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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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