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Mathematische Zeitschrift

, 262:881 | Cite as

Sobolev spaces related to Schrödinger operators with polynomial potentials

  • Jacek Dziubański
  • Paweł GłowackiEmail author
Article

Abstract

The aim of this note is to prove the following theorem. Let
$$Af(x)=P(D)f(x)+V(x)f(x),$$
where P(ix) is a nonnegative homogeneous elliptic polynomial on R d and V is a nonnegative polynomial potential. Then for every 1 < p < ∞ and every α > 0 there exist constants C 1, C 2 > 0 such that
$$\|P(D)^{\alpha}f\|_{L^p}+\|V^{\alpha}f\|_{L^p} \le C_1\|A^{\alpha}f\|_{L^p}$$
and
$$\|A^{\alpha}f\|_{L^p} \le C_2\|\left(P(D)^{\alpha}+V^{\alpha} \right)f\|_{L^p}$$
for f in the Schwartz class \({\mathcal{S}({\bf R}^d)}\) . We take advantage of the Christ inversion theorem for singular integral operators with a small amount of smoothness on nilpotent Lie groups, the maximal subelliptic L 2-estimates for the generators of stable semi-groups of measures, and the principle of transference of Coifman–Weiss.

Keywords

Unitary Representation Radon Measure Singular Integral Operator Homogeneous Norm Symmetric Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland

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