac{C_1(1+\lVert\xi\rVert^2)^s}{\lVert\xi\rVert^{2m}}. \end{align*} The leading order term on the right is $\lVert\xi\rVert^{2(s-m)}$ which is integrable because \begin{align*} 2(s-m) &= -d-1 < -d \end{align*} (note $\lVert\xi\rVert^\alpha\in L^1(\mathbb{R}^d)$ whenever $\alpha<-d$). Therefore $(1+\lVert\xi\rVert^2)^{s/2}\hat{\mu}(\xi)\in L^2$. By Remarks 2 and 3, $\mu\in H^s\subset C^k_0$; that is, $\mu$ has a $C^k$ density.