Littlewood-Paley-Stein functionals: an R-boundedness approach
Résumé
Let L = ∆ + V be a Schrödinger operator with a non-negative potential V on a complete Riemannian manifold M. We prove that the vertical Littlewwod-Paley-Stein functional associated with L is bounded on L p (M) if and only if the set { √ t ∇e −tL , t > 0} is R-bounded on L p (M). We also introduce and study more general functionals. For a sequence of functions m k : [0, ∞) → C, we define
H((f k)) := ( \sum_k \int_0^\infty |∇m k (tL)f _k |^2 dt )^1/2 + (\sum_k \int_0^\infty | √ V m k (tL)f _k | 2 dt )^1/2.
Under fairly reasonable assumptions on M we prove for certain functions m k the boundedness of H on L p (M) in the sense
\| H((f _k)) \|_p ≤ C \| (\sum_k |f _k | 2 )^1/2 \|_p for some constant C independent of (f _k) _k. A lower estimate is also proved on the dual space L p. We introduce and study boundedness of other Littlewood-Paley-Stein type functionals and discuss their relationships to the Riesz transform. Several examples are given in the paper.
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