A remarkable theorem of Léger asserts that if $\mu$ is a Radon measure in the Euclidean space and $B$ is a ball such that $\mu(B)=r(B)$ (where $r(B)$ stands for the radius of $B$) satisfying the linear growth condition $\mu(B(x,r))\leq C_0 r$ for all $x,r$, and so that the curvature of $\mu$ $$c^2(\mu)=\iiint \frac1{R(x,y,z)^2}\,d\mu(x)d\mu(y)d\mu(z)$$ is small enough, then a big piece of $\mu$ on $B$ is supported on a Lipschitz graph and is absolutely continuous with respect to arc length measure on the graph. In my talk I will present a version of this theorem which involves the $L^2$ norm of the codimension $1$ Riesz transform in the Euclidean space, and I will show an application (by Azzam, Mourgoglou and myself) of this result to an old problem on harmonic measure posed by C. Bishop.

Sobolev functions are known to be quasicontinuous, meaning that the restriction of a Sobolev function outside a set of small capacity is continuous. The same cannot hold for functions of bounded variation, or BV functions, since they can have jump sets with large 1-capacity. On a metric space equipped with a doubling measure supporting a Poincar\'e inequality, we show a weaker notion of quasicontinuity for BV functions. More precisely, we show that the restriction of a BV function outside a set of small 1-capacity is continuous outside the function's jump set and "one-sidedly" continuous in its jump set.