The K_t-functional for the interpolation couple L_1(A_0),L_infinity(A_1)

Complex interpolation of compact operators mapping into lattice couples

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…

The periodic complex method in interpolation spaces

We discuss a question which relates to Calderon's complex interpolation method. More precisely, we will consider the so-called "periodic" complex interpolation method, studied by Peetre. (Which also corresponds to the spaces obtained by Calderon's construction using Banach space valued analytic functions, but defined on an annulus instead of the strip used by Calderon.) Cwikel showed that, using functions with a given period iλin the complex method mechanism, one obtains the same interpolation spaces as in the original version of the complex method, up to equivalence of norms. He also showed t…

Calderón-Mityagin couples of Banach spaces related to decreasing functions

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…

On K-closedness, BMO-regularity and real interpolation of Hardy-type spaces

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…

Uniform homeomorphisms between spheres induced by interpolation methods

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].

Real interpolation of Hardy-type spaces and BMO-regularity

Let $(X, Y)$ be a couple of quasi-Banach lattices of measurable functions on $\mathbb T \times Ω$ satisfying some additional assumptions. The K-closedness of a couple of Hardy-type spaces $(X_A, Y_A)$ in $(X, Y)$ and the stability of the real interpolation $(X_A, Y_A)_{θ, p} = (X_A + Y_A) \cap (X, Y)_{θ, p}$ are shown to be equivalent to each other and to the BMO-regularity of the associated lattices $\left(L_1, \left(X^r\right)' Y^r\right)_{δ, q}$. The inclusion $\left(X^{1 - θ} Y^θ\right)_A \subset \left(X_A, Y_A \right)_{θ, \infty}$ is also characterized in these therms. New examples of cou…

Simplicity of reduced group Banach algebras

Let G be a discrete group. Suppose that the reduced group C*-algebra of G is simple. We use results of Kalantar-Kennedy and Haagerup, and Banach space interpolation, to prove that, for p in (1,infinity), the reduced group L^p operator algebra F^p_r(G) and its *-analog B^{p,*}_r(G) are simple. If G is countable, we prove that the Banach algebras generated by the left regular representations on reflexive Orlicz sequence spaces and certain Lorentz sequence spaces are also simple. We prove analogous results with simplicity replaced by the unique trace property. For use in the Orlicz sequence space…

Convexity in the interpolation spaces

In this work we study if the norms rotund, uniformly rotund, weakly uniformly rotund, locally uniformly rotund or weakly locally uniformly rotund interpolate in the complex or the real interpolation spaces. We will see that the properties uniformly rotund and weakly uniformly rotund interpolate, but the properties locally uniformly rotund or weakly locally uniformly rotund do not interpolate in general except for the dual interpolation couple.

Duality in a stability problem for some functionals arising in interpolation theory

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.

Complex interpolation of Orlicz sequence spaces and its higher order Rochberg spaces

We show that if $(\ell_{ϕ_0}, \ell_{ϕ_1})$ is a couple of suitable Orlicz sequence spaces then the corresponding Rochberg derived spaces of all orders associated to the complex interpolation method are Fenchel-Orlicz spaces. In particular, the induced twisted sums have the $(C[0, 1], \mathbb{C})$-extension property.

The k_t--functional for the interpolation couple L^\infty(dμ;L^1(dν)), L^\infty(dν;L^1(dμ))

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\ve…

Parabolic problems and interpolation with a function parameter

We give an application of interpolation with a function parameter to parabolic differential operators. We introduce the refined anisotropic Sobolev scale that consists of some Hilbert function spaces of generalized smoothness. The latter is characterized by a real number and a function varying slowly at infinity in Karamata's sense. This scale is connected with anisotropic Sobolev spaces by means of interpolation with a function parameter. We investigate a general initial--boundary value parabolic problem in the refined Sobolev scale. We prove that the operator corresponding to this problem se…

Complex interpolation of compact operators mapping into the couple (FL^{\infty},FL_{1}^{\infty})

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…

广义光滑性的插值空间及其在椭圆方程中的应用

Interpolation spaces of generalized smoothness and their applications to elliptic equations

本文通过插值方法构造以O-正则变化函数为正则性阶数的广义光滑性分布空间，推广经典Sobolev、Nikolskii-Besov和Triebel-Lizorkin空间，并将其应用于光滑流形上的椭圆方程与椭圆问题。

Arazy-Cwikel property for quasi-Banach couples

The main result of this paper establishes that the known Arazy-Cwikel property holds for classes of uniformly K-monotone spaces in the quasi-Banach setting provided that the initial couple is mutually closed. As a consequence, we get that the class of all quasi-Banach K-spaces (i.e., interpolation spaces which are described by the real K-method) with respect to an arbitrary mutually closed Banach couple enjoys the Arazy-Cwikel property. Another consequence complements some previous results by Bykov and Ovchinnikov, showing that this property holds also for the class of all interpolation quasi-…

Interpolation Hilbert spaces between Sobolev spaces

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.

Complemented subspaces of spaces obtained by interpolation

If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A_0,A_1) such that A_0 and A_1 are isometric to $X\oplus V$, and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype, having the Radon-Nikodym property, and having the analytic unconditional martingale difference sequence property, do not pass to intermediate spaces.

Two remarks on the interpolation space

Dans ce travail, on montre que $(M(\mathbb{T}),c_0(\mathbb{Z}))_θ= (L^1,c_0(\mathbb{Z}))_θ$, $0<θ<1$. Dans la suite on montre pour le couple d'interpolation $(C_0,C_1)$ trouvé par Garling-Smith qu'il existe un isomorphisme $U_θ: (C_0,C_0+C_1)_{θ,p}\rightarrow (C_1,C_0+C_1)_{θ, p}$ (resp. $U_θ: (C_0,C_0+C_1)_θ\rightarrow (C_1,C_0+C_1)_θ)$ tel que sa restriction à $C_{θ, p}$ (resp. à $C_θ)$ est un isomorphisme : $C_{θ, p} \rightarrow C_{1-θ, p}$ (resp. $C_θ\rightarrow C_{1-θ})$. -- In this work we show that $(M(\mathbb{T}),c_0(\mathbb{Z}))_θ= (L^1,c_0(\mathbb{Z}))_θ$, $0<θ<1.$ In the following w…

Une remarque sur les espaces d'interpolation faiblement localement uniformément convexes

Let $(A_0, A_1)$ be an interpolation couple, and let $B_j$ be the closure of $A_0^\ast \cap A_1^\ast$ in $A_j^\ast$, $j = 0, 1$. For every $θ\in \, ]0, 1[$, there exists a natural one to one contraction $R^θ: A^θ\rightarrow (B_0^\ast, B_1^\ast)^θ$. For some $β\in \, ]0, 1[$, the closure of $R^β(A^β)$ in $(B_0^\ast, B_1^\ast)^β$ is supposed to be weakly LUR. Then $A^θ= A_θ$ for every $θ\in \, ]0, 1[$.

Core Decreasing Functions（核递减函数）

Core Decreasing Functions

本文引入核递减函数定义Banach函数空间的下空间，推广相关构造方法研究其对偶与泛重排不变空间下空间的刻画，并证明相关下空间及其对偶偶为精确Calderón偶，给出对应精确插值空间的完整刻画。

Interpolation of compact Lipschitz operators

Let (A_0,A_1) and (B_0,B_1) be Banach couples such that A_0 is contained in A_1 and (B_0,B_1) satisfies Arne Persson's approximation condition (H). Let T:A_1 --> B_1 be a possibly nonlinear Lipschitz mapping which also maps A_0 into B_0 and satisfies the following quantitative compactnesss condition: Ta \in ||a||_{A_0} K for each a \in A_0, where K is a fixed compact subset of B_0. We show that T maps the real interpolation space (A_0,A_1)_{θ,p} compactly into its counterpart (B_0,B_1)_{θ,p} for each θ\in (0,1) and p \in [1,\infty].

Disjointly homogeneous rearrangement invariant spaces via interpolation

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 …

Interpolation of compact operators by the methods of Calderón and Gustavsson-Peetre

Let $ X=(X_0,X_1)$ and $ Y=(Y_0,Y_1)$ be Banach couples and suppose $T: X\to Y$ is a linear operator such that $T:X_0\to Y_0$ is compact. We consider the question whether the operator $T:[X_0,X_1]_θ\to [Y_0,Y_1]_θ$ is compact and show a positive answer under a variety of conditions. For example it suffices that $X_0$ be a UMD-space or that $X_0$ is reflexive and there is a Banach space so that $X_0=[W,X_1]_α$ for some $0<α<1.$

Interpolation between H^p spaces and non-commutative generalizations, I

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…

An alternative characterization of normed interpolation spaces between $\ell^{1}$ and $\ell^{q}$

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…

Some alternative definitions for the "plus-minus" interpolation spaces $\left\langle A_{0},A_{1}\right\rangle _θ$ of Jaak Peetre

The Peetre "plus-minus" interpolation spaces $\left\langle A_{0},A_{1}\right\rangle _θ$ are defined variously via conditions about the unconditional convergence of certain Banach space valued series whose terms have coefficients which are powers of 2 or, alternatively, powers of $e$. It may seem intuitively obvious that using powers of 2, or of $e$, or powers of some other constant number greater than 1 in such definitions should produce the same space to within equivalence of norms. To allay any doubts, we here offer an explicit proof of this fact, via a "continuous" definition of the same sp…

Real interpolation and transposition of certain function 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 …

P. Jones'Interpolation theorem for noncommutative martingale Hardy spaces

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 …

S-可分解 Banach 格，最优序列空间与插值

S-Decomposable Banach Lattices, Optimal Sequence Spaces and Interpolation

本文研究了 Banach 格上下估计与相对 s-可分解性的联系，通过为每个 Banach 格 X 分配两个由有限格可表示性决定的最优序列空间 XU 和 XL 进行刻画，并应用于相对 K-泛函估计的轨道分解。

Counterexamples for interpolation of compact Lipschitz operators

Let (A_0,A_1) and (B_0,B_1) be Banach couples with A_0 contained in A_1 and B_0 contained in B_1. Let T:A_1 --> B_1 be a possibly nonlinear operator which is a compact Lipschitz map of A_j into B_j for j=0,1. It is known that T maps the Lions-Peetre space (A_0,A_1)_θ,q boundedly into (B_0,B_1)_θ,q for each θin (0,1) and each q in [1,\infty), and that this map is also compact if if T is linear. We present examples which show that in general the map T:(A_0,A_1)_θ,q --> (B_0,B_1)_θ,q is not compact.

Interpolation of nonlinear maps

Let $(X_0, X_1)$ and $(Y_0, Y_1)$ be complex Banach couples and assume that $X_1\subseteq X_0$ with norms satisfying $\|x\|_{X_0} \le c\|x\|_{X_1}$ for some $c > 0$. For any $0<θ<1$, denote by $X_θ= [X_0, X_1]_θ$ and $Y_θ= [Y_0, Y_1]_θ$ the complex interpolation spaces and by $B(r, X_θ)$, $0 \le θ\le 1,$ the open ball of radius $r>0$ in $X_θ$, centered at zero. Then for any analytic map $Φ: B(r, X_0) \to Y_0+ Y_1$ such that $Φ: B(r, X_0)\to Y_0$ and $Φ: B(c^{-1}r, X_1)\to Y_1$ are continuous and bounded by constants $M_0$ and $M_1$, respectively, the restriction of $Φ$ to $B(c^{-θ}r, X_θ)$, $0…

On the stability of the Lions-Peetre method of real interpolation with functional parameter

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}. $

Besov-Morrey型空间的一些性质

Some properties of Besov-Morrey type spaces

本文研究Sobolev-Morrey和Besov-Morrey空间，证明了乘子定理、嵌入关系以及插值定理。

通过慢变函数定义的内插空间的对偶：0<q<1情形

Duality for Interpolation Spaces Defined Via Slowly Varying Functions: The Case 0<q<1

本文描述了当0<q<1且b为慢变函数时，极限实内插空间(A_0,A_1)^K_{1,q,b}的对偶空间，利用J-空间并建立了K和J空间的等价定理，推广了已知结果。

极限插值空间的对偶

Duals of limiting interpolation spaces

本文研究极限实插值 $K$-空间和 $J$-空间的对偶问题，针对参数 $\theta=0$ 且带缓变函数 $v$ 的情形建立对偶刻画，推广了 Lions 和 Peetre 在经典实插值情形下的结果。

Interpolation of quasi noncommutative $L_p$-spaces

Let $\mathcal{M}$ be a ($σ$-finite) von Neumann algebra associated with a normal faithful state $ϕ.$ We prove a complex interpolation result for a couple of two (quasi) Haagerup noncommutative $L_p$-spaces $L_{p_0} (\mathcal{M}, ϕ)$ and $L_{p_1} (\mathcal{M}, ϕ), 0< p_0 < p_1\leq \infty,$ which has further applications to the sandwiched $p$-Rényi divergence.

经典洛伦茨空间中振荡的插值

Interpolation of classical Lorentz spaces measuring oscillation

本文通过研究经典洛伦茨空间中振荡的插值问题，利用$K$-函数的显式刻画方法，推导出勒贝格空间和经典洛伦茨空间的$K$-函数公式，并建立了反向马卡乌不等式。

Majorization, Interpolation and noncommutative Khintchine inequalities

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.

关于算子非紧性测度的多线性扎弗兰定理

A multilinear Zafran theorem for the measure of noncompactness of operators

研究作用于拟巴拿赫空间乘积上算子的多线性扎弗兰定理，通过考虑拟巴拿赫对间多线性算子，依序列空间基本函数获插值估计，还研究插值多线性算子非紧性测度，得出结果并扩展了相关早期结论。

沿瓦瑟斯坦测地线的尖锐庞加莱插值

Sharp Poincaré interpolation along Wasserstein geodesics

研究沿二次瓦瑟斯坦测地线的庞加莱常数的尖锐插值不等式，设\(\mu_i\)为\(\kappa_i\)-强对数凹概率测度及最优位移插值\((\mu_t)\)，给出不等式估计并刻画等式情况，用两端点博赫纳方法证明，解决了相关奇函数问题。