Questions tagged [lp-spaces]
For questions about $L^p$ spaces. That is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.
5,707
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Finding a function which is $L^1$ not $L^2$ and the integral is bounded by the square root.
So I have been trying to solve the following this past exam problem:
Find $f\in L^1(\mathbb{R})$, not $L^2(\mathbb{R})$ with the property:
$$
\int_{A}|f(x)|dm\leq \sqrt{m(A)}\quad\text{ for all } A\...
- 394
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Applying the Lebesgue Dominated Convergence Theorem without explicitly building a sequence of functions.
Let $\Omega \subset \mathbb R^n$ denote an arbitrary non-empty open set.
The classical formulation of the Lebesgue Dominated Convergence Theorem (LDCT, for short) is as follows.
LDCT. Let $(f_n)_{n \...
1
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0
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Proof of Du Bois-Reymond Lemma using Riesz representation theorem
I’m working with this version of the fundamental calculus of variations lemma:
If $f\in L^p(\mathbb{R}^n)$ and $\int f\phi dx = 0 $ for all $\phi \in C^\infty_c(\mathbb{R}^n)$, then $f=0$ a.e.
My ...
- 133
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0
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Is every continuous function $g$ of the form $g(p) = \|f\|_p$ for some $f : \mathbb R \to \mathbb R$?
Let $1 \leq a \leq b \leq \infty$, and let $f \in L_a(\mathbb R) \cap L_b(\mathbb R)$. Then the function $g : [a,b] \to [0,\infty)$, $p \mapsto \|f\|_p$, is continuous. We also have that if there is $...
- 6,033
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1
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Notation $D^{m}$ in certain inequalities in $L^{p}$
In this wikipedia article, one can find the so-called "Gagliardo-Nirenberg inequality, which are of the form
$${\displaystyle \|D^{j}u\|_{L^{p}(\mathbb {R} ^{n})}\leq C\|D^{m}u\|_{L^{r}(\mathbb {...
- 2,876
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1
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Prove that Fourier transform is $L^p$
I am working on a problem, and I need to show that, if $f : \mathbb{R}^n \to \mathbb{C}$ is given by
$$f(x) = \frac{|x|^a}{(1+|x|^2)^{(a+b)/2}}$$
and $2a>-n$, and $b > n$, then the Fourier ...
- 591
2
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1
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How pathological are the $L^{p}$ functions?
The motivation for my question is simple: for Riemann integrable functions, we had, at worst, almost everywhere continuity. It's not that bad behavior wise, but it is integration wise. So we come up ...
- 161
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0
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Translation in $L^{\infty}$
Consider the translation operator $\tau_h$ defined on $L^\infty(\mathbb{R}^n)$ s.t. $\tau_hu(x)=u(x-h)$. I know that $\tau_h$ is not continuous with respect to $h$, I mean it’s not true that $h\to 0$ ...
- 133
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A sequence $(x^k) \subset l^p$ with $|x^k_n| \leq a_n$ for $(a_n)\in l^p$ has a convergent subsequence
I am trying to do the following exercise:
Let $1 \leq p < \infty$, $(a_n)\in l^p$, and $(x^k)\subset l^p$ a sequence such that
$$
|x^k_n| \leq a_n \qquad \forall k=1,2,3,4, \ldots
$$
Then $(x^k)$ ...
- 640
3
votes
2
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Multidimensional Hardy inequality
I am having a really hard time in finding references in which is stated rigorously and proved what I think is called the multidimensional Hardy inequality which is like :
$$\int_{\mathbb{R}^N}|\nabla ...
- 545
3
votes
1
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$X \approx \ell^p(I,X)$ $\Rightarrow$ $X \approx X \oplus X$ (used in The Pełczynski decomposition technique)
In "A. Pełczynski, Projections in certain Banach spaces. Stud. Math. ´ 19, 209–228" the following relation is used as a fact (for The Pełczynski decomposition technique):
$X \approx \ell^p(I,...
- 97
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0
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Boundedness of an Integral Operator on $L^p(\mu)$
I realize this question has been asked way too many times, for instance, it's this exact problem which I will put below for convivence:
Let $(X,\Omega, \mu)$ be a $\sigma$-finite measure space with $...
- 1,399
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0
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Isomorphism between $\ell^p(\mathbb{N} \times \mathbb{N}, \mathbb{R})$ and $\ell^p(\mathbb{N}, \mathbb{R})$
to apply the the Pełczynski decomposition technique I want to show that the infinte direct sum of $\ell^p$ is ismoetric isomorph to $\ell^p$. With infinte direct sum I mean:
$\ell^p(X) = \{(x_n)_{n=1}^...
- 97
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1
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Are Sobolev–Hölder functions continuous up to the boundary?
Let $U$ be an open subset of $\mathbb{R}^{n}$, let $k$ be a nonnegative integer, and let $W^{k,p}(U)$ ($1 \leq p < \infty$) be the Sobolev space consisting of all real-valued functions on $U$ whose ...
2
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0
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Can one construct an isometric embedding $\phi_p:\ell^p\to\ell^\infty$ without Hahn-Banach?
It is well-known (for example, see this question) that, as a consequence of the Hahn-Banach theorem, every separable Banach space can be isometrically embedded in $\ell^\infty$. In particular, for $1\...
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