In the graphical Sierpinski gasket, the reverse Riesz transform is unbounded on $L^p$, $p\in (1,2)$

Abstract: In this article, we proved that the reverse Riesz transform on the graphical Sierpinski gasket is unbounded on $L^p$ for $p\in (1,2)$. Together with previous results, it shows that the Riesz transform on the graphical Sierpinski gasket is bounded on $L^p$ if and only if $p\in (1,2]$ and the reverse Riesz transform is bounded on $L^p$ if and only if $p\in [2,\infty)$. Moreover, our method is quite flexible - but requires explicit computations - and hints to the fact that the reverse Riesz transforms is never bounded on $L^p$, $p\in (1,2)$, on graphs with slow diffusions.