We prove a variant of Tartar's first commutation lemma involving multiplier
operators with symbols not necessarily defined on a manifold of codimension
one.
We prove that that $L^p$, $p\in (1,\infty)$, bound of a multiplier operator
linearly depends on the $L^\infty$ bound of symbol of the multiplier operator.
We use the latter properties of the multiplier operators to extend the notion
of the $H$-measures in the $L^p$ framework.
We prove that that $L^p$, $p\in (1,\infty)$, bound of a multiplier operator
linearly depends on the $L^\infty$ bound of symbol of the multiplier operator.
We use the latter properties of the multiplier operators to extend the notion
of the $H$-measures in the $L^p$ framework.