Absolutely summing operators and atomic decomposition in bi-parameter Hardy spaces

الأصل · EN

For f ∈ Hᵖ(δ²), 0<p≤ 2, with Haar expansion f=∑ fᵢ × ⱼhᵢ× ⱼ we constructively determine the Pietsch measure of the 2-summing multiplication operator Mf:ℓ∞ → Hᵖ(δ²), (φᵢ× ⱼ) ∑ φᵢ× ⱼfᵢ × ⱼhᵢ × ⱼ. Our method yields a constructive proof of Pisier's decomposition of f ∈ Hᵖ(δ²) |f|=|x|¹⁻θ|y|θ and xₓ₀¹⁻θyθₕ₂₍δ₂₎≤ Cfₕₚ₍δ₂₎, where X₀ is Pisier's extrapolation lattice associated to Hᵖ(δ²) and H²(δ²). Our construction of the Pietsch measure for the multiplication operator Mf involves the Haar coefficients of f and its atomic decomposition. We treated the one-parameter Hᵖ-spaces in [P.F.X Müller, J.Penteker, p-summing multiplication operators, dyadic Hardy spaces and atomic decomposition, Houston Journal Math.,41(2):639-668,2015.].