Dziubański, Jacek; Zienkiewicz, Jacek Hardy space \(H^1\) associated to Schrödinger operator with potential satisfying reverse Hölder inequality. (English) Zbl 0959.47028 Rev. Mat. Iberoam. 15, No. 2, 279-296 (1999). Consider the Schrödinger operator \(-A=\Delta -V\) in \({\mathbb R}^d\), where \(V\) is nonnegative and satisfies the reverse Hölder inequality with exponent \(q>d/2\) (i.e., \((|B|^{-1}\int_B V^q dx)^{1/q}\leq C|B|^{-1}\int_b V dx\) for every ball \(B\)). Let \(T_t\) be the semigroup of linear operators generated by \(-A\), and let \(Mf(x)=\sup_{t>0} |T_t f(x)|\). The Hardy space \(H^1_A\) is defined to be \(\{ f: Mf\in L^1\}\). It is shown that \(H^1_A\) can be described in terms of atomic decomposition, much as in the case of the classical real variable \(H^1\), though the notion of an atom is different. The operators \(R_J=(\partial/\partial x_j)A^{-1/2}\) are analogs of the Riesz transforms. It is shown that \(H^1_A=\{ f\in L^1:R_j f\in L^1, 1\leq j\leq d \}\). Reviewer: S.V.Kislyakov (St.Peterburg) Cited in 5 ReviewsCited in 162 Documents MSC: 47F05 General theory of partial differential operators 46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces 42B30 \(H^p\)-spaces 43A80 Analysis on other specific Lie groups 47D06 One-parameter semigroups and linear evolution equations Keywords:Riesz transform; atomic decomposition; Schrödinger operator; Hölder inequality; Hardy space PDFBibTeX XMLCite \textit{J. Dziubański} and \textit{J. Zienkiewicz}, Rev. Mat. Iberoam. 15, No. 2, 279--296 (1999; Zbl 0959.47028) Full Text: DOI EuDML