Mathematische Zeitschrift

, 262:881 | Cite as

Sobolev spaces related to Schrödinger operators with polynomial potentials

  • Jacek Dziubański
  • Paweł GłowackiEmail author


The aim of this note is to prove the following theorem. Let
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
$$\|P(D)^{\alpha}f\|_{L^p}+\|V^{\alpha}f\|_{L^p} \le C_1\|A^{\alpha}f\|_{L^p}$$
$$\|A^{\alpha}f\|_{L^p} \le C_2\|\left(P(D)^{\alpha}+V^{\alpha} \right)f\|_{L^p}$$
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.


Unitary Representation Radon Measure Singular Integral Operator Homogeneous Norm Symmetric Kernel 
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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland

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