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Uniform Sobolev inequalities and absolute continuity of periodic operators. (English) Zbl 1133.35031
The authors prove uniform \(L^p\)-\(L^q\) inequalities for a family of second order elliptic operators \(H_\rho\) on the \(d\)-dimensional torus \(\mathbb T^d\). The operators \(H_\rho\) are of the form \(H_\rho=({\mathbf D} +(\delta + i\rho){\mathbf a}+{\mathbf b}) A({\mathbf D} +(\delta + i\rho){\mathbf a}+{\mathbf b})^T\), \({\mathbf D}=-i\nabla\), \({\mathbf a}, {\mathbf b}\in \mathbb R^d\), \(| {\mathbf a} | = 1\), \(<{\mathbf a},{\mathbf b}>=0\), \(A\) is a positive definite real matrix, \({\mathbf a}A=(s_0,0,\dots,0)\) for some \(s_0\not=0\), \(\delta=(1/2-{\mathbf b}_1)/{\mathbf a}_1\) and \(| \rho | \geq 2\). The approach used in proving the \(L^p\)-\(L^q\) inequalities was motivated by the paper of T. H. Wolff [Geom. Funct. Anal. 2, No. 2, 225–284 (1992; Zbl 0780.35015)] and the obtained inequalities extend results previously proved in Z. Shen’s paper [Int. Math. Res. Not. 2001, No. 1, 1–31 (2001; Zbl 0998.35007)].
These estimates are used to prove the absolute continuity of the Dirac operator on \(\mathbb R^d\) with periodic electric potential \(V\) belonging to the Lipschitz space \(\Lambda^{r,\infty}_\alpha(\mathbb T^d)\), for some \(r\geq d\) and \(\alpha > (d-1)/(2r)\). The absolute continuity of the operator \({\mathbf D} \omega(x)A {\mathbf D}^T\) is proved with \(\omega\) being a periodic function with respect to some lattice \(\Gamma\) of \(\mathbb R^d\), bounded by a positive constant and such that \(\nabla \omega \in \Lambda^{r,\infty}_\alpha(\mathbb R^d/\Gamma)\) for some \(r\geq d\) and \(\alpha > (d-1)/(2r)\).

MSC:
35J10 Schrödinger operator, Schrödinger equation
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B15 Multipliers for harmonic analysis in several variables
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