Riesz transforms on the Vicsek set

We study Riesz and reverse Riesz inequalities for the fractional powers of the Laplacian $\mathrm{\Delta }$ on the unbounded Vicsek set. The space carries its Hausdorff measure $m$. The gradient $\mathrm{\partial }$ is a weak gradient defined on the skeleton, and it is measured with respect to a length measure $\nu$ that is singular with respect to $m$. The relevant fractional order for the Riesz inequalities is not $1\mathrm{/}2$ but ${\gamma }_p=\frac{d_f+p-1}{p{d}_w}$, where $d_f={\log }_{3}5$ and $d_w=d_f+1$. For $p\in [1,2]$, we prove the weak-type reverse Riesz inequality $\mathrm{\parallel }(-\mathrm{\Delta }{)}^{{\gamma }_p}f{\mathrm{\parallel }}_{L^{p,\mathrm{\infty }}(m)}\lesssim \mathrm{\parallel }\mathrm{\partial }f{\mathrm{\parallel }}_{L^p(\nu )}$. The corresponding strong estimate fails for $1\le p<2$, whereas the weak estimate fails for $2<p<\mathrm{\infty }$, even after heat regularization. For the Riesz transform ${\mathcal{R}}_p=\mathrm{\partial }(-\mathrm{\Delta }{)}^{-{\gamma }_p}$, we show that ${\mathcal{R}}_1$ is not of weak type $(1,1)$ but that ${\mathcal{R}}_p$ is bounded from $L^p(m)$ to $L^{p,\mathrm{\infty }}(\nu )$ for $p\in (1,2)$, and from $L^{p,1}(m)$ to $L^p(\nu )$ for $p\in (2,\mathrm{\infty })$. However, if $p\ge 1$, ${\mathcal{R}}_p$ is bounded from $L^p(m)$ to $L^p(\nu )$ only for $p=2$.