# Questions tagged [interpolation-spaces]

The interpolation-spaces tag has no usage guidance.

55
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### Interpolation between projective and injective spaces

Suppose $(\Omega,\mu)$ be a $\sigma$-finite measure space. Suppose $X$ is a Banach space and $L_p(\Omega;X)$ be the corresponding Bochner space for $0<p\leq\infty.$ Is it true that the complex ...

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### Interpolation between (or: simultaneous Whitney extension for) $C^\alpha$ and $C^{1,\gamma}$ on a Lipschitz domain

I would like to know whether for a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ (in the weak Lipschitz, so a "Lipschitz manifold", sense, not necessarily a Lipschitz graph domain), ...

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### Real interpolation for vector-valued Sobolev spaces

I am having trouble finding any results concerning real interpolation of vector-valued Sobolev spaces. Namely, I would like to know if a continuous embedding of the type,
$$
L^p(0,T;X_1)\cap W^{1,p}(0,...

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### Aronszajn-Gagliardo Theorem: case of exponent $\theta$ (Exercise 2.8.4 from Bergh-Löfström)

I'm working on the book "Interpolation Spaces" by Bergh and Löfström. I'm interested in solving the exercise 2.8.4 on page 33. which asks:
Let $A$ be an interpolation space of exponent $\...

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### Motivation for considering the J and K-functionals of real interpolation

In interpolation theory, given a compatible couple of Banach spaces $(X_0, X_1)$ one considers the $J$ and $K$-functionals, defined as follows:
If $x \in X_0 + X_1$ and $t > 0$ then
$$K(t, x) = \...

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### Extreme case of K-interpolation

Suppose $X_0$ and $X_1$ are Banach spaces living in a larger Banach space
$X$. The $K$-functional is defined for each $f\in X_0+X_1$ and $t>0$ as
$$K(f,t,X_0,X_1)=\inf\{\|f_0\|_{X_0}+t\|f_1\|_{X_1}:...

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### Interpolation of normed spaces *vs* geometrical mean of positive matrices

Two $n\times n$ positive definite symetric matrices $A,B$ define two normed spaces $E_A=({\mathbb R}^n;\|\cdot\|_A)$ and $E_B=({\mathbb R}^n;\|\cdot\|_B)$, where
$$\|x\|_A=\sqrt{x^TAx},\qquad \|x\|_B=\...

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### Intersection of the kernel with the interpolation space

$\DeclareMathOperator\Ker{Ker}$Given two Banach spaces $X$ and $Y$ with a continuous inclusion $X\subset Y$, and another couple $X’ \subset Y’$ with the same properties. Take $f : Y \longrightarrow ...

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### Interpolation of Sobolev/Besov spaces in the limiting case q = ∞

I'm interested in the interpolation space ($1\le p_0,p_1\le\infty$, $0<\theta<1$)
$$
X=(L_{p_0}(0;1),W^1_{p_1}(0,1))_{\theta,q}\quad\text{with}\quad q=\infty\ \ \text{and}\ \ p_0\ne p_1 .
$$
It ...

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### Interpolation spaces defined by singular value decomposition

Let $ X $ and $Y$ be Hilbert space, $A:X \to Y $ compact and injective, $(\sigma_n;v_n,u_n)$ be its singular value decomposition, that is,
$$
Av_n = \sigma_n u_n \\
A^* u_n = \sigma_n v_n
$$
Since $\...

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128
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### Interpolation of embedded Hilbert spaces and intersection

I'm wondering under what hypothesis it is true a property like
$$[\mathcal{H}_1\cap X, \mathcal{H}_2\cap X]_{\theta}=\mathcal{H}_1\cap X\cap [\mathcal{H}_1, \mathcal{H_2}]_{\theta}$$
where $\mathcal{H}...

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231
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### Trace of a function

Let $T,L> 0$ two real numbers and we consider the Sobolev space $X := L^2(0,T; H^1(0,L))\cap H^{1}(0,T;H^{-1}(0,L))$. My question is:
Given $f \in X$, the trace $ t \mapsto f(t,L)$ belongs to what ...

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### Estimate involving Besov norm

When reading some old notes of my advisor on interpolation spaces, I bumped into a problem I can't quite wrap my head around. Here are the details.
For $p\in(0,\infty)$ a $p$-variation semi-norm of a ...

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### Vector-valued interpolation for sublinear operators

Grafakos in his $\textit{Classical Fourier Analysis}$ formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem.
$\textbf{Theorem}$
Let $1\...

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### Besov or Triebel-Lizorkin spaces versus Lorentz spaces

I first asked this question on math.stackexchange here but it seems it is more a research level question ...
At the $0$ order of derivatives of Sobolev spaces and for a fixed integrability order $p$, ...

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378
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### Dual space of the intersection of locally convex vector spaces

Let $S \neq \emptyset$ and let $\big((E_s,\mathcal{T}_s)\big)_{s \in S}$ be a family of locally convex vector subspaces of the same vector space. Denote by $E_s^*$ the dual space of $(E_s,\mathcal{T}...

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### Interpolation of product spaces

Suppose that $X_{\theta}$ is an interpolation space between the Banach spaces $X_0$ and $X_1$. Let $\mathcal{B}$ be another Banach space.
Is it true that $X_{\theta}\times\mathcal{B}$ is an ...

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### Interpolation for Sobolev spaces

How one can identify the following (complex) interpolation space
$$E_\theta :=[L^2(\Omega), H^2(\Omega)\cap H_0^1(\Omega)]_\theta,$$
where $\Omega$ is a regular domaine. After research, it seems that ...

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### Equivalent definition of real interpolation space

In one of his papers (On the Nash-Moser implicit function theorem, Ann. Acad. Sci. Fenn., Ser. A I, Math. 10, 255-259 (1985).), Lars Hörmander introduces a class of function spaces that seems related ...

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### Interpolation of $L^p$ spaces

Let $\Omega_x$ and $\Omega_y$ be sets of finite Lebesgue measure.
We can then look at the space $X_1:=L^2(\Omega_x \times \Omega_y).$
This space is contained in the larger space
$$X_0:=L^2(\...

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84
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### Extrapolate an Interpolation scale

Suppose $X$ and $Y$ are real Banach spaces with a continuous embedding $X\subset Y$. For given $0<\theta<1$ I am interested in constructing using the norms of $X$ and $Y$ a (Quasi-) Banach ...

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233
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### Relation between two different functionals: $\lVert p^{-\max}_{-\varepsilon}\rVert$ and $\kappa_{p}^{-1}(1-\varepsilon)$

Given a non-negative sequence $p=(p_i)_{i\in\mathbb{N}}\in \ell_1$ such that $\lVert p\rVert_1 = 1$,we define the two following quantities, for every $\varepsilon \in (0,1]$.
Assuming, without loss ...

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127
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### Dual Lorentz spaces

MO seems the perfect place to ask for the following question. Denote the Lorentz spaces on an arbitrary measure space $(E,\mu)$ by $L^{p,q}=L^{p,q}(E,\mu)$, and by $p'$ the conjugate index of $p$.
...

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### Interpolation of some Sobolev spaces

Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$
I am wondering what is
$$(X_0,X_2)_{1/2,2}=?$$
Would it be $H^2_0(0,...

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### Interpolation of some Lebesgue spaces

When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that
$$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...

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### Interpolation inequality related to the 5/3-Laplace operator

I'm having trouble with an estimate that would be helpful in information geometry.
The background is the following. Suppose we have a smooth positive function $g:X \to \mathbb{R}^+$ where $X$ is a ...

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### Relation between a norm and norm of Besov spaces

Let $(H, \|\cdot\|)$ be a Hilbert space, $A \colon D(A)\subset H \longrightarrow H$ generates an analytic semigroup $T(t)$ on $H$. We define the following Banach space with the respect norm
$$F=\{x\in ...

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### Interpolation space between $L^1\cap L^2$ and $L^1$

In the paper of Bourgain, the way equation (3.78) is deduced from (3.69) and (3.76) seems via the following interpolation result. Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces and let $T$ be a ...

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### Discrete Lions Peetre interpolation

In S. Heinrich's "Closed operator ideals and interpolation," Heinrich describes two equivalent descriptions of Lions-Peetre interpolation space $(X,Y)_{\theta, p}$ for $0<\theta<1$ and $1\...

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### What is interpolation between $L^2(\Omega)$ and $\mathring{H^2}(\Omega)$ when $\Omega$ is not smooth

Let $\mathring{H}^\theta$ be the closure of $\mathcal{D}(\Omega)$ under the norm $\mathring{H}^\theta$.
It is well-known that
$$
[L^2(\Omega), \mathring{H^{2}}(\Omega)]_{\theta}=\mathring{H^{2\theta}}...

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### Some questions on parabolic function spaces

I remember I read those problems some place, but I cannot find it. Does anyone have any idea where I can find it?
If $X$ is a Banach space, then $(L^1(a,b;X))^*\cong L^\infty(a,b;X^*)$?
$X, Y$ are ...

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### Interpolation spaces

In this paper, the authors claim that for $s\in [0,1]$, $\left [ H_0^1(\Omega), L^2(\Omega)\right ]_\theta=\text{dom}(-\Delta)^{\frac s 2}$, where $\Omega$ is a smooth and bounded domain in $\mathbb R^...

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### A modern reference for the "Intermediate Derivatives Theorem"

In the book Non-Homogeneous Boundary Value Problems and Applications I by Lions and Magenes, the Intermediate Derivative Theorem is stated as follows:
Intermediate Derivative Theorem: Let $X\subset ...

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### "Reversion" of class $J(\theta)$ interpolation property for Besov spaces

In (function space) interpolation theory, a Banach space $E$ is of class $J(\theta)$ (for $0 < \theta < 1$) if $$X \cap Y \subseteq E \subseteq X+Y,$$ where $(X,Y)$ are Banach spaces and form an ...

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### Do intermediate spaces imply the information about interpolation spaces couple?

Let $X,Y,Z$ be Banach spaces such that $X,Y\hookrightarrow Z$. We know that if $X\Subset Y$ (The symbol "$\hookrightarrow $" means continuous embedding), then $(Z,X)_{\theta,p}\hookrightarrow (Z,Y)...

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### Trace theorem for boundary value problem

Consider the inhomogeneous boundary value problem on the infinite strip $(x,y)\in \mathbb{R}\times [0,1]$ defined by
$$\begin{cases}\partial_{x}u + \partial_{y}v=f & {(x,y)\in \mathbb{R}\times (0,...

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### Interpolation space of mixed Lorentz-Lebesgue spaces

Is it true that the real interpolation space
$$ (L^{q_0}_{r_0} L^{p_0}, L^{q_1}_{r_1} L^{p_1})_{\theta,q} = L^q L^p_q $$
with
$$ \frac{1}{q} = \frac{1-\theta}{q_0} + \frac{\theta}{q_1}, \text{ and } \...

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### Interpolation in Sobolev spaces

Let $H^s$, $0\leq s<\infty$ be the $L^2$ based Sobolev spaces such that
$$
\hat{f}(\xi)(1+|\xi|^2)^{s/2} \in L^2.
$$
Let $r_1,r_2,p_1,p_2>0$ be given parameters. Assume that a linear operator $...

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### Quotients in complex interpolation of Banach spaces

Let $(X_0,X_1)$ be an admissible pair of complex Banach spaces with $X_0$ continuously embedded in $X_1$. For $0<\theta<1$, let us denote by $X_\theta =(X_0,X_1)_\theta$ the complex ...

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### An interpolation estimate involving fractional derivatives

I have come across various instances of interpolation estimates of the following type:
$ \| f \|_{L^r(\mathbb{R^n})} \lesssim \| |\nabla|^{-s_1} f \|_{L^{r_1}(\mathbb{R^n})}^\theta \| |\nabla|^{s_2} ...

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### Properties of the optimal decomposition for the $K$-functional between $\ell_1$ and $\ell_2$

Background: For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$:
$$
\lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\...

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### Characterization of the interpolation space $(X,D(A^\alpha))_{\theta,p}$ with semigroup $A$ generates?

Let $X$ be a Banach space (could work for over $\mathbb{R}$ as well?)
Let $A\colon D(A)\subset X\to X$ be a sectorial operator, and $e^{tA}$ be the semigroup generated by $A$.
It is well-known that ...

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### Interpolation inequality for fractional Sobolev spaces

In Theorem 5.2 of the book
Robert A. Adams and John J. F. Fournier, MR 2424078 Sobolev spaces, ISBN: 0-12-044143-8.
is stated that, for any integers $0 \leq j \leq m$ and $u \in W^{m,p}(\Omega)$ (...

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### Interpolation between $L_1^0$ and $L_2^0$

Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...

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### Real interpolation of weighted Sobolev spaces with different weights

Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...

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### Interpolation functional for BV spaces?

Recently, while trying to tackle a problem, I found that it would be convenient that I could find some sort of 'interpolation theorem' for Bounded Variation Spaces. More specifically, let's define, ...

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### The real method of interpolation and operator ideals

Let $\overline{A} \mbox{ and } \overline{B}$ be n+1-tuples of Banach spaces and $T:\overline{A}\rightarrow \overline{B}$ be an interpolation operator; let $J(\overline{A})$ and (the corresponding, ...

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### What is $(L^2(M), H^1_0(M))_{\frac 12}$ on a smooth manifold with boundary?

Let $M$ be a smooth compact manifold. If $M$ is closed, we have that the interpolation space
$$(L^2(M), H^1(M))_{\frac 12}=H^{\frac 12}(M)$$
(see Taylor's book on PDE for example). Suppose $M$ has a ...

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### Traces of fractional Sobolev spaces $W^{s,p}$ with $0<s<1/p$

I've stumbled upon a problem involving the trace of a function in a fractional Sobolev space of the form $W^{s,2}(H)$, where $H$ is a half-plane in $\mathbb{R}^2$. Would it be possible to define a ...

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### Interpolation of the row and column operator spaces

If $R$ and $C$ are respectively the row and column operator spaces, and $\theta \in (0, 1)$, we denote by $R(\theta)$ the interpolation operator space $(R, C)_{\theta}$ (with $R(0) = R$ and $R(1) = C$)...