14 votes
Accepted
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?
Here is a Littlewood-Paley + interpolation style proof. If $P_k$ denotes a Littlewood-Paley projection to frequencies $|\xi| \sim 2^k$, then the standard Littlewood-Paley characterisations of Sobolev ...
- 120k
9 votes
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?
I know this direct proof which works also for $\infty$. Assume $2/q=1/p+1/r$ and $q \geq 2$. Then $$ \int|D_ku|^q=\int D_k u D_k u |D_k u|^{q-2}=-(q-1)\int u|D_k u|^{q-2} D_{kk}u \leq (q-1)\int |u||...
- 6,265
6 votes
Accepted
Interpolation of product spaces
Yes, interpolation on product spaces works componentwise, so $$\Bigl(\prod_{i=1}^n X_i,\prod_{i=1}^n Y_i\Bigr) = \prod_{i=1}^n (X_i,Y_i)$$ for any interpolation functor $(\cdot,\cdot)$ even with equal ...
- 2,790
6 votes
Accepted
Unit ball of the sum space
Let $B_+,B_1,B_2$ denote the closed unit balls w.r. to $\|\cdot\|_+,\|\cdot\|_1,\|\cdot\|_2$, respectively. Let $C$ be the convex hull of $B_1\cup B_2$. Let $\bar C$ be the closure of $C$ (w.r. to the ...
- 142k
4 votes
Accepted
Some questions on parabolic function spaces
This is true if and only if $X$ has the Radon-Nikodym property, see Diestel & Uhl: Vector measures, Chapter IV.1, Theorem 1. This would be very useful indeed, but unfortunately, no. The standard ...
- 2,790
4 votes
Accepted
Real interpolation for vector-valued Sobolev spaces
The desired embedding is indeed correct for $\theta = 1-1/p$. This is a classical result in interpolation theory and the theory of evolution equations. See for example the book of Amann [2], Theorem ...
- 2,790
4 votes
Unit ball of the sum space
Iosif Pinelis neatly answered the question. I just want to add another viewpoint. The Minkowski functional (or gauge) $p_B(x)=\inf\{t>0: x\in tB\}$ of an absolutely convex absorbing set is a semi-...
- 16.9k
3 votes
Accepted
Complex interpolation with a reflexive Banach space yields reflexive Banach spaces
It seems like Calderón organizes his paper by presenting results all together, and proofs all together: The presentation of the material is arranged as follows: paragraphs $1$ to $14$ contain the ...
- 336
3 votes
Accepted
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$
This is to extend Christian Remling's comment to all real $t>0$, with an explicit lower bound on $K/M$, where $K:=K_n(a,t)$ and $M:=M_n(a,t)$. $\newcommand\norm[1]{\lVert#1\rVert}$The key here, as ...
- 142k
3 votes
Interpolation for Sobolev spaces
This is a result by R. Seeley: Interpolation in Lp with boundary conditions, Studia Mathematica, 1972. The main ingredient of the proof is that step functions are pointwise multipliers in Hs for s&...
- 101
3 votes
Accepted
Interpolation of $L^p$ spaces
No. Notice that since you assumed finiteness of the $\Omega_x$ measure, we can restrict to looking at functions that are constant in $x$. Let $\Omega_y = [0,1]$ for example, and take $\chi_n(y) = \...
- 41.6k
3 votes
Accepted
Interpolation of normed spaces *vs* geometrical mean of positive matrices
Yes. The proof of Theorem 1.1 from John E. McCarthy's "Geometric interpolation between Hilbert spaces," Ark. Mat. 30, 321-330 (1991) works for this case. Let $A_i$, $B_i$, be SPD matrices, $...
- 621
3 votes
Accepted
Interpolation space between $L^1\cap L^2$ and $L^1$
As requested, I post my comment as an answer (although this is not a true answer, just a possibly useful reference; feel free to edit it if this approach works out). In Section 3 of the article ...
- 17.4k
3 votes
Interpolation space between $L^1\cap L^2$ and $L^1$
In the case of a set of finite measure (Bourgain in the quoted paper deals with the case of finite measure (torus)) we have that $\Vert f\Vert_1\leq C\Vert f\Vert_2$ so we actually have $(\infty,1)$ ...
- 28.7k
3 votes
Interpolation spaces
Please see the answer of Mateusz for some fundamental problems with your question. With a bit of good faith however, your question HAS a positive answer for $m=2$ in the sense that $$\bigl[H^2(\Omega)...
- 2,790
2 votes
Interpolation spaces
No. A short argument is that the domain of $-\Delta$ on $\Omega$ with Dirichlet boundary condition is not $H^2_0(\Omega)$. To be specific, $\sin x$ is the eigenfunction of $\Delta$ on $\Omega = (0, \...
- 17.4k
2 votes
Accepted
"Reversion" of class $J(\theta)$ interpolation property for Besov spaces
Here is a negative answer for a certain range of $p$ and $\theta$. It shows that one can have convergence to zero in a $\gamma$-Besov norm, where $\gamma$ may be larger than $\theta$. Hope I didn't ...
- 101
2 votes
Interpolation of some Lebesgue spaces
$E \in L^\infty_t(L^1_y) \cap L^\infty_y (L^1_t)$ is insufficient to give any information about whether $E$ belongs to $L^p_t L^q_y$ when $p < \infty$. Example: for simplicity, let's work with ...
- 41.6k
2 votes
Trace of a function
The spatial evaluation (or trace) operator $\mathrm{tr}_L$ at $L$ is well defined and continuous on $H^s(0,L)$ for $s>1/2$; a classic reference is [LM, Chapter 1.9]. (Of course the range $s>1/2$ ...
- 2,790
2 votes
Dual space of the intersection of locally convex vector spaces
Let me suggest a slightly more structured situation in a first attempt at a solution: I would consider the one where we have lcs’s $E_0$ and $E_\infty$ with $E_0 \subset E_s\subset E_\infty$ and each ...
- 2,492
2 votes
Accepted
Motivation for considering the J and K-functionals of real interpolation
I found the motivation in Luc Tartar's book An Introduction to Sobolev Spaces and Interpolation Spaces. It is in the first page of Chapter 24. I quote: The K-method is the natural result of ...
- 565
2 votes
Accepted
Separability is an interpolation property
The answer is no. Let us work with spaces on $[0,1]$. Then the weak Lebesgue space $L^{2,\infty}$ can be obtained using the real interpolation functor from the compatible couple $(L^1, L^3)$ (this ...
- 56
2 votes
Lemma about the weighted interpolation inequality
Take polar coordinates, and you find $$\int |x|^{p\alpha} |u|^p dx = \int_{\mathbb{S}^{n-1}} \int_0^\infty r^{p\alpha + n - 1} |u|^p ~dr ~dA$$ This you rewrite as $$ C \int_{\mathbb{S}^{n-1}} \int_0^\...
- 41.6k
1 vote
Accepted
Interpolation between $L^\infty$ and Lipschitz that pinches Dini on $\mathbb{T}^2$
I think I can get the logarithmic dependence you asked for. For $t \geq 0$, we have the bound $$D(\rho) = \int_0^{tM/\text{Lip}(\rho)} \frac{\omega_\rho (r)}{r} \, dr + \int_{tM/\text{Lip}(\rho)}^1 \...
- 9,448
1 vote
Accepted
Lemma about the weighted interpolation inequality
$\newcommand\R{\mathbb R}\newcommand{\al}{\alpha}$In view of conditions (0.1) and (0.4) in the linked paper, $p\in[1,\infty)$ and $u\in C_0^\infty(\R^n)$. To prove the lemma in question (Lemma 2.2 in ...
- 142k
1 vote
Motivation for considering the J and K-functionals of real interpolation
The idea of introducing K-functional is described in R. A. DeVore's article Nonlinear approximation p. 85.
- 11
1 vote
Relation between two different functionals: $\lVert p^{-\max}_{-\varepsilon}\rVert$ and $\kappa_{p}^{-1}(1-\varepsilon)$
$\newcommand{\vp}{\varepsilon}$ $\Psi$ is not majorized by $\Phi$. Indeed, let $p=(1,0,0,\dots)$ and $\vp\in(0,1)$. Then $\Phi(\vp,p)=0$, whereas $\kappa_p(t)=1\wedge t:=\min(1,t)$ for $t>0$ and $\...
- 142k
1 vote
Accepted
Interpolation inequality related to the 5/3-Laplace operator
We have $$ DF(g)=\frac{\|\nabla g \|_{L^{5/3}(X)}}{ \|g\|_{L^{10}(X)} -\|g\|_{L^1(X)}},$$ which is a homogeneous expression. Hence we can drop the condition $\int_X g^{10}dx=1$ and replace it by $\|g\...
- 893
1 vote
Accepted
Relation between a norm and norm of Besov spaces
Yes, your identities are correct. Theorem 1.14.5 in Triebel's book [T] says that $$F = (H,D(A))_{1/2,2},$$ and $(1/2,2)$-real interpolation spaces between Hilbert spaces are in fact exactly the $1/2$-...
- 2,790
1 vote
"Reversion" of class $J(\theta)$ interpolation property for Besov spaces
This is a well known and classical argument of Lions & Peetre and has been used by numerous mathematicians since. An up to date proof can also be found here on page 124/125.
- 29
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