| Popis: |
We prove some nonhomogeneous versions of the div-curl lemma in the context of weighted spaces. Namely, assume the vector fields \(\mathbf{V},\mathbf{W}\!\!: \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\), along with their distributional divergence and curl, respectively, lie in L μ p and L ν q , \(\frac{1} {p} + \frac{1} {q} = 1\), where μ and ν are in certain Muckenhoupt weight classes. Then the resulting scalar product V ⋅ W is in the weighted local Hardy space \(h_{\omega }^{1}(\mathbb{R}^{n})\), for \(\omega =\mu ^{\frac{1} {p} }\nu ^{\frac{1} {q} }\) in \(A_{1+ \frac{1} {n} }\). |
