An Interpolation Theorem with Perturbed Continuity

Studia Mathematica, 2005

We prove an interpolation theorem for weak-type operators. This is closely related to interpolation between weak-type classes. Weak-type classes at the ends of interpolation scales play a similar role to that played by BMO with respect to the L p interpolation scale. We also clarify the roles of some of the parameters appearing in the definition of the weak-type classes. The interpolation theorem follows from a K-functional inequality for the operators, involving the Calderón operator. The inequality was inspired by a K-J inequality approach developed by Jawerth and Milman. We show that the use of the Calderón operator is necessary. We use a new version of the strong fundamental lemma of interpolation theory that does not require the interpolation couple to be mutually closed. 1. Introduction. Weak-type classes were defined in [9], [10], and [11]. Interpolation theorems between these classes give stronger versions of classical interpolation theorems, and calculations of the K-functionals between these classes give a systematic way of proving rearrangement-function inequalities for classical operators. These classes, when constructed at the natural ends of interpolation scales, turn out to be interesting in their own right, and imply useful generalizations of interpolation theorems. The definition of the weak-type classes, Definition 1.2 below, is quite general, and among other things yields an interpolation theorem, Theorem 1.3, that is valid only for a subinterval of the interpolation scale. In this paper we generalize the interpolation theorem. We define weak-type operators between interpolation pairs of Banach groups, and prove an interpolation theorem for these operators. Moreover we prove an inequality between the K-functionals of T a and of a, which implies the interpolation theorem and is of independent interest. We also show that the interpolation

Arkiv för matematik, 1984

Given an interpolation couple (A0, A~), the approximation functional is dcfined

Journal of the London Mathematical Society, 1998

Izvestiya: Mathematics, 1998

In this paper we study linear forms Λ = b 1 ln α 1 + • • • + bn ln αn with rational integer coefficients b j (bn = 0, n 2), where the α j are algebraic numbers satisfying the so-called strong independence condition. In standard notation, we prove an explicit estimate of the form |Λ| > exp −C n D n+2 Ω ln C n D n+2 Ω ln(eB). Its novel feature is that it contains no factors of the form n n .

Journal of Approximation Theory, 1969

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.

arXiv, 2017

Given a constant q ∈ (1, ∞), we study the following property of a normed sequence space E: If {x n } n∈N is an element of E and if {y n } n∈N is an element of ℓ q such that ∞ n=1 |x n | q = ∞ n=1 |y n | q and if the nonincreasing rearrangements of these two sequences satisfy N n=1 |x * n | q ≤ N n=1 |y * n | q for all N ∈ N, then {y n } n∈N ∈ E and {y n } n∈N E ≤ C {x n } n∈N E for some constant C which depends only on E. We show that this property is very close to characterizing the normed interpolation spaces between ℓ 1 and ℓ q. More specificially, we first show that every space which is a normed interpolation space with respect to the couple (ℓ p , ℓ q) for some p ∈ [1, q] has the above mentioned property. Then we show, conversely, that if E has the above mentioned property, and also has the Fatou property, and is contained in ℓ q , then it is a normed interpolation space with respect to the couple ℓ 1 , ℓ q. These results are our response to a conjecture of Galina Levitina, Fedor Sukochev and Dmitriy Zanin in [11].

Arkiv for Matematik, 1989

This originated the question of whether there are generalizations of Theorem L-P to the case Ao~A~ and BomB 1.

Journal of Functional Analysis, 1979

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.