Abstract
LetL=-Δ+Vbe a Schrödinger operator acting onL2(Rn),n≥1, whereV≢0is a nonnegative locally integrable function onRn. In this paper, we will first define molecules for weighted Hardy spacesHLp(w) (0<p≤1)associated withLand establish their molecular characterizations. Then, by using the atomic decomposition and molecular characterization ofHLp(w), we will show that the imaginary powerLiγis bounded onHLp(w)forn/(n+1)<p≤1, and the fractional integral operatorL-α/2is bounded fromHLp(w)toHLq(wq/p), where0<α<min{n/2,1},n/(n+1)<p≤n/(n+α), and1/q=1/p-α/n.
Highlights
Let n ≥ 1 and V be a nonnegative locally integrable function defined on Rn, not identically zero
It is well known that this symmetric form is closed
Note that it was shown by Simon [1] that this form coincides with the minimal closure of the form given by the same expression cboumt pdaecfitnseudppoonrtCs)0∞
Summary
Let n ≥ 1 and V be a nonnegative locally integrable function defined on Rn, not identically zero. Since the kernel pt(x, y) of {e−tL}t>0 satisfies the Gaussian upper bounds (4), it is easy to check that |L−α/2(f)(x)| ≤ C ⋅ Iα(|f|)(x) for all x ∈ Rn, where Iα denotes the classical fractional integral operator (see [5]): Iα. For more information about the fractional integrals L−α/2 associated with some classes of operators, one can see [6,7,8,9]. We first define molecules for the weighted Hardy spaces HLp(w) associated with L and establish their molecular characterizations. For any γ ∈ R, the imaginary power Liγ is bounded from HLp(w) to the weighted Lebesgue space Lp(w). The fractional integral operator L−α/2 is bounded from HLp(w) to Lq(wq/p). The fractional integral operator L−α/2 is bounded from HLp(w) to HLq(wq/p)
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