For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a ...

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10
votes
0answers
540 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" ...
9
votes
0answers
174 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded Lipschitz domain and $u$ is a measurable function. A sufficient condition for the integral $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in ...
7
votes
0answers
2k views

Proof of separability of Lp spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof. Questions: It says 'it is easy to construct a function $f_{2} ...
6
votes
0answers
124 views

Solving a functional equation in $L_2(\mathbb{R})$

Let $e\left(x\right)=e^{2\pi ix}$ and let $F$ be an arbitrary complex-valued function in $L^2 (\mathbb R)$. I am trying to solve the following functional equation (or rather family of equations): ...
5
votes
0answers
118 views

Cyclic vector in $\ell^2(\mathbb{Z})$ space

Suppose we look at $\ell^2(\mathbb{Z})$, which contains vectors $c=(\dots,c_{-2},c_{-1},c_0,c_1,\dots)$ with $\sum|c(n)|^2<\infty$. Define the right-translation operator by $$R:\{c(n)\}\mapsto ...
5
votes
0answers
43 views

Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure.

Let $(X,M, μ)$ be a measure space and $0 < p < q ≤ ∞$. Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure. My work: I proved the ...
5
votes
0answers
114 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 ...
5
votes
0answers
144 views

Gradient on Sobolev spaces

Suppose I have a function $\Phi \in L_p(\Omega)$ and $\nabla\Phi \in W^{-\alpha}_p(\Omega)$, $1<p<\infty$ and where $\Omega\subset\mathbb{R}^n$ is a bounded domain. Can I conclude that $\Phi\in ...
5
votes
0answers
86 views

Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
5
votes
0answers
222 views

Is there an orthonormal basis for $L_2[0,1]$ consisting of convex functions?

Is there an orthonormal basis $\{\phi_{\alpha}\}$ for the space $L_2[0,1]$ of square-integrable functions from $[0,1]$ to $\mathbb{R}$ such that every $\phi_{\alpha}$ is convex? Edit: A helpful ...
4
votes
0answers
84 views

Limit of $L_p$ norm as $ p \rightarrow 0$

I have reviewed Ayman Houreih's proof for the limit of the $L_p$ norm as $ p \rightarrow 0$ at "Scaled $L^p$ norm" and geometric mean. While I have found the outline of the proof very ...
4
votes
0answers
62 views

Why $\|f-g\|=0$ if and only if $f=g$?

I'm learning Fourier Transformation lately, and in the Course Reader page 23, it defines $\|f\|=\left(\int_0^1 \left|f(t)\right|^2 dt\right)^{1/2}$. And then $\|f-g\|=0$ if and only if $f=g$. My ...
4
votes
0answers
148 views

Is $L^p$ separable?

Whether a $L^p(X,\mu)$ space is separable? I understand that the answer depends on $p$ and $X$. It seems to me that it is separable when $1\leq p < \infty, X=\mathbb{R}^n$ or $X=\mathbb{N}$. ...
4
votes
0answers
162 views

When does $|f*g|_{p}=|f|_{1}|g|_{p}$?

From Rudin, Real and Complex Analysis, 1st edition, Chapter 7, Problem 4 Suppose $1\le p\le \infty$, $f\in L^{1}(\mathbb{R}^{1})$, $g\in L^{p}(\mathbb{R}^{1})$. Show that the the integral defining ...
4
votes
0answers
137 views

A question about functions in $L^p(E)$

I've been working on a problem. I found a couple of related questions on the site, but I was having a little bit of trouble clarifying everything. The goal is to show that given $f\notin L^p(E)$, ...
3
votes
0answers
43 views

Is there a way to find the operator norm in this case?

If $k:[a,b]\times[a,b]\rightarrow \mathbb{R}$ is in $L^2([a,b]\times[a,b])$, we can show that the linear operator: $T_k: L[a,b]\rightarrow L[a,b]$, given by ...
3
votes
0answers
48 views

$L^p$ norm converges to $L^\infty$ norm

The question is: Let $ (X,\mathcal M,\mu) $ be an arbitrary measure space. Let $f$ be a function in $L^r$ for some $0<r<\infty$. Show that $||f||_p$ converges to $||f||_\infty$ as $p\to \infty$. ...
3
votes
0answers
76 views

Universal property of l^p-spaces

The category $\mathsf{Ban_1}$ of Banach spaces together with short linear maps (i.e. those of norm $\leq 1$) seems to have a natural construction which interpolates between coproduct and product: Let ...
3
votes
0answers
21 views

Equivalent descriptions of Sobolev spaces on compact manifolds

While reading through a set of lectures on the Laplacian on manifolds, I encountered two descriptions of Sobolev spaces. The first one, valid only for compact manifolds (because it needs to globalize ...
3
votes
0answers
54 views

Doob's inequalities for not necessarily right-continuous martingales

In Revuz and Yor, they denote $\mathbb{H}^2$ the space of $L^2$-bounded martingales, and $H^2$ the space of continuous $L^2$-bounded martingales. They state "... by Doob's inequality ... ...
3
votes
0answers
78 views

Weak convergence and trace operator

Suppose that $u_j\rightharpoonup u$ in $W^{1,p}(\Omega)$ (notice the weak convergence), with $\Omega\subset \mathbb{R}^3$ regular enough. Let $v_j=Tu_j$, and $v=Tu$, where $T:W^{1,p}(\Omega)\to ...
3
votes
0answers
52 views

Properties of $L^{\infty}$

I'm trying to get a better grasp on the idea of $L^{\infty}$. What are the implications if we are given that $f \in L^{\infty}$? Also, how do we write $\|f\|_{\infty}$ in terms of the inf of a set of ...
3
votes
0answers
142 views

Dense subset of $L^p[a,b]$

It is true that $C_{0}^{\infty}[a,b]$, (the space of all smooth functions f with the property that f and all its derivatives vanish at a and b) is dense in $L^p[a,b]$ with $1\leq p< \infty$ ? ...
3
votes
0answers
32 views

lp and c0 little question

i want to explain that the inclusion $\bigcup_{p < \infty} l^p \subset c_0$ is true. that comes very quickly from the definition of $l^p$. The problem is, that Im sure that this inclusion is ...
3
votes
0answers
120 views

Dual of $l^p$ Direct sum

I am asked to show that the $l^p$-direct sum of a sequence of Banach Spaces $X_n$ is isometrically isomorphic to the $l^q$ direct sum of $X_n^*$ where $X_n^*$ is the dual of $X_n$ for each $n$ ...
3
votes
0answers
46 views

Approximation of $L^2$ function by smooth functions on a manifold

Let $M$ be a $C^2$ compact Riemannian manifold with boundary. Suppose $f \in L^2(M)$ is such that $0 \leq f \leq 1$. Is it possible to find $f_n \in C^\infty(M)$ such that $0 \leq f_n \leq 1$ for ...
3
votes
0answers
139 views

Do these limits commute?

Given a sequence of functions $f_{n,m}:\mathbb{R}^{n} \to \mathbb{R}$, suppose that $$\displaystyle\lim_{m} f_{n,m}(x)$$ exists almost everywhere (for any fixed n) and also suppose that ...
3
votes
0answers
49 views

Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
3
votes
0answers
72 views

$\|\phi_{\lambda}- \phi_{\lambda} \ast f \|_{L^{2}(\mathbb R)}\to 0$ as $\lambda \to \infty$? ($\phi_{\lambda}(x)=\lambda^{-1} \phi(x/\lambda).$)

For $f\in L^{1}(\mathbb R),$ we define its Fourier transform as follows: $\hat{f}(t)=\int_{\mathbb R} f(x) e^{-ix\cdot t} dx ,(t\in \mathbb R).$ Suppose that $f\in L^{1}(\mathbb R)$ with ...
3
votes
0answers
49 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
3
votes
0answers
72 views

Relationship between complex and real Lp spaces.

The theory of integration of real functions is (as far as I know) usually extended to the complex case as follows: Let $X$ be a set. Given a function $f:X\to\mathbb C$, define ...
3
votes
0answers
95 views

Mapping $G$ into its group algebra as left multiplication. Continuous?

I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove: Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
3
votes
0answers
292 views

The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
2
votes
0answers
26 views

Closed subset for $L^2$ strong and weak convergence

I was trying to solve the following exercise. Let $K$ a closed subset of $\mathbb{R}$. $$X:=\{f\in L^2[0,1]:f(x)\in K \:a.e.\:x\in [0,1] \}$$ Then: 1)X is closed under strong convergence in $L^2$. ...
2
votes
0answers
22 views

Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$, $\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$ where the scalar complex function ...
2
votes
0answers
29 views

A question about general Marcinkiewicz interpolation theorem

The general Marcinkiewicz interpolation theorem states as following: If $T$ is a linear operator of weak type $(p_0,q_0)$ and of weak type $(p_1,q_1)$ where $q_0\neq q_1$, then for each ...
2
votes
0answers
52 views

approximations in lp spaces

If $f$ is a bounded measurable function, then, on every ball, the functions $f*\rho_{\epsilon}$ converge to $f$ in the mean and in measure. This corollary is from V.I.Bogachev, page 253 volume 1. I ...
2
votes
0answers
16 views

sum of uniformly bounded projections acting on a Lorentz sequence space

It is known that for every $k\in\mathbb{N}$ there is $N_k\in\mathbb{N}$ such that every $N_k$-dimensional subspace of $\ell_p$, $1<p<\infty$, contains uniformly complemented copies of $\ell_2^k$ ...
2
votes
0answers
29 views

Is the set of linear combinations dense in the set of the dual space of $l_p$?

Good day, Right now I'm working with the book "Functional Analysis" by Bachman and Narici, it is available on Google Books, see ...
2
votes
0answers
19 views

Fréchet differentiability of Nemyckij operator defined on $L^2$

I have been told the following. Suppose $\Omega\subseteq\mathbb{R}^n$ is a bounded borel set, $f$ is Carathéodory function on $\Omega\times\mathbb{R}=\{(x,s):x\in\Omega,s\in\mathbb{R}\}$, $f_s$, ...
2
votes
0answers
45 views

Disproving that a particular space is Banach

Let $E$ be a measurable set of finite measure and $1\leq p_1<p_2<\infty$. Consider the linear space $L^{p_2}(E)$ normed by $\|.\|_{p_1}$. Is this normed linear space a Banach space? My ...
2
votes
0answers
35 views

The sequence of functions $f_n=\frac{n}{1+n\sqrt{x}}$ in $L^1(0,1)$ and $L^2(0,1)$

During a lecture on $L^p$ spaces, the lecturer made a few comments about the sequence of functions $f_n=\frac{n}{1+n\sqrt{x}}$ that I am not sure I fully understand. 1) First he said $f_n\in ...
2
votes
0answers
39 views

Averages of integral and$ L^p$ space problem

Let $f: \mathbb R \to \mathbb R$ be an integrable function, for each $h>0$ let $$f_h(t)=\dfrac{1}{h}\int_{t-\frac{h}{2}}^{t+\frac{h}{2}}f(x)dx$$ Suppose $f \in L^P$, prove the following (1) $f_h ...
2
votes
0answers
21 views

Is such a function in $L^2$?

Let $L^2$ denote $L^2([0,T];\mathbb{R}^N)$ and let $A:[0,T]\times L^2 \to (L^2)^\prime =L^2$ be a Carathéodory function satisfying the following estimate: $$\exists \alpha \in ...
2
votes
0answers
20 views

Find K s.t. $\int_{\Omega}|K-f(x)|(K-f(x))=0$

I'm trying to solve the following problem: Let $\Omega$ a bounded set of $\mathbb{R}$, $f\in L^3(\Omega)$. Find $K\in\mathbb{R}$ such that $\int_{\Omega}|K-f(x)|(K-f(x))=0.$ I'dont know how to ...
2
votes
0answers
34 views

Khintchine inequality (question about Holder inequality)

I'm reading a proof of a Khintchine inequality : Let $(r_{1}, \dots , r_{n})$ be iid random variables with $P(r_{i} = \pm1) = \frac{1}{2}$. Let $f = \sum\limits_{j=1}^{n}a_{j}r_{j}$, where ...
2
votes
0answers
37 views

$L^p(\mathbb{R})$ separable.

I'm trying to prove $L^p(\mathbb{R})$, $p \in [1,\infty)$ is separable by showing the collection$$ S:= \{\sum_{i=1}^nr\chi_{(a_i,b_i)}\}_{(a_i,b_i,r) \in \mathbb{Q}^3}$$ is dense in $L^p$. So, since ...
2
votes
0answers
38 views

Examples of Lp spaces in Applied Math

I was wondering if there are examples of exotic Lp spaces being used in applied mathematics. I know that the "special" p's (1,2 , infinity ) are of use, for example in statistics, L1 is mean, L2 is ...
2
votes
0answers
38 views

Factorization of $L^{p}$ spaces

Is the following theorem true and if so does it have a name? Theorem: Let $p$, $q$, $r\in\left(0, \infty\right]$ with $\frac{1}{q}=\frac{1}{p}+\frac{1}{r}$. Every $L^{q}$-function is the product of a ...
2
votes
0answers
34 views

convergence in $L^p$ implies convergence in measure

I am trying to show that if $f_n$ converges to $f$ in $L^p(X,\mu)$ then $f_n\to f$ in $L^p$ in measure, where $1\le p \le \infty$. Here is my attempt for $p>1$ - Let $\varepsilon>0$ and define ...