5
votes
0answers
43 views

Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
1
vote
1answer
42 views

Banach valued sequence spaces $\ell^p(X)$

Let $X$ be a Banach space and $\ell^p(X)$ denote the space of sequences $x_i\in X$ for which the norm $\big(\sum_{i=1}^\infty\|x_i\|^p\big)^\frac1p$ is finite, when $X=\mathbb{R}$ we get the usual ...
1
vote
1answer
35 views

Reference for $f \in L^{p,\infty} \cap L^{q}$ then $f \in L^r$ for $p < r \leq q$

Okay, so I think I've shown that if $f \in L^{p,\infty} \cap L^{q}$ with $p < q$ then $f \in L^r$ for $p < r \leq q$ where $L^{p, \infty}$ denotes the weak $L^p$ space. what I did was I wrote $$ ...
0
votes
0answers
49 views

Generalisation of vector-valued Marcinkiewicz interpolation theorem

Given a compatible couple $(X,Y)$ of Banach spaces and some measure space $\Omega$ Lions and Peetre identified the real interpolation space between the vector-valued Lebesgue spaces $L_1(\Omega;X)$ ...
2
votes
0answers
27 views

References on Weak Convervenge

I am looking for a good reference on weak convergence in L^p spaces, can anyone recommend anything? Thanks a lot in advance!
2
votes
0answers
41 views

uniform equivalence to unit vector basis of $\ell_p$

Let $(e_n)$ be the unit vector basis of $\ell_p$, $1\leq p<\infty$. It is well-known that if $(x_n)\subset\ell_p$ is seminormalized and weakly null then it contains a subsequence equivalent to ...
0
votes
1answer
67 views

Compactness of Sobolev Space in L infinty

I want to show that if $u_{m} \rightharpoonup u$ in $W^{1,\infty}(\Omega)$ then $u_{m} \rightarrow u$ in $L^{\infty}(\Omega)$. I know that I can't directly use the compactness of Rellich Kondrachov ...
9
votes
2answers
798 views

$L^1$ and $L^{\infty}$ are not reflexive

I want some proof for the following statement : $L^1$ and $L^{\infty}$ are not reflexive. Can anyone help me, please? or reference me?
3
votes
1answer
69 views

Is there an easy/elementary way to show that if $f_k\to f \in L^p(\Omega)$ we can conclude that $\int |f_k|^p\to \int |f|^p$

Hey just stumbled upon this problem. Often one finds the Lebesgue dominated convergence theorem presented for $L^1$ spaces with the statement that the integrals do converge, i.e. $\int f_k \to \int ...
2
votes
0answers
44 views

A system of ODEs, what existence results are there?

Let $u(t) \in \mathbb{R}^n$. Are there existence results for the ODE $$C(t)u'(t) = A(t)u(t) + f(t)$$ where $A(t), C(t) \in L^\infty(0,T;\mathbb{R}^{n\times n})$, $f(t) \in L^2(0,T;\mathbb{R}^n).$ In ...
1
vote
1answer
227 views

Proof of existence of Schauder basis for $L^p(\Omega)$?

There are a statements around, see [Brezis 2011, p. 146], like All classical (separable) Banach spaces used in analysis have a Schauder basis . I was wondering where to find a proof confirming ...
7
votes
1answer
344 views

Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
1
vote
1answer
115 views

Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$

I'm looking for articles describing (or proving nonexistence) of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$. Since $\ell_q^m$ is finite ...