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Uniform $$L^ 2$$-weighted Sobolev inequalities. (English) Zbl 0745.35007
The authors give two conditions on weight functions $$\omega$$ so that an inequality $$\| u\|_{L^ 2(\omega)}\leq C\| P(D)u\|_{L^ 2(\omega^{-1})}$$ holds for the constant coefficient operator in $$\mathbb{R}^ N$$, $$P(D)=\Delta+\sum a_ i(\partial/\partial x_ i)+b$$, where $$n\geq 3$$ and the constant $$C$$ is independent of the lower order terms $$a_ i$$, $$b$$, in $$\mathbb{C}$$. One of these conditions is a one- dimensional $$A_ 2$$ Muckenhoupt’s requirement depending only on the direction of the vector $$({\mathfrak R}a_ j)$$ and the other is an estimate $\left(r^{-n}\int_{B_ r}\omega^ \alpha(x)dx\right)\left(r^{- n}\int_{B_ r}\omega^{ -\alpha}(x)dx\right)^{-1}\leq Cr^{- 4\alpha},$ where $$B_ r$$ denotes a ball of radius $$r$$ and $$C$$ is a constant independent of $$B_ r$$.
As a consequence they prove a version of a result due to S. Chanillo and E. Sawyer [Trans. Am. Math. Soc. 318, No. 1, 275-300 (1990; Zbl 0702.35034)] on unique continuation of solutions of Schrödinger operator with the potential belonging to the Fefferman- Phong class.

##### MSC:
 35B60 Continuation and prolongation of solutions to PDEs 35J20 Variational methods for second-order elliptic equations 42B99 Harmonic analysis in several variables 35J10 Schrödinger operator, Schrödinger equation
##### Keywords:
unique continuation theorem
Zbl 0702.35034
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