Let $(M,μ)$ and $(N,ν)$ be measure spaces. In this paper, we study the $K_t$--\,functional for the couple $$A_0=L^\infty(dμ\,; L^1(dν))\,,~~A_1=L^\infty(dν\,; L^1(dμ))\,. $$
Here, and in what follows the vector valued $L^p$--\,spaces $L^p(dμ\,; L^q(dν))$ are meant in Bochner's sense.
One of our main results is the following, which can be viewed as a refinement of a lemma due to Varopoulos [V].
\proclaim Theorem 0.1. Let $(A_0,A_1)$ be as above. Then for all $f$ in $A_0+A_1$ we have $${1\over 2}\,K_t(f;\,A_0\,,A_1)\leq \sup\,\bigg\{ \Big(μ(E)\vee t^{-1}ν(F)\Big)^{-1} \int_{E\times F} \vert f\vert\,dμ\,dν\,\bigg\} \leq K_t(f;\,A_0\,,A_1)\,,$$ where the supremum runs over all measurable subsets $E\subset M\,,~ F\subset N$ with positive and finite measure and $u\!\vee\!v$ denotes the maximum of the reals $u$ and $v$.
In this paper, equivalence between interpolation properties of linear operators and monotonicity conditions are studied, for a pair $(X_0,X_1)$ of rearrangement invariant quasi Banach spaces, when the extreme spaces of the interpolation are $L^\infty$ and a pair $(A_0,A_1)$ under some assumptions. Weak and restricted weak intermediate spaces fall in our context. Applications to classical Lorentz and Lorentz-Orlicz spaces are given.
The monograph contains the detailed exposition of the results obtained by the author during the last several years. In particular it contains an improvement of the well known Calderon - Ryff interpolation theorem and description of "verifying" operators of conditional mathematical expectations {if such an operator leaves an order ideal in $L^1% invariant, then the ideal is an interpolation space}.
The real and complex interpolation spaces for the classical Hardy spaces $H^1$ and $H^\infty$ were determined in 1983 by P.W. Jones. Due to the analytic constraints the associated Marcinkiewicz decomposition gives rise to a delicate approximation problem for the $L^ 1$ metric. Specifically for $ f \in H^p$ the size of $$ {\rm{inf}} \{ \| f - f_1 \| _1 \,:\, f_1 \in H^\infty ,\, \|f_1\|_\infty \le λ\}$$ needs to be determined for any $ λ>0 $. In the present paper we develop a new set of truncation formulae for obtaining the Marcinkiewicz decomposition of $(H^1, H^\infty) $. We revisit the real …
Given a finite set σof the unit disc \mathbb{D}={z\in\mathbb{C}:, |z|<1} and a holomorphic function f in \mathbb{D} which belongs to a class X, we are looking for a function g in another class Y (smaller than X) which minimizes the norm |g|_{Y} among all functions g such that g_{|σ}=f_{|σ}.
General results of interpolation (eg. Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebra $F^\infty$ (resp. noncommutative disc algebra $A_n$) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ${\bf C}^n$ are obtained. Non-commutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebra $F^\infty/J$ on Hilbert spaces, where $J$ is any $w^*$-closed, 2-sided ideal of $F^\infty$, are obtained and used to construct a $w^*$-continuous, $F^…
The interpolation property of Ces{à}ro sequence and function spaces is investigated. It is shown that $Ces_p(I)$ is an interpolation space between $Ces_{p_0}(I)$ and $Ces_{p_1}(I)$ for $1 < p_0 < p_1 \leq \infty$ and $1/p = (1 - θ)/p_0 + θ/p_1$ with $0 < θ< 1$, where $I = [0, \infty)$ or $[0, 1]$. The same result is true for Ces{à}ro sequence spaces. On the other hand, $Ces_p[0, 1]$ is not an interpolation space between $Ces_1[0, 1]$ and $Ces_{\infty}[0, 1]$.
Given a finite set $σ$ of the unit disc $\mathbb{D}$ and a holomorphic function $f$ in $\mathbb{D}$ which belongs to a class $X$ we are looking for a function $g$ in another class $Y$ which minimizes the norm $|g|_{Y}$ among all functions $g$ such that $g_{|σ}=f_{|σ}$. Generally speaking, the interpolation constant considered is $c(σ,\, X,\, Y)={sup}{}_{f\in X,\,\parallel f\parallel_{X}\leq1}{inf}\{|g|_{Y}:\, g_{|σ}=f_{|σ}\} \,.$ When $Y=H^{\infty}$, our interpolation problem includes those of Nevanlinna-Pick (1916), Caratheodory-Schur (1908). Moreover, Carleson's free interpolation (1958) has…
We prove embeddings and identities for real interpolation spaces between mixed Lorentz spaces. This partly relies on Minkowski's (reverse) integral inequality in Lorentz spaces $L^{p,r}(X)$ under optimal assumptions on the exponents $(p,r)\in (0,\infty)\times (0,\infty]$.
We prove a weighted version of a classical inequality of Johnson and Schechtman from which we derive a decomposition theorem for $p$-th moments ($0<p\leq 1$) of nonnegative generalized $U$-statistics with constant not dependent on $p$. In particular, for $1\leq p\leq 2$, the norm in the subspace $U^p_{\leq m}\left(Ω^\infty\right)$ of $L^p\left(Ω^\infty\right)$ spanned by functions dependent on at most $m$ variables is equivalent to the norm in a suitable interpolation sum of $L^p\left(L^2\right)$ spaces. As a consequence, we obtain some interpolation properties of $U^1_m\left(Ω^\infty,\ell^p\r…
Given an inner function $B$ we classify the invariant subspaces of the algebra $H^\infty_B:=\mathbb{C}+BH^\infty$. We derive a formula in terms of these invariant subspaces for the distance of an element in $L^\infty$ to a certain weak*-closed ideal in $H^\infty_B$ and use this to prove an analogue of the Nevanlinna-Pick interpolation theorem.
By using duality, it is shown that there exist near-minimizers for the distance functionals for the couple $(L^\infty, L^p)$, $1<p<\infty$, that are stable under the action of singular integral operators.
The condition numbers $CN(T)==||T|| .||T^{-1}|| $ of Toeplitz and analytic $n\times n$ matrices $T$ are studied. It is shown that the supremum of $CN(T)$ over all such matrices with $||T|| \leq1$ and the given minimum of eigenvalues $r=min_{i=1..n}|λ_{i}|>0$ behaves as the corresponding supremum over all $n\times n$ matrices (i.e., as $\frac{1}{r{}^{n}}$ (Kronecker)), and this equivalence is uniform in $n$ and $r$. The proof is based on a use of the Sarason-Sz.Nagy-Foias commutant lifting theorem.
Given a collection $K$ of positive integers, let $H^{\infty}_K(\mathbb{D})$ denote the set of all bounded analytic functions defined on the unit disk $\mathbb{D}$ in $\mathbb{C}$ whose $k^{\text{th}}$ derivative vanishes at zero, for all $k \in K$. In this paper, we establish a Nevanlinna-Pick interpolation result for the subalgebra $H^{\infty}_K(\mathbb{D})$, where $K = \{1,2,\dotsc,k\}$, which is a slight generalization of the interpolation theorem that Davidson, Paulsen, Raghupathi, and Singh proved for the algebra $H^{\infty}_{\{1\}}(\mathbb{D})$. Furthermore, we provide a sufficient condi…
If (A_0,A_1) and (B_0,B_1) are Banach couples and a linear operator T from A_0 + A_1 to B_0 + B_1 maps A_0 compactly into B_0 and maps A_1 boundedly into B_1, does T necessarily also map [A_0,A_1]_s compactly into [B_0,B_1]_s for s in (0,1)? After 42 years this question is still not answered, not even in the case where T is also compact from A_1 to B_1. But affirmative answers are known for many special choices of (A_0,A_1) and (B_0,B_1). Furthermore it is known that it would suffice to resolve this question in the special case where (B_0,B_1) is the special couple (l^\infty(FL^\infty), l^\inf…
We give an elementary proof that the $H^p$ spaces over the unit disc (or the upper half plane) are the interpolation spaces for the real method of interpolation between $H^1$ and $H^\infty$. This was originally proved by Peter Jones. The proof uses only the boundedness of the Hilbert transform and the classical factorisation of a function in $H^p$ as a product of two functions in $H^q$ and $H^r$ with $1/q+1/r=1/p$. This proof extends without any real extra difficulty to the non-commutative setting and to several Banach space valued extensions of $H^p$ spaces. In particular, this proof easily e…
A Banach lattice E is called p-disjointly homogeneous, 1< p< infty, when every sequence of pairwise disjoint normalized elements in E has a subsequence equivalent to the unit vector basis of l_p. Employing methods from interpolation theory, we clarify which rearrangement invariant (r.i.) spaces on [0,1] are p-disjointly homogeneous. In particular, for every 1<p< infty and any increasing concave function f on [0,1], which is not equivalent neither 1 nor t, there exists a p-disjointly homogeneous r.i. space with the fundamental function f. Moreover, in the class of all interpolation r.i. spaces …
Our starting point is a lemma due to Varopoulos. We give a different proof of a generalized form this lemma, that yields an equivalent description of the $K$-functional for the interpolation couple $(X_0,X_1)$ where $X_0=L_{p_0,\infty}(μ_1; L_q(μ_2))$ and $X_1=L_{p_1,\infty}(μ_2; L_q(μ_1))$ where $0<q<p_0,p_1\le \infty$ and $(Ω_1,μ_1), (Ω_2,μ_2)$ are arbitrary measure spaces. When $q=1$, this implies that the space $(X_0,X_1)_{θ,\infty}$ ($0<θ<1$) can be identified with a certain space of operators. We also give an extension of the Varopoulos Lemma to pairs (or finite families) of conditional …
We give a set of test functions for the interpolation problem on $H_1^\infty$ , the constrained interpolation problem studied by Davidson, Paulsen, Raghupathi and Singh. We show that this set of test functions is minimal.
Let (A_0,A_1) be a compatible couple of Banach spaces in the interpolation theory sense. We give a formula for the K_t-functional of the interpolation couples (l_1(A_0),c_0(A_1)) or (l_1(A_0),l_infinity(A_1)) and (L_1(A_0),L_infinity(A_1)).
Given a finite set σof the unit disc \mathbb{D}=\{z\in\mathbb{C}:,\,| z|<1\} and a holomorphic function f in \mathbb{D} which belongs to a class X, we are looking for a function g in another class Y (smaller than X) which minimizes the norm ||g||_{Y} among all functions g such that g_{|σ}=f_{|σ}. For Y=H^{\infty}, and for the corresponding interpolation constant c(σ,\, X,\, H^{\infty}), we show that c(σ,\, X,\, H^{\infty})\leq aϕ_{X}(1-\frac{1-r}{n}) where n=#σ, r=max_{λ\inσ}|λ| and where ϕ_{X}(t) stands for the norm of the evaluation functional f\mapsto f(λ) on the space X. The upper bound is…
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-called $K(p,q)$-monotonicity for some $0<q\leq p\leq 1$, which allows us to get their description by the real $K$-method. Moreover, we obtain a refined version of the $K$-divisibility property for Banach lattice couples and then prove an appropriate version of this proper…
Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into "good" and "bad" parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(\cdot)}(\mathbb R^n)$ and the space $L^{\infty}(\mathbb R^n)$: \begin{equation*} (H^{p(\cdot)}(\mathbb R^n),L^{\infty}(\mathbb R^n))_{θ,\infty} =W\!H^{p(\cdot)/(1-θ)}(\mathbb R^n),\quad θ\in(0,1), \end{equation*} where $W\!H^{p(\cdot)…
We study the relationship between exact interpolation spaces for positive, linear operators, for order preserving, Lipschitz continuous operators, and for positive Gagliardo-Peetre operators, and exact partially $K$-monotone spaces in interpolation couples of compatible Banach lattices. By general Banach lattice theory we recover a characterisation of exact interpolation spaces for order preserving, Lipschitz continuous operators in the couple $(L^1,L^\infty )$ due to Bénilan and Crandall.
Recent results of A. Lerner concerning certain properties of the Fefferman-Stein maximal function are applied to show that $(\BMO, X)_θ= X^θ$, $0 < θ< 1$, for a Banach lattice $X$ of measurable functions on $\mathbb R^n$ satisfying the Fatou property such that $X$ has order continuous norm and the Hardy-Littlewood maximal operator $M$ is bounded in $(X^α)'$ for some $0 < α\leqslant 1$.
Let ${\mathcal M}$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $τ$. We show that the symmetrically $Δ$-normed operator space $E({\mathcal M},τ)$ corresponding to an arbitrary symmetrically $Δ$-normed function space $E(0,\infty)$ is an interpolation space between $L_0({\mathcal M},τ)$ and ${\mathcal M}$, which is in contrast with the classical result that there exist symmetric operator spaces $E({\mathcal M},τ)$ which are not interpolation spaces between $L_1({\mathcal M},τ)$ and ${\mathcal M}$. Besides, we show that the ${\mathcal K}$-functional of every $X\in L…
Let $\mathcal{M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal{M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\M$. For $0<p \leq\infty$, let $\h_p^c(\mathcal{M})$ denote the noncommutative column conditioned martingale Hardy space associated with the filtration $(\mathcal{M}_n)_{n\geq 1}$ and the index $p$. We prove that for $0<p<\infty$, the compatible couple $\big(\h_p^c(\mathcal{M}), \h_\infty^c(\mathcal{M})\big)$ is $K$-closed in the couple $\big(L_p(\mathcal{N}), L_\infty(\mathcal{N}) \big)$ for an appropriate amplified semifinite von …
Given an inner function $θ$, the associated star-invariant subspace $K^\infty_θ$ is formed by the functions $f\in H^\infty$ that annihilate (with respect to the usual pairing) the shift-invariant subspace $θH^1$ of the Hardy space $H^1$. Assuming that $B$ is an interpolating Blaschke product with zeros $\{a_j\}$, we characterize the traces of functions from $K^\infty_B$ on the sequence $\{a_j\}$. The trace space that arises is, in general, non-ideal (i.e., the sequences $\{w_j\}$ belonging to it admit no nice description in terms of the size of $|w_j|$), but we do point out explicit -- and sha…
We prove novel results on interpolation of Fredholm operators including an abstract factorization theorem. The main result of this paper provides sufficient conditions on the parameters $θ\in (0,1)$ and $q\in \lbrack 1,\infty ]$ under which an operator $A$ is a Fredholm operator from the real interpolation space $(X_{0},X_{1})_{θ,q}$ to $(Y_{0},Y_{1})_{θ,q} $ for a given operator $A\colon (X_{0},X_{1})\rightarrow (Y_{0},Y_{1})$ between compatible pairs of Banach spaces such that its restrictions to the endpoint spaces are Fredholm operators. These conditions are expressed in terms of the corre…
We investigate the properties of a class of weighted vector-valued $L_p$-spaces and the corresponding (an)isotropic Sobolev-Slobodetskii spaces. These spaces arise naturally in the context of maximal $L_p$-regularity for parabolic initial-boundary value problems. Our main tools are operators with a bounded $\calH^\infty$-calculus, interpolation theory, and operator sums.
We prove that under the extended Carleson's condition, a sequence $(x_n) \subset B_H$ is linear interpolating for $H^{\infty}(B_H)$ for an infinite dimensional Hilbert space H. In particular, we construct the interpolating functions for each sequence and find a bound for the constant of interpolation.
We consider linear operators $T$ mapping a couple of weighted $L_p$ spaces $\{L_{p_0}(U_0), L_{p_1}(U_1)\}$ into $\{L_{q_0}(V_0), L_{q_1}(V_1)\}$ for any $1\le p_0$, $ p_1$, $q_0$, $q_1\le\infty$, and describe the interpolation orbit of any $a\in L_{p_0}(U_0)+ L_{p_1}(U_1)$ that is we describe a space of all $\{Ta\}$, where $T$ runs over all linear bounded mappings from $\{L_{p_0}(U_0), L_{p_1}(U_1)\}$ into $\{L_{q_0}(V_0),L_{q_1}(V_1)\}$. We present in this paper the proofs of results which were announced in V.I.Ovchinnikov, C. R. Acad. Sci. Paris Ser. I 334 (2002) 881--884. We show that inte…
In this paper, we present the complex interpolation of Besov and Triebel-Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel-Lizorkin spaces. An application, we obtain the complex interpolation between the weighted Triebel-Lizorkin spaces $\dot{F}_{p_{0},q_{0}}^{s_{0}}(ω_{0})$ and $\dot{F}_{\infty ,q_{1}}^{s_{1}}(ω_{1})\ $with suitable assumptions on the parameters $\ s_{0},s_{1},p_{0},$ $q_{0}$ and$\ q_{1}$, and the pair of weights $(ω_{0},ω_{1})$.
Motivated by recent applications of weighted norm inequalities to maximal regularity of first and second order Cauchy problems, we study real interpolation spaces on the basis of general Banach function spaces and, in particular, weighted rearrangement invariant Banach function spaces. We show equivalence of the trace method and the $K$-method, identify real interpolation spaces between a Banach space and the domain of a sectorial operator, and reprove an extension of Dore's theorem on the boundedness of $H^\infty$-functional calculus to this general setting.
The validity of the von-Neumann inequality for commuting $n$ - tuples of $3\times 3$ matrices remains open for $n\geq 3$. We give a partial answer to this question, which is used to obtain a necessary condition for the Carathéodory-Fejér interpolation problem on the polydisc $\mathbb D^n.$ In the special case of $n=2$ (which follows from Ando's theorem as well), this necessary condition is made explicit. An alternative approach to the Carathéodory-Fejér interpolation problem, in the special case of $n=2,$ adapting a theorem of Korányi and Pukánzsky is given. As a consequence, a class of polyno…
Let $\vec{X}=(X_0, X_1)$ be a compatible couple of Banach spaces, $ 1\le p \le \infty$ and let $ φ$ be positive quasi-concave function. Denote by $\overline{X}_{φ,p}=(X_0,X_1)_{φ,p}$ the real interpolation spaces defined by S. Janson (1981). We give necessary and sufficient conditions on $ φ_{0}$, $φ_{1}$ and $φ$ for the validity of \begin{equation*} \left(\overline{X}_{φ_{0},1},\overline{X}_{φ_{1},1} \right) _{φ,p}= \left(\overline{X}_{φ_{0},\infty},\overline{X}_{φ_{1},\infty}\right)_{φ,p} \end{equation*} for all $ 1\le p\le \infty$, and all Banach couples $\overline{X}. $
In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz-Zygmund spaces or more generally $GΓ$-spaces. As a direct consequence of our results any Lorentz-Zygmund space $L^{a,r}({\rm Log}\, L)^β$, is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that $ 1<a<\infty, β\not= 0$. The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm.
We prove that an arbitrary Banach couple is uniquely determined by the collection of intermediate spaces that are interpolation spaces for the operators of rank one. From this we deduce a confirmation to the conjecture by Yu. A. Brudnyi and N.Ya. Kruglyak that a Banach couple is uniquely determined by the collection of all its interpolation spaces. Some relevant problems are also analized.
Given a constant $q\in(1,\infty)$, we study the following property of a normed sequence space $E$: ===================== If $\left\{ x_{n}\right\}_{n\in\mathbb{N}}$ is an element of $E$ and if $\left\{ y_{n}\right\}_{n\in\mathbb{N}}$ is an element of $\ell^{q}$ such that $\sum_{n=1}^{\infty}\left|x_{n}\right|^{q}=\sum_{n=1}^\infty \left|y_{n}\right|^{q}$ and if the nonincreasing rearrangements of these two sequences satisfy $\sum_{n=1}^{N}\left|x_{n}^{*}\right|^{q}\le\sum_{n=1}^{N}\left|y_{n}^{*}\right|^{q}$ for all $N\in\mathbb{N}$, then $\left\{ y_{n}\right\}_{n\in\mathbb{N}}\in E$ and $\lef…
Let $\mathcal{M}$ be a semifinite von Neumann algebra. We equip the associated noncommutative $L_p$-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for $1<p<\infty$ let $$L_{p,p}(\mathcal{M})=\big(L_{\infty}(\mathcal{M}),\,L_{1}(\mathcal{M})\big)_{\frac1p,\,p}$$ be equipped with the operator space structure via real interpolation as defined by the second named author ({\em J. Funct. Anal}. 139 (1996), 500--539). We show that $L_{p,p}(\mathcal{M})=L_{p}(\mathcal{M})$ completely isomorphically if and only if $\mathcal{M}$ is f…
On any complete Riemannian manifold $M$ and for all $p\in [2,\infty)$, we prove a family of second order $L^{p}$-interpolation inequalities that arise from the following simple $L^{p}$-estimate valid for every $u \in C^{\infty}(M)$: $$ \|\nabla u\|_{p}^p \leq \|u Δ_{p} u\|_1\in [0,\infty], $$ where $Δ_p$ denotes the $p$-Laplace operator. We show that these inequalities, in combination with abstract functional analytic arguments, allow to establish new global Sobolev regularity results for $L^p$-solutions of the Poisson equation for all $p\in (1,\infty)$, and new global Sobolev regularity resul…
Suppose that (A_0,A_1) and (B_0,B_1) are Banach couples, and that T is a linear operator which maps A_0 compactly into B_0 and A_1 boundedly (or even compactly) into B_1. Does this imply that T maps [A_0,A_1]_s to [B_0,B_1]_s compactly for 0<s<1 ? (Here, as usual, [A_0,A_1]_s denotes the complex interpolation space of Alberto Calderon.) This question has been open for 44 years. Affirmative answers are known for it in many special cases. We answer it affirmatively in the case where (A_0,A_1) is arbitrary and (B_0,B_1) is a couple of Banach lattices having absolutely continuous norms or the Fato…
Let $0<p<q\leq\infty$ and $α\in (0,\infty]$. We give a characterization of quasi-Banach interpolation spaces for the couple $(L_p(0,α),L_q(0,α))$ in terms of two monotonicity properties, extending known results which mainly dealt with Banach spaces. This enables us to recover recent results of Cwikel and Nilsson on sequence spaces and to solve a conjecture of Levitina, Sukochev and Zanin in the setting of function spaces. We apply the results obtained to characterize symmetric spaces in which the standard forms of the noncommutative Khintchine inequalities hold.
We generalize Abrahamse's interpolation theorem from the setting of a multiply connected domain to that of a more general Riemann surface. Our main result provides the scalar-valued interpolation theorem for the fixed-point subalgebra of $H^\infty$ associated to the action of a Fuchsian group. We rely on two results from a paper of Forelli. This allows us to prove the interpolation result using duality techniques that parallel Sarason's approach to the interpolation problem for $H^\infty$. In this process we prove a more general distance formula, very much like Nehari's theorem, and obtain rel…
We continue an investigation started in a preceding paper. We discuss the classical results of Carleson connecting Carleson measures with the $\d$-equation in a slightly more abstract framework than usual. We also consider a more recent result of Peter Jones which shows the existence of a solution of the $\d$-equation, which satisfies simultaneously a good $L_ı$ estimate and a good $L_1$ estimate. This appears as a special case of our main result which can be stated as follows: Let $(Ω,\cal{A},μ)$ be any measure space. Consider a bounded operator $u:H^1\ra L_1(μ)$. Assume that on one hand $u$ …
We explicitly describe all Hilbert function spaces that are interpolation spaces with respect to a given couple of Sobolev inner product spaces considered over $\mathbb{R}^{n}$ or a half-space in $\mathbb{R}^{n}$ or a bounded Euclidean domain with Lipschitz boundary. We prove that these interpolation spaces form a subclass of isotropic Hörmander spaces. They are parametrized with a radial function parameter which is OR-varying at $+\infty$ and satisfies some additional conditions. We give explicit examples of intermediate but not interpolation spaces.
We prove that the $Z$-spaces $Z^{p,q}_s$ form a complex interpolation scale for all $0 < p,q \leq \infty$ and $s \in \mathbb{R}$, filling a gap in recent work with Pascal Auscher.
Diagrams generated by three interpolators in an abstract Kalton-Montgomery complex like interpolation scheme. We will consider in detail the case of the first three Schechter interpolators associated to the usual Calderón complex interpolation method in two especially interesting cases: weighted $\ell_2$ spaces, i.e., interpolation pairs $(\ell_2(w^{-1}), \ell_2(w))_θ$, and $\ell_p$ spaces, i.e., the interpolation pair $(\ell_\infty, \ell_1)_θ$, both at $θ=1/2$.
Given a von Neumann algebra $M$ and a $W^{\ast}$-correspondence $E$ over $M$, we construct an algebra $H^{\infty}(E)$ that we call the Hardy algebra of $E$. When $M=\mathbb{C}=E$, then $H^{\infty}(E)$ is the classical Hardy space $H^{\infty}(\mathbb{T})$ of bounded analytic functions on the unit disc. We show that given any faithful normal representation $σ$ of $M$ on a Hilbert space $H$ there is a natural correspondence $E^σ$ over the commutant $σ(M)^{\prime}$, called the $σ$-dual of $E$, and that $H^{\infty}(E)$ can be realized in terms of ($B(H)$-valued) functions on the open unit ball $\ma…
The connection between the standard $H^\infty$-problem in control theory and Nevanlinna-Pick interpolation in operator theory was established in the 1980s, and has led to a fruitful cross-pollination between the two fields since. In the meantime, research in $H^\infty$-control theory has moved on to the study of robust control for systems with structured uncertainties and to various types of multidimensional systems, while Nevanlinna-Pick interpolation theory has moved on independently to a variety of multivariable settings. Here we review these developments and indicate the precise connection…
Let $\mathcal{M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal{M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal{M}$. For $0<p <\infty$, let $\mathsf{h}_p^c(\mathcal{M})$ denote the noncommutative column conditioned martingale Hardy space and $\bmo^c(\M)$ denote the column \lq\lq little\rq\rq \ martingale BMO space associated with the filtration $(\mathcal{M}_n)_{n\geq 1}$. We prove the following real interpolation identity: if $0<p <\infty$ and $0<θ<1$, then for $1/r=(1-θ)/p$, \[ \big(\mathsf{h}_p^c(\mathcal{M}), \bmo^c(\mathcal{M})\…
A number of Calderón-Mityagin couples and relative Calderón-Mityagin pairs are identified among Banach function spaces defined in terms of the least decreasing majorant construction on the half line. The interpolation structure of such spaces is shown to closely parallel that of the rearrangement invariant spaces, and it is proved that a couple of these spaces is a Calderón-Mityagin couple if and only if the corresponding couple of rearrangement invariant spaces is a Calderón-Mityagin couple. Consequently, the class of all interpolation spaces for any couple of spaces of this type admits a com…
We prove noncommutative Khintchine inequalities for all interpolation spaces between $L_p$ and $L_2$ with $p<2$. In particular, it follows that Khintchine inequalities hold in $L_{1,\infty}$. Using a similar method, we find a new deterministic equivalent for the $RC$-norm in all interpolation spaces between $L_p$-spaces which unifies the cases $p > 2$ and $p < 2$. It produces a new proof of Khintchine inequalities for $p<1$ for free variables. To complete the picture, we exhibit counter-examples which show that neither of the usual closed formulas for Khintchine inequalities can work in $L_{2,…
Let $1\le p<q\le\infty$ and let $T$ be a subadditive operator acting on $L^p$ and $L^q$. We prove that $T$ is bounded on the Orlicz space $L^ϕ$, where $ϕ^{-1}(u)=u^{1/p}ρ(u^{1/q-1/p})$ for some concave function $ρ$ and \[ \|T\|_{L^ϕ\to L^ϕ}\le C\max\{\|T\|_{L^p\to L^p},\|T\|_{L^q\to L^q}\}. \] The interpolation constant $C$, in general, is less than 4 and, in many cases, we can give much better estimates for $C$. In particular, if $p=1$ and $q=\infty$, then the classical Orlicz interpolation theorem holds for subadditive operators with the interpolation constant C=1. These results generalize o…
Let $(X, Y)$ be a suitable couple of quasi-Banach lattices of measurable functions on $\mathbb T \times Ω$, and let $(X_A, Y_A)$ be the couple of the corresponding Hardy-type spaces. It has long been suspected that the BMO-regularity property of $(X, Y)$ is not only sufficient for the $\mathrm K$-closedness of $(X_A, Y_A)$ in $(X, Y)$ but also necessary. We establish the equivalence of these two properties for a general couple of Banach lattices having the Fatou property when $Ω$ is a discrete measurable space, and also for couples $(X, Y)$ where $X$ is allowed to be quasi-Banach but $Y$ is as…
M. Daher [9] showed that if $(X_0, X_1)$ is a regular couple of uniformly convex spaces then the unit spheres of the complex interpolation spaces $X_θ$ and $X_η$ are uniformly homeomorphic for every $0 < θ, η< 1$. We show that this is a rather general phenomenon of interpolation methods described by the discrete framework of interpolation of [15].
