Riesz transforms on the Vicsek set

Source abstract

We study Riesz and reverse Riesz inequalities for the fractional powers of the Laplacian $Δ$ on the unbounded Vicsek set. The space carries its Hausdorff measure $m$. The gradient $\partial$ is a weak gradient defined on the skeleton, and it is measured with respect to a length measure $ν$ that is singular with respect to $m$. The relevant fractional order for the Riesz inequalities is not $1/2$ but $γ_{p}=\frac{d_f+p-1}{pd_w}$, 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 $\|(-Δ)^{γ_p}f\|_{L^{p,\infty}(m)}\lesssim\|\partial f\|_{L^p(ν)}$. The corresponding strong estimate fails for $1\le p<2$, whereas the weak estimate fails for $2<p<\infty$, even after heat regularization. For the Riesz transform $\mathcal{R}_p=\partial(-Δ)^{-γ_p}$, 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 $L^{p,\infty}(ν)$ for $p\in(1,2)$, and from $L^{p,1}(m)$ to $L^p(ν)$ for $p\in(2,\infty)$. However, if $p \ge 1$, $\mathcal{R}_p$ is bounded from $L^p(m)$ to $L^p(ν)$ only for $p=2$.