Interpolation for intersections of Hardy-type spaces
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INTERPOLATION FOR INTERSECTIONS OF HARDY-TYPE SPACES∗
BY
Sergei V. Kislyakov and Ilya K. Zlotnikov St. Petersburg Department of V. A. Steklov Mathematical Institute of the Russian Academy of Sciences 27 Fontanka, St. Petersburg 191023, Russia e-mail: [email protected], [email protected]
ABSTRACT
Let (X, µ) be a space with a finite measure µ, let A and B be w ∗ -closed subalgebras of L∞ (µ), and let C and D be closed subspaces of Lp (µ) (1 < p < ∞) that are modules over A and B, respectively. Under certain additional assumptions, the couple (C ∩ D, C ∩ D ∩ L∞ (µ)) is K-closed in (Lp (µ), L∞ (µ)). This statement covers, in particular, two cases analyzed previously: that of Hardy spaces on the two-dimensional torus and that of the coinvariant subspaces of the shift operator on the circle. Furthermore, many situations when A and B are w ∗ -Dirichlet algebras also fit in this pattern.
1. Introduction Let (X0 , X1 ) be a compatible couple of Banach spaces, and let Y0 , Y1 be closed subspaces of X0 and X1 , respectively. We remind the reader that the couple (Y0 , Y1 ) is said to be K-closed in (X0 , X1 ) if, whenever Y0 + Y1 x = x0 + x1 with xi ∈ Xi , i = 0, 1, we also have x = y0 + y1 with yi ∈ Yi and yi Cxi , i = 0, 1. ∗ This research was supported by the Russian Science Foundation (grant No. 18-
11-00053). Received March 23, 2019 and in revised form July 30, 2019
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S. V. KISLYAKOV AND I. K. ZLOTNIKOV
Isr. J. Math.
Obviously, this property implies that the interpolation spaces of the real method for the couple (Y0 , Y1 ) are equal to the intersection of the corresponding interpolation spaces for (X0 , X1 ) with the sum Y0 + Y1 . So, whenever we know the interpolation spaces for the latter couple, K-closedness makes interpolation for the former one quite easy. We recall (see [10] or the survey [6]) that Kclosedness does occur in the scale of the Hardy spaces on the unit circle (viewed as subspaces of the corresponding Lebesgue spaces), but now we are interested in the following two more complicated results (in them we assume that 1 < p < ∞, though, in fact, some information beyond this condition is available). (i) The couple (H p (T2 ), H ∞ (T2 )) is K-closed in the couple (Lp (T2 ), L∞ (T2 )) (see [8]). (ii) For an inner function θ on the unit circle, the couple (H p ∩ θH p , H ∞ ∩ θH ∞ ) is K-closed in (Lp (T), L∞ (T)) (see [9]). It should be noted that an analog of (i) in dimensions n > 2 is an open problem. Surprisingly, the proofs of Statements (i) and (ii) have turned out to be quite similar, signalizing that these facts might be particular cases of a more general claim. Such a claim exists indeed and looks roughly like this. Again, here 1 < p < ∞. Theorem: Let (X, μ) be a space with a finite measure μ, let A and B be w∗ closed subalgebras of L∞ (μ), and let C and D be w∗ -closed subspaces of L∞ (μ) that are Banach modules over A and B, respectively. Under certain additional assumptions, the couple clos C ∩ clos D, C ∩ D p p L (μ)
p
L (μ)
∞
is K-closed in (L (μ), L (μ)). The above-mentio
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