I am trying to prove that for a given Hölder parameter $\alpha \in (0, 1)$ and a distribution $f \in \mathcal{D}'(\mathbb{R}^d)$ the following are equivalent:
- $f \in C^{\alpha}$
- For any $x$ there exists a polynomial $P_x$ such that $| \langle f - P_x, \phi_x^{\lambda} \rangle | \le C \lambda^{\alpha.}$
Where the latter estimate holds uniformly over all$x$ and $\phi \in \mathcal{D}$ with compact support in the unit ball, and: $$\phi_x^{\lambda}(\cdot) = \lambda^{-d} \phi\left( \frac{\cdot \ - \ x}{\lambda} \right)$$
Proving that the first implies the second is fairly easy. The other way around gives me some problems.
- The first step I took is to realize that we care only about the order zero term of $P_x,$ since all other terms vanish at a order higher than $\alpha.$ Here we assume that the polynomial is centered in $x.$
- Then I would like to prove that $P_x(x) = g(x)$ defines a $\alpha$ - Hölder function.
- Thus we would get that $g \in \mathcal{D}'.$
- Eventually I would like to prove that $g = f$ in $\mathcal{D}'.$
My problem is that I can't prove any implications between 2,3,4 nor any of 2,3,4 starting from 1. Any hints, help, suggestions?