# Questions tagged [interpolation-spaces]

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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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### 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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### 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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### 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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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 $$\|x\|_A=\sqrt{x^TAx},\qquad \|x\|_B=\... 0 votes 1 answer 189 views ### Intersection of the kernel with the interpolation space \DeclareMathOperator\Ker{Ker}Given two Banach spaces X and Y with a continuous inclusion X\subset Y, and another couple X’ \subset Y’ with the same properties. Take f : Y \longrightarrow ... 2 votes 0 answers 75 views ### Interpolation of Sobolev/Besov spaces in the limiting case q = ∞ I'm interested in the interpolation space (1\le p_0,p_1\le\infty, 0<\theta<1)$$ 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 . $$It ... 1 vote 0 answers 24 views ### Interpolation spaces defined by singular value decomposition Let  X  and Y be Hilbert space, A:X \to Y  compact and injective, (\sigma_n;v_n,u_n) be its singular value decomposition, that is,$$ Av_n = \sigma_n u_n \\ A^* u_n = \sigma_n v_n $$Since \... 2 votes 0 answers 159 views ### Interpolation of embedded Hilbert spaces and intersection I'm wondering under what hypothesis it is true a property like$$[\mathcal{H}_1\cap X, \mathcal{H}_2\cap X]_{\theta}=\mathcal{H}_1\cap X\cap [\mathcal{H}_1, \mathcal{H_2}]_{\theta}$$where \mathcal{H}... 2 votes 1 answer 318 views ### Trace of a function Let T,L> 0 two real numbers and we consider the Sobolev space X := L^2(0,T; H^1(0,L))\cap H^{1}(0,T;H^{-1}(0,L)). My question is: Given f \in X, the trace  t \mapsto f(t,L) belongs to what ... 2 votes 0 answers 132 views ### Estimate involving Besov norm When reading some old notes of my advisor on interpolation spaces, I bumped into a problem I can't quite wrap my head around. Here are the details. For p\in(0,\infty) a p-variation semi-norm of a ... 5 votes 0 answers 412 views ### Vector-valued interpolation for sublinear operators Grafakos in his \textit{Classical Fourier Analysis} formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem. \textbf{Theorem} Let 1\... 1 vote 0 answers 146 views ### Besov or Triebel-Lizorkin spaces versus Lorentz spaces I first asked this question on math.stackexchange here but it seems it is more a research level question ... At the 0 order of derivatives of Sobolev spaces and for a fixed integrability order p, ... 2 votes 2 answers 533 views ### Dual space of the intersection of locally convex vector spaces Let S \neq \emptyset and let \big((E_s,\mathcal{T}_s)\big)_{s \in S} be a family of locally convex vector subspaces of the same vector space. Denote by E_s^* the dual space of (E_s,\mathcal{T}... 2 votes 1 answer 236 views ### Interpolation of product spaces Suppose that X_{\theta} is an interpolation space between the Banach spaces X_0 and X_1. Let \mathcal{B} be another Banach space. Is it true that X_{\theta}\times\mathcal{B} is an ... 6 votes 1 answer 342 views ### Interpolation for Sobolev spaces How one can identify the following (complex) interpolation space$$E_\theta :=[L^2(\Omega), H^2(\Omega)\cap H_0^1(\Omega)]_\theta,$$where \Omega is a regular domaine. After research, it seems that ... 2 votes 0 answers 59 views ### Equivalent definition of real interpolation space In one of his papers (On the Nash-Moser implicit function theorem, Ann. Acad. Sci. Fenn., Ser. A I, Math. 10, 255-259 (1985).), Lars Hörmander introduces a class of function spaces that seems related ... 1 vote 1 answer 156 views ### Interpolation of L^p spaces Let \Omega_x and \Omega_y be sets of finite Lebesgue measure. We can then look at the space X_1:=L^2(\Omega_x \times \Omega_y). This space is contained in the larger space$$X_0:=L^2(\... 102 views

### 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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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 $H^2_0(0,... 5 votes 1 answer 189 views ### Interpolation of some Lebesgue spaces When dealing with time-dependent PDEs, one often obtain that some quantity$E(t,x)$belongs to a Lebesgue space$L^p_t(L^q_y)$, which means that $$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ... 3 votes 1 answer 138 views ### Interpolation inequality related to the 5/3-Laplace operator I'm having trouble with an estimate that would be helpful in information geometry. The background is the following. Suppose we have a smooth positive function g:X \to \mathbb{R}^+ where X is a ... 1 vote 1 answer 236 views ### Relation between a norm and norm of Besov spaces Let (H, \|\cdot\|) be a Hilbert space, A \colon D(A)\subset H \longrightarrow H generates an analytic semigroup T(t) on H. We define the following Banach space with the respect norm$$F=\{x\in ... 6 votes 2 answers 560 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 votes 0 answers 86 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 vote
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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 ...
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 $u \in W^{m,p}(\Omega)$ (...