The authors provide weak \(M^{1,q,a}(X)\)-estimates for the maximal and Riesz potential operators. Under certain assumptions on the growth of the measures of balls in \(X\) (which hold in the Euclidean setting), it is shown in Theorems 3.3 and 4.6 that the maximal operator and the Riesz potential operator are bounded from \(M^{1,q,a}(X)\) to \(WM^{\varphi,q,a}(X)\). Moreover, quantitative estimates for the boundedness of these operators are provided in Theorems 3.5 and 4.10 for functions \(f\) satisfying \[ \Vert f \Vert_{N^{p,q,a}(X)} := \Vert f \Vert_{L^p(B(x_0,2))} + \left( \int_1^{\infty} \left( r^a \Vert f \Vert_{L^p(X \setminus B(x_0,r))}\right)^q \frac{dr}{r} \right)^{1/q} < 1 \] for \(p = 1\). The paper concludes with a discussion of the duality between \(M^{1,q,a}(X)\) and the space \(N^{\infty,q',a}(X)\) consisting of those measurable functions on \(X\) which satisfy \(\Vert f \Vert_{N^{\infty,q',a}(X)} < \infty\). In particular, \(N^{\infty,q',a}(X)\) is precisely the associate space of \(M^{1,q,a}(X)\), i.e., \[ \Vert f \Vert_{N^{\infty,q',a}(X)} = \sup_{g \in M^{1,q,a}(X) \, : \, \Vert g \Vert_{M^{1,q,a}(X)}\leq 1} \int_X |f(x)g(x)| \, d \mu (x). \]