Operator-valued fourier multipliers on multi-dimensional hardy spaces

Abstract

The author establishes operator-valued Fourier multiplier theorems on multi-dimensional Hardy spaces H ^p($\mathbb{T}$ ^d;X), where 1 ≤ p < ∞, d ∈ ℕ, and X is an AUMD Banach space having the property (α). The sufficient condition on the multiplier is a Marcinkiewicz type condition of order 2 using Rademacher boundedness of sets of bounded linear operators. It is also shown that the assumption that X has the property (α) is necessary when d ≥ 2 even for scalar-valued multipliers. When the underlying Banach space does not have the property (α), a sufficient condition on the multiplier of Marcinkiewicz type of order 2 using a notion of d-Rademacher boundedness is also given.

Keywords

H ^p-Spaces / Fourier multiplier / Rademacher boundedness / d-Rademacher boundedness

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