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We extend the notion of $H$-measures on test functions defined on $\R^d\timesP$, where $P\subset \R^d$ is an arbitrary compact simply connected Lipschitzmanifold such that there exists a family of regular nonintersecting curvesissuing from the manifold and fibrating $\R^d$. We introduce a concept ofquasi-solutions to purely fractional scalar conservation laws and apply ourextension of the $H$-measures to prove strong $L^1_{loc}$ precompactness ofsuch quasi-solutions.

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