Some remarks on Wente’s inequality and the Lorentz-Sobolev space

Abstract

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 H¹ 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.