Tagged Questions
2
votes
1answer
48 views
Weak $L^{p}$ spaces are quasi-normed?
Let $(X,\mathcal{B}, \mu)$ be a measure space. Then for $0< p < \infty$ by definition
$L^{p,\infty}(X,\mathcal{B}, \mu)$ is the class of all measureable functions $f$ such that
...
3
votes
1answer
32 views
Are the continuous functions on $G$ dense in $L^{1}(G)$?
If $G$ is a locally compact group, is the set $C_{c}(G)$ of all continuous functions on $G$ with compact support dense in $L^{1}(G)$?
3
votes
1answer
32 views
Why are Haar measures finite on compact sets?
I'm working through the answer by t.b. to another user's question here:
A net version of dominated convergence?
because I am trying to work through a related problem and I think it will be ...
3
votes
0answers
20 views
Are Haar measures complete?
If $G$ is a locally compact group and $\mu$ is a left Haar measure for $G$, then is the measure space $(G,B(G),\mu)$ complete (where $B(G)$ is the set of Borel subsets of $G$)?
Or do we have to take ...
1
vote
0answers
16 views
Convergence of an Integral in a locally compact group
I'm trying to finish an exercise which I asked about earlier here:
Mapping $G$ into its group algebra as left multiplication. Continuous?
$\bf{\text{The setting:}}$
Let $G$ be a locally compact ...
2
votes
0answers
36 views
A few questions about Measure Algebras
I've written up some of my understanding as well as I can of the Measure Algebra, trying to see the details behind a very brief treatment. There a couple places where I cannot make see how to make ...
4
votes
1answer
67 views
Involution in $L^{1}(G)$ is isometric.
(Sorry for asking so many questions of the same type. There is an underlying issue that I think once resolved will allow me to understand them all at once.)
Let $G$ be a locally compact group, and ...
2
votes
1answer
35 views
How to use the modular function of a locally compact group?
Let $G$ be a locally compact group with left Haar measure $\mu$.
Let $\delta:G\to(0,\infty)$ the unique modular function such that $\delta(x)\mu(E) = \mu(Ex)$ for all Borel sets $E$ and $x\in G$.
...
2
votes
1answer
51 views
Why does this inequality for all characteristic functions imply it for simple functions?
This question is probably obvious, but I'm not seeing how to obtain it.
A simple function is said to be finitely simple if its support is of finite measure. Let $(X_1,\mu_1)$, $(X_2,\mu_2)$, and ...
1
vote
1answer
38 views
Lorentz Space property: $\left\| f \right\|_{L^{q,s}} \leq \lim\limits_{n\to\infty} \| f_n \|_{L^{q,s}}$
I would like to understand a statement similar to Fatou's Lemma in the Lorentz space setting. It is as follows.
Suppose $0 < q,s < \infty$ and $f_n,f$ are measurable functions on a
...
4
votes
1answer
182 views
Hardy-Littlewood-Sobolev inequality for $p=1$
Let $\mu $ be a positive Borel measure on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{d}$ such that $\mu \left( B\left( a,r\right) \right) \leq Cr^{n}$ for some
$n\in (0,d]$ ...
6
votes
1answer
183 views
$1/|x|^n$ is not integrable
Let $\mu $ be a positive Borel measure on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{d}$ such that $\mu \left( B\left( a,r\right) \right) \leq Cr^{n}$ for some
$n\in (0,d]$ ...
1
vote
1answer
103 views
How to evaluate integrals with respect to Lebesgue measure on the unit sphere?
Let $\sigma$ be the "normalized Lebesgue" (Haar, really...) measure on the unit sphere $S=S^{n-1} \subset \mathbb R^n$. That is, $\sigma$ has support $S$, it is uniformly distributed, and $\int_S ...
7
votes
1answer
189 views
How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function
We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$.
...
4
votes
1answer
135 views
Radial limits of harmonic conjugate and Hilbert transform
Let $\mu$ be a real measure on the circle $\mathbf{T}$. Then the function
$$f(z)=\int_\mathbf{T} \mathrm{Im}\left(\frac{\zeta+z}{\zeta-z}\right) d\mu(\zeta)$$
is harmonic on the unit disc and its ...
6
votes
1answer
147 views
Basics of Haar measure
I feel totally confused about the sentence "therefore, ... because $dgh^{-1}$ is another left-invariant measure." What is the reason for "therefore"? Why is $dgh$ a left-invariant measure? (It seems ...
3
votes
0answers
173 views
Computing the modularity function of upper triangular matrices
Put $B_p := \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \in GL_2(Q_p) : a, b, c \in Q_p \right\}$
the subgroup of upper triangular matrices in $GL_2(Q_p)$, $Q_p$ denoting the $p$-adic ...
3
votes
1answer
67 views
Reference for: $G$ discrete iff the measure algebra $M(G)$ is weakly amenable.
I search the reference for the proof of the following theorem:
Let $G$ be a locally compact group. Then
the group $G$ is discrete if and only if the measure algebra $M(G)$ is weakly amenable.
The ...
3
votes
2answers
147 views
Shifting a function is continuous
I'm slightly puzzled by the following: if $g(t)$ is a function in $L^q(X)$ then we can show that $g(t-x)$ is continuous function of $t$, i.e. for $\varepsilon > 0$ we can find $\delta$ such that ...
1
vote
1answer
99 views
Truncation in Lorentz spaces
I am reading a paper, whose author state the following: if $f \in L^{(q,\infty)}(\mathbb{R}^N)$, then $f_\delta \in L^p(\mathbb{R}^N)$ for every $p \in [1,q)$, where $\delta > 0$ and
$$
f_\delta = ...
6
votes
1answer
289 views
A net version of dominated convergence?
Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
1
vote
0answers
142 views
A function in BMO space
Let $\psi:[0;1]\to\mathbb R$ is a nonnegative measurable function. Let $b_d(x)=1_{B(0,1)}\cdot{\rm sgn}(\sin (\pi d|x|))$, where $d\in\mathbb N$. Here $1_{B(0,1)}$ is the charateristic function of the ...
3
votes
2answers
208 views
Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space
Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has ...
4
votes
1answer
408 views
Approximation of the identity and Hardy-Littlewood maximal function
The inequality seems to be simple but I could not find the right limits of integration.
$$\sup_{\delta>0} |f*K_{\delta}|(x)\leq c f^*(x)$$
Where is some positive constant, $f$ is integrable, ...
11
votes
1answer
328 views
Properties of Haar measure
Let $G$ be a locally compact group (but not discrete) and let $m$ be its left Haar measure. Is it true that $\forall \epsilon$ $\exists$ $C$ such that $C$ is a compact neighborhood of the identity and ...
2
votes
1answer
128 views
Surjective endomorphism preserves Haar measure
How to prove the following statement:
Let $G$ be a compact topological group and let $m$ be the Haar measure on it. Let $\varphi$ be a continuous endomorphism of $G$ onto $G$, i.e., the map $\varphi$ ...
6
votes
1answer
498 views
Theorem of Steinhaus
The Steinhaus theorem says that if a set $A \subset \mathbb R^n$ is of positive inner Lebesgue measure then $\operatorname{int}{(A+A)} \neq \emptyset$. Is it true that also ...
1
vote
0answers
157 views
Harmonic measure
could anybody will help me to do this problems:
Let $\mathcal D$ be the unit disk a Set $E\subseteq\partial\mathcal D$ has harmonic measure identically $0$ with respect to $\mathcal D$. What can you ...
2
votes
1answer
116 views
Restricted Direct Products in Koch's Number Theory
On p.353 of Number Theory: Algebraic Numbers and Functions by Helmut Koch, he considers a group $G$ which is the restricted direct product of the locally compact abelian groups $G_i$ with respect to ...
1
vote
3answers
157 views
Why is it useful to express PDE solutions as $L^2$-convergent series?
The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the ...
