Let $\rho:\mathbb R^n\to\mathbb R$ be a function supported on a ball of radius $r$ with $$\begin{array}{rcl} \int_{\mathbb R^n}|\rho(y)|dy & \lesssim & r^m \\ \int_{\mathbb R^n}|\nabla \rho(y)|dy & \lesssim & r^{m-2}. \end{array}$$ Then for $0<\delta\leq1$, we have the holder norm $$\|\rho\|_{L_\delta^1}=\sup_{t\in\mathbb R^n}|t|^{-\delta}\int_{\mathbb R^n}|\rho(y+t)-\rho(y)|dy\lesssim r^{m-2\delta}.$$
I think the intermediate value theorem or mean value theorem should be applied here, but I got a worse bound $r^{m-1-\delta}$. How do I get $m-2\delta$ in the exponent? Could you show the details?
Edit1: This is from page 62-63 of Ricci and Stein, Harmonic analysis on nilpotent groups and singular integrals: II. Singular kernels supported on submanifolds. Journal of Functional Analysis. 78 (1988), 56-84.
Edit2: add one more condition on the integral on $|\rho|$. I don’t know if it’s useful here, but maybe that’s why I can’t get the bound.
Suppose that $|t| \le r^2$. Then
$$ \begin{align*} |\rho(y+t) - \rho(y)| &= \left|\int_0^1 \frac{\mathrm{d}}{\mathrm{d}s} \rho(y + st) \ \mathrm{d} s\right|\\ &= \left|\int_0^1 (\nabla\rho)(y + st) \cdot t \ \mathrm{d} s\right|\\ &\le |t| \int_0^1 |(\nabla\rho)(y + st)| \mathrm{d} s. \end{align*} $$ Utilizing Fubini's theorem, we obtain $$ \begin{align*} |t|^{-\delta}\int_{\mathbb{R}^n} |\rho(y+t) - \rho(y)| \ \mathrm{d} y &\le |t|^{1-\delta} \int_0^1 \int_{\mathbb{R}^n} |(\nabla\rho)(y + st)| \ \mathrm{d} y \ \mathrm{d} s \\ &= |t|^{1-\delta} \int_0^1 \int_{\mathbb{R}^n} |\nabla\rho(y)| \ \mathrm{d} y \ \mathrm{d} s \\ &\lesssim |t|^{1-\delta} r^{m-2} \le r^{m-2\delta},\\ \end{align*} $$ where the equality follows from translation invariance of the $L^1$ norm.
Suppose now that $|t| > r^2$. Then $$ \begin{align*} |t|^{-\delta}\int_{\mathbb{R}^n} |\rho(y+t) - \rho(y)| \ \mathrm{d} y &\le r^{-2\delta} \left( \int_{\mathbb{R}^n}|\rho(y+t)|\ \mathrm{d} y + \int_{\mathbb{R}^n}|\rho(y)|\ \mathrm{d} y\right) \\ &= 2r^{-2\delta}\int_{\mathbb{R}^n}|\rho(y)|\ \mathrm{d} y \\ &\lesssim r^{m-2\delta}. \end{align*} $$ It follows that $$ \sup_{t \in \mathbb{R}^n}|t|^{-\delta}\int_{\mathbb{R}^n} |\rho(y+t) - \rho(y)| \ \mathrm{d} y \lesssim r^{m-2\delta} . $$