We prove an analogue of Y. Meyer's wavelet characterization of the Hardy space H¹(ℝⁿ) for the space H¹(ℝⁿ,X) of X-valued functions. Here X is a Banach space with the UMD property. The proof uses results of T. Figiel on generalized Calderón-Zygmund operators on Bochner spaces and some new local estimates.
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Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant 𝔓 ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: {-1,1}ⁿ → X satisfies $∫_{{-1,1}ⁿ} ||∑_{j=1}^{n} ∂_{j}f_{j}(ε)||^{p} dμ(ε) ≤ 𝔓^{p} ∫_{{-1,1}ⁿ} ∫_{{-1,1}ⁿ} ||∑_{j=1}^{n} δ_{j} Δf_{j}(ε)||^{p} dμ(ε)dμ(δ)$, where μ is the uniform probability measure on the discrete hypercube {-1,1}ⁿ, and ${∂_j}_{j=1}^{n}$ and $Δ = ∑_{j=1}^{n}∂_{j}$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by $𝔓ⁿ_{p}(X)$, we show that $𝔓ⁿ_{p}(X) ≤ ∑_{k=1}^{n} 1/k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special case $f_{j} = Δ^{-1}∂_{j} f$ for some f: {-1,1}ⁿ → X. We show that $sup_{n∈ ℕ }𝔓ⁿ_{p}(X) < ∞$ if either the dual X* is a UMD⁺ Banach space, or for some θ ∈ (0,1) we have $X = [H,Y]_{θ}$, where H is a Hilbert space and Y is an arbitrary Banach space. It follows that $sup_{n∈ ℕ}𝔓ⁿ_{p}(X) < ∞$ if X is a Banach lattice of nontrivial type.
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A number of recent results in Euclidean harmonic analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making these tools available for analysis on metric spaces. The results include a new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen and the first author. We illustrate the usefulness of these constructions with applications to weighted inequalities and the BMO space; further applications will appear in forthcoming work.
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