Questions tagged [interpolation-spaces]

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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\...
Clement C.'s user avatar
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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 ...
user41467's user avatar
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2 votes
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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 $u \in W^{m,p}(\Omega)$ (...
Paglia's user avatar
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10 votes
1 answer
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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 $...
Bill Johnson's user avatar
5 votes
0 answers
318 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 ...
Leonardo Figueroa's user avatar
1 vote
0 answers
108 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, ...
João Ramos's user avatar
1 vote
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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, ...
Alexi Quevedo S.'s user avatar
2 votes
0 answers
109 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 ...
C_Al's user avatar
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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 ...
B.R. Smith's user avatar
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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 $R(1) = C$)...
Seven9's user avatar
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0 answers
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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 ...
Upin's user avatar
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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 $\...
Bazin's user avatar
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7 votes
2 answers
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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 ...
Pit's user avatar
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1 vote
0 answers
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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_{...
GuestTwo's user avatar
2 votes
2 answers
314 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 ...
timur's user avatar
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