Weak-$A_{\mathrm{\infty }}$ packing and estimates for fractional sparse forms on spaces of homogeneous type

We prove two-weight estimates for fractional sparse forms on spaces of homogeneous type. The relevant transformed weights are assumed to belong to weak-$A_{\mathrm{\infty }}$. The weak reverse H"older inequality is normalized on enlarged balls, while sparse testing is normalized on dyadic cubes. To measure this loss, we introduce ${\mathfrak{d}}_Q^w:=w(Q)\mathrm{/}w(2\mathrm{\Lambda }{B}_Q)$. A packing estimate for the defect levels, combined with two-weight testing, gives the required mixed weak-$A_{\mathrm{\infty }}$ bounds.