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Let $\Omega\subset\mathbb{R}^n$ be a bounded open set and $u\in C^\alpha_{\mathrm{loc}}(\Omega)$. For $h>0$, $1\leq k\leq n$, let $$D_k^hu(x)=\frac{u(x+he_k)-u(x)}{h}$$ where $e_k$ is the $k$-th coordinate vector. Suppose for each $\Omega_0\Subset\Omega$ and $k$, there is a constant $C$ s.t. $\|D^h_ku\|_{C^\alpha(\Omega_0)}\leq C$ for all small $h$. Then $u\in C^{1,\alpha}_{\mathrm{loc}}(\Omega)$ and in fact $\|D_ku\|_{C^\alpha(\Omega_0)}\leq C$.

Is this statement true? If so, how can I prove it?

By Arzelà–Ascoli we know that there is a sequence $h_j\to0$ such that $D^{h_j}_ku$ converges uniformly. But why does this even imply $u\in C^1$?

Yuxiao Xie
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  • I suspect that $D^{h_{j}}{k}u \to \partial{k}u$. But somehow the details aren't working out. Does anyone know how to see it? – Prakhar Gupta Nov 17 '20 at 17:29

1 Answers1

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This follows immediately from the following Proposition, which can be found in Proposition 8 and 9, Chapter 5 of the book "Stein, E. M. 1970. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J."

Proposition: Let $u\in C(\Omega)\cap L^\infty(\Omega)$. Suppose there exists $C>0$ and $\alpha\in(0,1)$ such that \begin{align*} \sup_{B_r(x_0)}|u(x_0+x)-2u(x_0)+u(x_0-x)|\leq C r^{1+\alpha}, \end{align*} for every $0<r<1$ and every $x_0\in \Omega$ with $B_r(x_0)\subset \Omega $. In addition, for every $\Omega' \Subset \Omega$ there exists $C_0>0$ for which \begin{align*} \|u\|_{C^{1,\alpha}(\Omega')}\leq C_0\left(\|u\|_{L^\infty(\Omega)}+C\right). \end{align*}

TronyJavolta
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