  This article provides quantitative proofs of the results obtained in [Cabre,
Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)], a work that proved the
H\"older regularity of stable solutions to semilinear elliptic equations in the
optimal range of dimensions $n \leq 9$. Some results in that paper were
established through compactness proofs, arguing by contradiction -- the
boundary regularity arguments being specially delicate. Here we provide proofs
that do not rely on compactness and, in particular, allow to quantify the
H\"older regularity exponent. For this, we establish two new estimates which
control the solution in $L^1$ by the $L^1$ norm of its radial derivative.
  For expository purposes, we provide self-contained proofs of the full
H\"older regularity results. They only involve basic Analysis tools.
