# Questions tagged [interpolation-spaces]

The interpolation-spaces tag has no usage guidance.

71
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### Support function of the intersection of two $\ell_p$ balls

Denote $\|\cdot \|_p$ for the norm in $\ell_p^n$, where $1 \leq p \leq \infty$, and $n \geq 1$.
Let $(x^\star_i)$ denote a nonincreasing arrangement of the sequence $(|x_i|) \in \mathbb{R}^n$.
We ...

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### $H^{s,\infty}$ via Triebel Lizorkin spaces

I know that
$F^s_{p,2} = H^{s,p}$
for $p \in (1, \infty)$, but what about $H^{s,1}$ and $H^{s,\infty}$? I know one extension for $F^0_{\infty,2}$ which should be BMO. But how do I get the $H^{s,\infty}...

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### Proof of the equivalence between Triebel Spaces and Bessel Potential

I've encountered a question regarding the relationship between Triebel-Lizorkin spaces and Bessel potential spaces. Specifically, I understand that
$F^s_{p,2} = H^{s,p}$, for $p \in (1,\infty)$.
...

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93
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### Dual of closure

Currently I'm studying about abstract interpolation theory for my research. One of the basic ways to construct new interpolation spaces, given an interpolation space $E$ with respect to a compatible ...

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### Does the complex interpolation space $(L^1(\mathbb{R}),W^{2,1}(\mathbb{R}))_{\frac{1}{2}}$ continuously embed into $L^\infty(\mathbb{R})$?

The complex interpolation space between $(L^p(\mathbb{R}),W^{2,p}(\mathbb{R}))_\theta$ with interpolation parameter $\theta=\frac{1}{2}$ is known to be $W^{1,p}(\mathbb{R})$ for $1<p<\infty$. As ...

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### About the J-method of interpolation

In the classic text of J.Bergh and J.Lofstrom, Interpolation Spaces, the $J$-method of real interpolation defines a functor in the following way: For $0<\theta<1$ and $1\leq q\leq \infty$ we ...

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48
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### Continuity of the constant in maximal Sobolev regularity

Let $\Omega \subset \mathbb R^n$ be a smooth, bounded domain. For each pair $(p, q) \in (1, \infty)^2$, maximal regularity asserts that there is some $\widetilde K(p, q) > 0$ such for all $f \in L^...

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### Find reasonable definition for endpoint Lorentz function spaces $L^{\infty,q}$ via the idea from endpoint Triebel-Lizorkin ${\scr F}_{\infty,q}^s$

On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\...

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114
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### Dependence of Sobolev embedding theorem constant on smoothness

Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...

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114
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### Uniqueness in interpolation of Hilbert spaces

I am wondering under what condition it is true that for Hilbert spaces $H$ and $H_0$ such that $H \hookrightarrow H_0$ there is uniqueness in the existence of a Hilbert space $H_1 \hookrightarrow H$ ...

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113
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### Lemma about the weighted interpolation inequality

In this article Interpolation inequalities with weights
Chang Shou Lin the following lemma is stated and proved.
Lemma: Suppose $\dfrac{1}{p}+\dfrac{\alpha}{n} > 0$, then there exists a constant $C$...

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106
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### Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces

The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is,
$[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$.
Now I ...

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796
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### Is it possible to obtain the inequality $\|\nabla f\|_{L^{2p}} \leq C (\|f\|_{L^\infty} \|f\|_{W^{2, p}})^{1/2}$ from interpolation/harmonic analysis?

Nirenberg's paper On elliptic PDEs claims that if a function $f$ on $\mathbb{R}^n$ tends to zero at infinity or is in $L^q$ for any $q < \infty$ then the "interpolation" inequality
$$
\...

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### Interpolation between Sobolev spaces

In the classical book $Interpolation$ $Spaces$ by Joran Bergh and Jorgen Lofstrom, the Sobolev norm is defined by
$$\|f\|_{H_p^s}=\|D^sf\|_{L^p}$$
where $D^sf$ is defined by the Fourier transform
$$(D^...

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49
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### A question of interpolation space on homogeneous Carnot group

Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows:
A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...

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1
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105
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### Fix positive $t$. Construct $a_n \in \mathbb R^n$ such that $(\inf_x \|x-a_n\|_2 + t\|x\|_1 )/\min(\|a_n\|_2,t\|a_n\|_1) \to 0$

For any positive number $t$, nonnegative integer $n$, and nonzero vector $a \in \mathbb R^n$, define
$$
\begin{split}
K_n(a,t) &:= \inf_{x \in \mathbb R^n} \|x-a\|_2 + t\|x\|_1,\\
M_n(a,t) &:= ...

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### Unit ball of the sum space

Let $V$ be a vector space and $\|\cdot \|_1$ and $\|\cdot\|_2$ two norms on $V$.
Let $\|\cdot\|_+$ be given by
$$ \|v\|_+ := \inf_{v = v_1 + v_2} \|v_1\|_1 + \|v_2\|_2 $$
It is well-known that $\|\...

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0
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84
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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), ...

4
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1
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384
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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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votes

2
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141
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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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0
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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}:...

2
votes

1
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77
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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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1
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246
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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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198
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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}...

2
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1
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386
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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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155
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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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435
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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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votes

2
answers

667
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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}...

2
votes

1
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291
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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 ...

6
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1
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385
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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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1
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172
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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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111
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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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### 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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### 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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141
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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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1
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256
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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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534
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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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156
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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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530
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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 ...