  This note is devoted to exploring some analytic-geometric properties of the
regularity and capacity associated to the so-called fractional dissipative
operator $\partial_t+(-\Delta)^\alpha$, naturally establishing a diagonally
sharp Hausdorff dimension estimate for the blow-up set of a weak solution to
the fractional dissipative equation
$(\partial_t+(-\Delta)^\alpha)u(t,x)=F(t,x)$ subject to $u(0,x)=0$.
