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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11
votes
0answers
647 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
226 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 H_0^...
6
votes
0answers
124 views

Inequality for Lévy SDE

Let $X_{s}^{t,x}$ denote the solution at time $s$ of an Ito SDE whose coefficients are Lipschitz continuous with initial condition $X_t=x$. Let $t\leq s\leq T<\infty$. The inequality $$ \mathbb{E}\...
6
votes
0answers
132 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): $e\...
6
votes
0answers
95 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
351 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 \{c(...
5
votes
0answers
95 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 ...
5
votes
0answers
48 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
130 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
156 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
245 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 ...
5
votes
0answers
152 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 $f\notin L^p(E)$, ...
4
votes
0answers
27 views

Is it natural how $L^p$ spaces measure local and global sizes the same?

This is a continuation of my question Spaces of functions similar to $L^p$ but with different local and global sizes. I have been bothered by the fact that the $L^p$ norm on $\mathbb R^n$, which is ...
4
votes
0answers
111 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 L^p(\...
4
votes
0answers
65 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
211 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
178 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 $f*...
3
votes
0answers
28 views

Some sort of generalized Jensen inequality?

Let $(X, \mathcal{A},\mu)$ a measure space such that $\mu(X) > 0$ and let $f, g : X \rightarrow (0,\infty)$ be such that $f, g, f\log(f), f\log(g) \in L^1(\mu)$. Show that $$ \|f\|_1\log \|f\|_1 ...
3
votes
0answers
31 views

Lebesgue-integrability of derivatives

Let $f:\mathbb R\to\mathbb [0,\infty)$ be a non-negative, twice-differentiable function. Suppose that $\int_{-\infty}^{\infty}f(x)\,\mathrm dx<\infty$, $\int_{-\infty}^{\infty}|f''(x)|\,\mathrm ...
3
votes
0answers
87 views

continuous linear functional on $l^{\infty}$ space

Let $l_{\infty}$ be the space of all bounded complex-valued sequences equipped with the supremum norm. Consider the natural standard basis $\{e_n\}_{n \in \mathbb{N}}$ of $l_{\infty}$. For any ...
3
votes
0answers
43 views

Proving a trivial bound on $L_2$ norm of the error in a sparse approximation of a vector

Trying to understand this supposedly 'trivial' bound from a paper: If $\theta_N$ denotes the vector $\theta$ with everything except $N$ largest coefficients set to $0$ then we have $$ || \theta - \...
3
votes
0answers
33 views

$L^p$-bounding inequality

Do we have that$$\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over2}$$for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$? Here, $U$ denotes an open subset of $\mathbb{R}^n$. ...
3
votes
0answers
51 views

If $\mathcal{F}$ is the Fourier transform, what can be said about $\mathcal{F}(L^1(\mathbb{R})) \cap L^1(\mathbb{R})$?

The Fourier transform gives a map of the Schwartz space to itself which turns out to be a linear homeomorphism of period 4. However, when the domain is extended to $L^1(\mathbb{R})$, the situation is ...
3
votes
0answers
52 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 $T_k(f)(s)=\int_{[a,b]}k(s,t)f(t)d\lambda(...
3
votes
0answers
65 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
79 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
34 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
57 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 ... $M_\...
3
votes
0answers
68 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
153 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
41 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
163 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
51 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
146 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
50 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
76 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 $\hat{f}(0)=...
3
votes
0answers
50 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
80 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 $\Re(f),\Im(f):X\to\...
3
votes
0answers
100 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
347 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 \mathrm{C}_{0}...
2
votes
0answers
13 views

Poincaré's inequality proof for $u \in W_0^{1,p}(\Omega)$.

I am trying to prove Poincare's inequality for $u \in W_0^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is an open bounded set and $1 \leq p < \infty$. This is Poincare's inequality: $||...
2
votes
0answers
26 views

Show linearity of a functional if it holds for nonnegatives

Consider a functional $G^+:L_p \to \mathbb{R}$. Here $L_p = L_p (X,\textbf{X}, \mu)$ is the collection of all integrable fns (f s.t. $\int \vert f \vert^p d \mu < \infty$ on the measure space $(X,\...
2
votes
0answers
24 views

Embedding of Schwartz space onto L^p related spaces

So, i'm trying to prove that a Banach quasinormed space E (with its quasinorm based on $L^p$ and Weak-$L^p$ norms) is between $\mathcal{S}$ and $\mathcal{S'}$, on respect to embeddings, that is, $$ \...
2
votes
0answers
26 views

How fast can Sobolev functions grow?

It is a simple fact that $L^p$-functions cannot grow arbitrarily fast. More precisely, one has for every $\ell>0$ $$ |\{f\geq\ell\}|\leq \frac{\|f\|_{L^p}^p}{\ell^p} $$ for every $f\in L^p$. My ...
2
votes
0answers
28 views

Show that there exists $ \lambda \ge 0$ such that $v=\lambda u$

Let $\Omega \subset \mathbb{R}^n$ be open. Let $u,v \in L^1_\text{loc}(\Omega)$ with $u \ne 0$ a.e on a set of positive measure. Assume that $$\phi \in C_c^\infty(\Omega), \int u\phi > 0 \implies \...
2
votes
0answers
43 views

What is the motivation for $l^p$ space?

I'm studying $L^p$,$l^p$ spaces recently, but I don't see the motivation for this. The only application I know is that $l^2$ can be used to characterize Hilbert spaces up to dimension. Is $L^p$ spaces ...
2
votes
0answers
41 views

If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$

Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$. Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
2
votes
0answers
81 views

$\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$

Let $\Omega\subset \mathbb R^d$ ($d=2,3$) is a bounded Lipschitz domain. Question: Is it true that for each function $g(x)\in L^2(\Omega)$ one can find a sequence $\{g_n\}_1^\infty$ of $H_0^1(\...
2
votes
0answers
33 views

Volume of n-dimensional ball in L1 norm with change of variables

For a homework problem, I need to find a recursive equation that relates the volume of an $n$-dimensional ball $V_n(r)$ of radius $r$ to that of an $(n-2)$-dimensional ball, expressed by $V_{n-2}(r)$. ...
2
votes
0answers
23 views

Lebesgue Space/Bochner Space interpolation Theorem

I need the embedding, for $I\subset\mathbb{R}$ is a bounded intervall and $\Omega\subset\mathbb{R}^n$ is a bounded domain, $$L^{q_1}(I;L^{p_1}(\Omega))\cap L^{p_2}(I;L^{p_2}(\Omega))\hookrightarrow L^...