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Questions tagged [interpolation-spaces]

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4
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1answer
132 views

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 ...
3
votes
1answer
113 views

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 ...
1
vote
1answer
66 views

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 ...
8
votes
2answers
367 views

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 ...
3
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0answers
79 views

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\...
1
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0answers
37 views

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}}...
2
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1answer
84 views

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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0answers
60 views

Shrinking bases and real interpolation

Suppose that the canonical $c_{00}$ basis is a normalized, $1$-unconditional Schauder basis for some Banach space $E$. Then $\ell_1\subset E$ and we may use real interpolation to produce spaces $(\...
3
votes
2answers
244 views

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^...
3
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0answers
93 views

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 ...
5
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1answer
173 views

“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 ...
3
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1answer
104 views

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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0answers
116 views

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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0answers
72 views

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 } \...
3
votes
1answer
303 views

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 $...
2
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0answers
80 views

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 ...
2
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1answer
152 views

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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0answers
162 views

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\...
4
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1answer
144 views

Charcterization 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 ...
0
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0answers
79 views

interpolation inequalities and embeddings

When $Z$ is an interpolation space between two Banach spaces $X$ and $Y$ (say real / complex method), we have a norm inequality $$ \| x \|_Z \le C \| x \|_X^\theta \| x \|_Y^{1-\theta} $$ My question ...
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0answers
433 views

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 $u \in W^{m,p}(\Omega)$ (...
10
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1answer
352 views

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 $...
3
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0answers
165 views

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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0answers
53 views

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, ...
1
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0answers
158 views

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, ...
2
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0answers
69 views

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 ...
3
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0answers
155 views

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 ...
1
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0answers
54 views

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 $R(1) = C$)...
2
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0answers
42 views

About norm on $H^{\frac 12}(M \times \{0,1\})$

Let $X=M \times \{0,1\}$ with $M$ a smooth compact manifold without boundary. Define the fractional Sobolev space $H^{\frac 12}(X) = (L^2(X), H^1(X))_{\frac 12}$, as the real interpolation space ...
6
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0answers
124 views

Real interpolation space between the Wiener algebra and $L^2$

The Wiener algebra $W_n$ is the image by the Fourier transform of $L^1(\mathbb R^n)$. What is the (complex) interpolation space between $W_n$ and $L^2(\mathbb R^n)$? It is probably not true that for $\...
7
votes
2answers
556 views

Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains

I am not really familiar with the topic, thus I am looking for some references about the following problem. Let $s>0$ be a positive real and let $p\in(1,+\infty)$. We define the Bessel Potential ...
1
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0answers
84 views

Discrete J-method of interpolation [closed]

The discrete $J$ method is, given Banach spaces $A_0$ and $A_1$: The interpolationn space $[A_0, A_1]_\theta$ is defined by: $a \in [A_0, A_1]_\theta$ if and only if $a$ can be written as $a=\sum_{...
3
votes
2answers
229 views

Interpolation between $L_p$ and $B^s_{q,q}$

I am looking for a reference or a direct argument that shows the real interpolation space between $L_p$ and $B^s_{q,q}$ is $B^\alpha_{r,r}$, with the usual conditions on the indices. This result is ...