A generalization of $H$-measures and application on purely fractional scalar conservation laws

We extend the notion of $H$-measures on test functions defined on $\R^d\times P$,where$P\subset \R^d$ is an arbitrary compact simply connected Lipschitz manifold such that there exists a family of regular nonintersecting curves issuing from the manifold and fibrating ${\mathbb{R}}^d$. We introduce a concept of quasi-solutions to purely fractional scalar conservation laws and apply our extension of the $H$-measures to prove strong $L_{loc}^1$ precompactness of such quasi-solutions.