In this note we consider Wente’s type inequality on the Lorentz-Sobolev space. If
$$\nabla f \in {L^{{p_1},{q_1}}}{\left( R \right)^n},\;G \in {L^{{p_2},{q_2}}}{\left( R \right)^n}$$
and div *G* ≡ 0 in the sense of distribution where
$$\frac{1}
{{p_1 }} + \frac{1}
{{p_2 }} = \frac{1}
{{q_1 }} + \frac{1}
{{q_2 }} = 1,1 < p_1 ,p_2 < \infty ,$$
it is known that *G* · ∇*f* belongs to the Hardy space H^{1} and furthermore
$${\left\| {G \cdot \nabla f} \right\|_{{H^1}}} \leqslant C{\left\| {\nabla f} \right\|_{{L^{{p_1},{q_1}}}\left( {{R^2}} \right)}}{\left\| G \right\|_{{L^{{p_2},{q_2}}}\left( {{R^2}} \right)}}$$
. Reader can see [9] Section 4.

Here we give a new proof of this result. Our proof depends on an estimate of a maximal operator on the Lorentz space which is of some independent interest. Finally, we use this inequality to get a generalisation of Bethuel’s inequality.