Questions tagged [interpolation-spaces]
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79 questions
7 votes
1 answer
143 views
Complex interpolation with a reflexive Banach space yields reflexive Banach spaces
Let $(A,B)$ be a compatible pair of Banach spaces. A result of Calderón states that if at least one of the Banach spaces $A$ or $B$ is reflexive and if $0<s<1$, then the interpolation space $[A,...
2 votes
0 answers
91 views
An interpolation scale for uniformly local Sobolev spaces?
Are there any known results about the interpolation of uniformly local Sobolev spaces on the real line? For $s \geq 0$, the space $H_{\mathrm{u,l}}^s(\mathbb{R})$ is defined as follows, $$ H^s_{\...
2 votes
1 answer
121 views
Interpolation between $L^\infty$ and Lipschitz that pinches Dini on $\mathbb{T}^2$
Let $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ be the flat $2$D torus with geodesic distance $d(\cdot,\cdot)$. For a bounded function $\rho:\mathbb{T}^2\to\mathbb{R}$, define the modulus of continuity $$...
1 vote
0 answers
134 views
Real interpolation of negative Sobolev spaces $(L^p,W^{-k,p})_{\theta,p}$ for $1\le p\le\infty$
For an open set $U\subset\mathbb R^n$, $k\ge1$ and $1\le p\le\infty$, the space $W^{-k,p}(U)$ consists of all distributions of the form $\sum_{|\alpha|\le k}\partial^\alpha g_\alpha$ where $g_\alpha\...
4 votes
0 answers
85 views
Generalized real method of interpolation for Orlicz spaces
Consider the Orlicz spaces $L^A(\mathbb{R}^n)$. Under suitable conditions for the $N$-function $A(x,t)$, namely $(Inc)_p$, $(Dec)_q,$, for $1<p<q<\infty$, it is known that $$L^p\cap L^q \...
2 votes
0 answers
131 views
Bessel spaces and Triebel Lizorkin
It is known that bessel potential spaces $H^{s,p}$ coincide with Triebel-Lizorkin spaces $F^{s}_{p,2}$ for $s\in \mathbb{R}$ and $1<p<\infty$. Im wondering what can be said por $p=1$ and $p=\...
1 vote
0 answers
218 views
Fractional Sobolev embedding
Let $s\in (0,1)$ and $1<p<\infty$. Let $H^{s,p}(\mathbb{R}^n)=H^{s,p}$ the Bessel potential space, defined as the image of $L^p(\mathbb{R^n})$ by the Bessel potential. It is known that these ...
4 votes
0 answers
120 views
Interpolation-extrapolation scales of H. Amann
I am currently reading H. Amann's notes titled "Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems", and I have a question regarding the abstract ...
2 votes
1 answer
175 views
Separability is an interpolation property
I'm trying to prove that certain space, which can be obtained as an interpolation space, is separable. The fact that is separable is well known but i want to simplify it via interpolation. I haven't ...
1 vote
0 answers
221 views
Interpolation of Sobolev spaces with constraints
Let us consider a real interval $[0, L]$, with $a\in (0, L)$, and let $I_1=(0, a)$ and $I_2=(a, L)$. We denote by $H^k(I_1)$ and $H^k(I_2)$ the usual Sobolev spaces, defined for $k\in \mathbb{N}$. Now,...
0 votes
0 answers
96 views
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 ...
0 votes
0 answers
141 views
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 ...
2 votes
0 answers
192 views
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 ...
3 votes
0 answers
141 views
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 ...
1 vote
0 answers
84 views
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^...
4 votes
0 answers
101 views
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^\...
1 vote
0 answers
180 views
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 $$ \|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
2 votes
0 answers
141 views
Uniqueness in interpolation of Hilbert spaces
I am wondering under what condition it is true that for Hilbert spaces $H$ and $H_0$ such that $H \hookrightarrow H_0$ there is uniqueness in the existence of a Hilbert space $H_1 \hookrightarrow H$ ...
1 vote
2 answers
195 views
Lemma about the weighted interpolation inequality
In this article Interpolation inequalities with weights Chang Shou Lin the following lemma is stated and proved. Lemma: Suppose $\dfrac{1}{p}+\dfrac{\alpha}{n} > 0$, then there exists a constant $C$...
3 votes
0 answers
143 views
Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces
The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is, $[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$. Now I ...
11 votes
2 answers
959 views
Is it possible to obtain the inequality $\|\nabla f\|_{L^{2p}} \leq C (\|f\|_{L^\infty} \|f\|_{W^{2, p}})^{1/2}$ from interpolation/harmonic analysis?
Nirenberg's paper On elliptic PDEs claims that if a function $f$ on $\mathbb{R}^n$ tends to zero at infinity or is in $L^q$ for any $q < \infty$ then the "interpolation" inequality $$ \...
3 votes
0 answers
256 views
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 $$(D^...
1 vote
0 answers
69 views
A question of interpolation space on homogeneous Carnot group
Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows: A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...
2 votes
1 answer
155 views
Fix positive $t$. Construct $a_n \in \mathbb R^n$ such that $(\inf_x \|x-a_n\|_2 + t\|x\|_1 )/\min(\|a_n\|_2,t\|a_n\|_1) \to 0$
For any positive number $t$, nonnegative integer $n$, and nonzero vector $a \in \mathbb R^n$, define $$ \begin{split} K_n(a,t) &:= \inf_{x \in \mathbb R^n} \|x-a\|_2 + t\|x\|_1,\\ M_n(a,t) &:= ...
4 votes
2 answers
697 views
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 $\|\...
1 vote
0 answers
105 views
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 ...
6 votes
0 answers
259 views
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 votes
1 answer
605 views
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, $$ L^p(0,T;X_1)\cap W^{1,p}(0,...
3 votes
2 answers
241 views
Motivation for considering the J and K-functionals of real interpolation
In interpolation theory, given a compatible couple of Banach spaces $(X_0, X_1)$ one considers the $J$ and $K$-functionals, defined as follows: If $x \in X_0 + X_1$ and $t > 0$ then $$K(t, x) = \...
1 vote
0 answers
106 views
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 $$K(f,t,X_0,X_1)=\inf\{\|f_0\|_{X_0}+t\|f_1\|_{X_1}:...
2 votes
1 answer
105 views
Interpolation of normed spaces *vs* geometrical mean of positive matrices
Two $n\times n$ positive definite symetric matrices $A,B$ define two normed spaces $E_A=({\mathbb R}^n;\|\cdot\|_A)$ and $E_B=({\mathbb R}^n;\|\cdot\|_B)$, where $$\|x\|_A=\sqrt{x^TAx},\qquad \|x\|_B=\...
1 vote
1 answer
370 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
166 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
52 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 $\...
3 votes
0 answers
292 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
469 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
185 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 ...
6 votes
0 answers
506 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\...
3 votes
0 answers
279 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
832 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
391 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
476 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
102 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
323 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 $$X_0:=L^2(\...
0 votes
0 answers
142 views
Extrapolate an Interpolation scale
Suppose $X$ and $Y$ are real Banach spaces with a continuous embedding $X\subset Y$. For given $0<\theta<1$ I am interested in constructing using the norms of $X$ and $Y$ a (Quasi-) Banach ...
5 votes
1 answer
297 views
Relation between two different functionals: $\lVert p^{-\max}_{-\varepsilon}\rVert$ and $\kappa_{p}^{-1}(1-\varepsilon)$
Given a non-negative sequence $p=(p_i)_{i\in\mathbb{N}}\in \ell_1$ such that $\lVert p\rVert_1 = 1$,we define the two following quantities, for every $\varepsilon \in (0,1]$. Assuming, without loss of ...
3 votes
0 answers
213 views
Dual Lorentz spaces
MO seems the perfect place to ask for the following question. Denote the Lorentz spaces on an arbitrary measure space $(E,\mu)$ by $L^{p,q}=L^{p,q}(E,\mu)$, and by $p'$ the conjugate index of $p$. ...
6 votes
0 answers
132 views
Interpolation of some Sobolev spaces
Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$ I am wondering what is $$(X_0,X_2)_{1/2,2}=?$$ Would it be $H^2_0(0,...
5 votes
1 answer
291 views
Interpolation of some Lebesgue spaces
When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that $$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...
3 votes
1 answer
166 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 ...