Uniform Sobolev inequalities and absolute continuity of periodic operators

Abstract

We establish certain uniform L p − L q L^{p}-L^{q} inequalities for a family of second order elliptic operators of the form ( D + k ) A ( D + k ) T ( {\mathbf {D}} + {\mathbf {k}} ) A ( {\mathbf {D}}+ {\mathbf {k} })^{T} on the d d -torus, where D = − i ∇ , k ∈ C d {\mathbf {D}} =-i\nabla , {\mathbf {k}}\in {\mathbb {C}} ^{d} and A A is a symmetric, positive definite d × d 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 R d {\mathbb R}^{d} with singular potential. The absolute continuity of the elliptic operator div ( ω ( x ) ∇ ) (\omega ( {\mathbf {x}})\nabla ) on R d {\mathbb R}^{d} with a positive periodic scalar function ω ( x ) \omega ( {\mathbf {x}} ) is also studied.