Regularity and Capacity for the Fractional Dissipative Operator

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