Uniform Sobolev inequalities and absolute continuity of periodic operators

We establish certain uniform $L^{p}-L^{q}$ inequalities for a family of second order elliptic operators of the form $( {\bold {D}} + {\bold {k}} ) A ( {\bold {D}}+ {\bold {k} })^{T}$ on the $d$-torus, where ${\bold {D}} =-i\nabla , {\bold {k}}\in {\Bbb {C}} ^{d}$ and $A$ is a symmetric, positive definite $d\times d$ matrix with real constant entries. Using these Sobolev type inequalities, we obtain the absolute continuity of the spectrum of the periodic Dirac operator on ${\Bbb R}^{d}$ with singular potential. The absolute continuity of the elliptic operator div $(\omega ( {\bold {x}})\nabla )$ on ${\Bbb R}^{d}$ with a positive periodic scalar function $\omega ( {\bold {x}} )$ is also studied.