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Hardy space \(H^1\) associated to Schrödinger operator with potential satisfying reverse Hölder inequality. (English) Zbl 0959.47028

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 \(H^1_A=\{ f\in L^1:R_j f\in L^1, 1\leq j\leq d \}\).

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
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