Interpolation of Calderon and Ovcinnikov Pairs

2021

The main aim of this paper is to develop a general approach, which allows to extend the basics of Brudnyi-Kruglyak interpolation theory to the realm of quasi-Banach lattices. We prove that all K-monotone quasi-Banach lattices with respect to a L-convex quasi-Banach lattice couple have in fact a stronger property of the so-calledK(p, q)-monotonicity for some 0 < q ≤ p ≤ 1, which allows us to get their description by the real K-method. Moreover, we obtain a refined version of theK-divisibility property for Banach lattice couples and then prove an appropriate version of this property for L-convex quasiBanach lattice couples. The results obtained are applied to refine interpolation properties of couples of sequence land function L-spaces, considered for the full range 0 < p < ∞.

2000

Lorentz and Shimogaki [2] have characterized those pairs of Lorentz A spaces which satisfy the interpolation property with respect to two other pairs of A spaces. Their proof is long and technical and does not easily admit to generalization. In this paper we present a short proof of this result whose spirit may be traced to Lemma 4.3 of [4] or perhaps more accurately to the theorem of Marcinkiewicz [5, p. 112]. The proof involves only elementary properties of these spaces and does allow for generalization to interpolation for n pairs and for M spaces, but these topics will be reported on elsewhere. The Banach space A^ [1, p. 65] is the space of all Lebesgue measurable functions ƒ on the interval (0, /) for which the norm is finite, where </> is an integrable, positive, decreasing function on (0, /) and/* (the decreasing rearrangement of |/|) is the almost-everywhere unique, positive, decreasing function which is equimeasurable with \f\. A pair of spaces (A^, A v) is called an interpolation pair for the two pairs (A^, A Vl) and (A^2, A V2) if each linear operator which is bounded from A^ to A v (both /== 1, 2) has a unique extension to a bounded operator from A^ to A v. THEOREM (LORENTZ-SHIMOGAKI). A necessary and sufficient condition that (A^, A w) be an interpolation pair for (A^, A Vi) and (A^2, A V2) is that there exist a constant A independent of s and t so that (*) ^(0/0(5) ^ A max(TO/^(a)) t=1.2 holds, where O 00=ƒ S <j>{r) dr,-" , VaC'Wo Y a (r) dr.

Journal of Computational and Applied Mathematics, 1997

By using a norm generated by the error series of a sequence of interpolation polynomials, we obtain in this paper ~ertain Banach spaces. A relation between these spaces and the space (Co, S) with norm generated by the error series of the best polynomial approximations (minimax series) is established.

2016

e is compact and show a positive answer under a variety of conditions. For example it suffices that X o be a UMD-space or that X o is reflexive and there is a Banach space so that X 0 = [W,X l~\I for some 0<ot< 1. 1991 Mathematics subject classification: 46M35.

Proceedings of The London Mathematical Society, 1988

We show that the equivalence of the K and J methods of interpolation for 2 d Banach spaces holds in the case of the Calder6n spaces X(E). Consequently, the reiteration theorem also holds. As an application we determine interpolation spaces among families of non-isotropic spaces of Besov-Nikol'skij and Hardy-Sobolev-Nikol'skij type.

Advances in Mathematics, 1982

A detailed development is given of a theory of complex interpolation for families of Banach spaces which extends the well-known theory for pairs of spaces. 203 000 I-8708/82/030203-27$05.00/O Copyri%t 0 1982 by Academic Rew. Inc. All &hts of reproduction i n my fm resawd.

Bulletin of the American Mathematical Society, 1974

Lorentz and Shimogaki [2] have characterized those pairs of Lorentz A spaces which satisfy the interpolation property with respect to two other pairs of A spaces. Their proof is long and technical and does not easily admit to generalization. In this paper we present a short proof of this result whose spirit may be traced to Lemma 4.3 of [4] or perhaps more accurately to the theorem of Marcinkiewicz [5, p. 112]. The proof involves only elementary properties of these spaces and does allow for generalization to interpolation for n pairs and for M spaces, but these topics will be reported on elsewhere. The Banach space A^ [1, p. 65] is the space of all Lebesgue measurable functions ƒ on the interval (0, /) for which the norm is finite, where </> is an integrable, positive, decreasing function on (0, /) and/* (the decreasing rearrangement of |/|) is the almost-everywhere unique, positive, decreasing function which is equimeasurable with \f\. A pair of spaces (A^, A v) is called an interpolation pair for the two pairs (A^, A Vl) and (A^2, A V2) if each linear operator which is bounded from A^ to A v (both /== 1, 2) has a unique extension to a bounded operator from A^ to A v. THEOREM (LORENTZ-SHIMOGAKI). A necessary and sufficient condition that (A^, A w) be an interpolation pair for (A^, A Vi) and (A^2, A V2) is that there exist a constant A independent of s and t so that (*) ^(0/0(5) ^ A max(TO/^(a)) t=1.2 holds, where O 00=ƒ S <j>{r) dr,-" , VaC'Wo Y a (r) dr.

Annali di Matematica Pura ed Applicata, 1982

In the present paper we study a general ]orm of Peetre's J-and K-methods o] interpolation. Special emphasis is given to the equivalence theorem for J-and K-spaces and to reiteration theorems.

Journal of Mathematical Analysis and Applications, 1991

Beauzamy in [l] proved that the Lions-Peetre spaces A,, (0 < 0 < 1, 1 <p < co) are reflexive if and only if the imbedding i: d(A) + E(A) is weakly compact. Later on, the Davis-Figiel-Johnson-Pelczynski factorization technique was again used in [lo] to prove that the operator T: A,, + Be,p is weakly compact if and only if T: d(A)-+ Z(g) is weakly compact, thereby generalizing the aforesaid result of Beauzamy. In this paper, a modification of this technique is used to give a characterization of the reflexivity for the more general spaces AH,t.

m-hikari.com

LetĀ = (A 1 , A 2 , • • • , A n) be a compatible n-tuple of Banach spaces. We may define the interpolation method in R n , and prove some related lemma and theorem.