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Questions tagged [interpolation-spaces]

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,...
P. P. Tuong's user avatar
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_{\...
Theleb's user avatar
  • 263
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 $$...
user43389's user avatar
  • 277
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\...
Liding Yao's user avatar
  • 1,481
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 \...
Guillermo García Sáez's user avatar
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=\...
Guillermo García Sáez's user avatar
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 ...
Guillermo García Sáez's user avatar
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 ...
Michelangelo's user avatar
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 ...
Guillermo García Sáez's user avatar
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,...
rebo79's user avatar
  • 91
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 ...
Drew Brady's user avatar
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 ...
Guillermo García Sáez's user avatar
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 ...
vmist's user avatar
  • 1,141
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 ...
Guillermo García Sáez's user avatar
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^...
Keba's user avatar
  • 338
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^\...
Liding Yao's user avatar
  • 1,481
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)}...
user515999's user avatar
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 $H_1 \hookrightarrow H$ ...
rafoub92's user avatar
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 $C$...
Ilovemath's user avatar
  • 687
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 ...
heppoko_taroh's user avatar
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 $$ \...
Carlos Esparza's user avatar
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^...
kuuga's user avatar
  • 71
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 ...
pxchg1200's user avatar
  • 267
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) &:= ...
dohmatob's user avatar
  • 7,033
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 $\|\...
Willie Wong's user avatar
  • 41.6k
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 ...
A beginner mathmatician's user avatar
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), ...
Hannes's user avatar
  • 2,790
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,...
Theleb's user avatar
  • 263
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) = \...
Seven9's user avatar
  • 565
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}:...
pipenauss's user avatar
  • 319
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=\...
Denis Serre's user avatar
  • 53.1k
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 ...
M.Oud's user avatar
  • 11
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 ...
Lev's user avatar
  • 61
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 $\...
Yidong Luo's user avatar
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}...
rebo79's user avatar
  • 91
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 ...
Raul Kazan's user avatar
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 ...
Tony419's user avatar
  • 421
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\...
Tony419's user avatar
  • 421
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 $p$, ...
LL 3.14's user avatar
  • 240
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}...
Henning's user avatar
  • 143
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 ...
Capublanca's user avatar
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 ...
Migalobe's user avatar
  • 395
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 ...
Romain Gicquaud's user avatar
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(\...
user avatar
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 ...
Philip's user avatar
  • 23
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 ...
Clement C.'s user avatar
  • 1,412
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 $p$. ...
Piero D'Ancona's user avatar
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,...
Saj_Eda's user avatar
  • 405
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 ...
Denis Serre's user avatar
  • 53.1k
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 ...
Gabe K's user avatar
  • 6,366