Real Interpolation of Hardy-Type Spaces and BMO-Regularity
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(2020) 26:61
Real Interpolation of Hardy-Type Spaces and BMO-Regularity Dmitry V. Rutsky1 Received: 4 April 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let be a σ -finite measurable space. Suppose that (X , Y ) is a couple of quasi-Banach lattices of measurable functions on T × satisfying some additional assumptions. The Hardy-type spaces X A consist of functions on D × belonging to the Smirnov class N+ in the first variable such that their boundary values are in X . Here T is the unit circle and D is the open unit disc of the complex plane. Couple (X A , Y A ) is said to be K -closed in (X , Y ) with constant C if for any f ∈ X , g ∈ Y such that H = f + g ∈ X A + Y A there exist some F ∈ X A , G ∈ Y A satisfying H = F + G, F X ≤ C f X and GY ≤ CgY . This property is shown to be equivalent to the stability of the real interpolation (X A , Y A )θ, p = (X A + Y A ) ∩ (X , Y )θ, p and to the BMO-regularity of the associated lattices L1 , (X r ) Y r δ,q under fairly broad assumptions. The inclusion X 1−θ Y θ A ⊂ (X A , Y A )θ,∞ is also characterized in these therms. New examples of couples (X A , Y A ) with this stability are given, proving that this property is strictly weaker than the usual BMO-regularity of (X , Y ). Keywords Hardy-type spaces · Real interpolation · K -closedness · AK-stability · BMO-regularity Mathematics Subject Classification Primary 30H10 · 46B70 · 46B42 · 46E30
Communicated by Mieczyslaw Mastylo. This research was supported by the Russian Science Foundation (Grant No. 18-11-00053).
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Dmitry V. Rutsky [email protected] St. Petersburg Department of V.A.Steklov Institute of Mathematics, Fontanka 27, St. Petersburg 191023, Russia 0123456789().: V,-vol
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Journal of Fourier Analysis and Applications
(2020) 26:61
1 Introduction This work is concerned with the stability of the real interpolation for one-dimensional Hardy-type spaces and the related properties. See [13] for a comprehensive overview with the appropriate references, and e.g. [1] for the generalities on the interpolation spaces. In this rather informal introduction we do not attempt to trace the detailed history of the developments mentioned here, and the formal definitions and statements of the results will be given independently of this introduction in Sect. 2 below. For a slightly different description of the results see the announcement [35] of the present work. It also elaborates on certain details omitted here for the sake of clarity. To give an illustrative example right away, for classical Hardy spaces the stability of the real interpolation is the formula H p , Hq θ,r = H p + Hq ∩ L p , Lq θ,r = Hr θ with 0 < θ < 1, 0 < p < q ≤ ∞, r1 = 1−θ p + q . That is, for the Lebesgue spaces X = L p and Y = Lq the real interpolation functor F ((·, ·)) = (·, ·)θ, p commutes with the intersection by a suitable class of analytic functions S = H p +Hq :
F ((X ∩ S, Y ∩ S)) = S ∩ F ((X , Y )) . For weighted Hardy spaces the validity of a similar fo
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