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.


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


Zbl 0702.35034
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