All Questions
18 questions with no upvoted or accepted answers
9
votes
0
answers
240
views
What is known about when $vN(G)$ is a factor, for a locally compact group $G$?
When $G$ is a discrete group, it is an elementary result in the theory of von Neumann algebras that the group von Neumann algebra $vN(G)$ is a factor if and only if $G$ is an ICC group.
What is known ...
- 146
5
votes
0
answers
215
views
Explicit description of the Plancherel measure for $GL_n(\mathbb{R})$
Let $G:=\mathrm{GL}_n(\mathbb{R})$ and $f\in C_c^\infty(G)$. One can uniquely determine the Plancherel measure $d\mu_p$ on $\hat{G}$, the unitary (actually tempered) dual of $G$, by the equation
$$f(g)...
- 1,692
4
votes
0
answers
68
views
Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$
For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that
$$
|f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
- 61
4
votes
0
answers
264
views
Is the Gelfand transform strictly continuous?
Let $M$ be the Banach algebra of measures on the circle with $L_1$ naturally sitting as a closed ideal of $M$. Then $M$ carries the strict topology implemented by the family of seminorms $\|\mu\|_f = \...
- 113
3
votes
0
answers
317
views
Best constant for maximal function for locally compact groups
Maximal functions for locally compact groups have been studied. In particular papers of K. Phillips and Phillips-Taibleson (here and here) contain nice results in this direction. It might be too much ...
- 1,583
3
votes
0
answers
140
views
convergence of $e^{it\Delta}f$
I heard of a conjecture that
$e^{it\Delta}f\rightarrow f$ a.e. as $t\rightarrow 0$ for $f\in H^{1/4+\epsilon}$
but couldn't find a proper reference.
- 31
2
votes
0
answers
169
views
Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?
Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
- 839
1
vote
0
answers
53
views
Reference for Density question
Let $ B $ be a reflexive, separable Banach space and $ p \in (1,\infty)$. Then denote by $L^{p}(B)$ the space of all functions $$ f : \mathbb{R}^{n} \to B $$ with
$$
\int_{\mathbb{R}^{n}} \vert f \...
- 31
1
vote
0
answers
82
views
For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$
For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such ...
- 835
1
vote
0
answers
148
views
BMO estimates of singular integral operators on torus
I have the following elliptic problem:
$$ \Delta u = \operatorname{div}\operatorname{div}S, $$
where $S=(S_{i,j})\colon \mathbb{T}^n\to \mathbb{R}^{n\times n} $ is bounded and $\mathbb{T}^n$ is the $n$...
- 123
1
vote
0
answers
62
views
Properties of the Fourier Transform of Countably Supported Functions on $[0,1)$
Identifying $\mathbb{R}/\mathbb{Z}$ with the interval $\left[0,1\right)$, let $C_{\textrm{coun}}\left(\mathbb{R}/\mathbb{Z}\right)$ denote the set of all functions $f:\mathbb{R}/\mathbb{Z}\rightarrow\...
- 1,284
1
vote
0
answers
74
views
Dimension dependence: boundedness result of the fractional Riesz integral
I am looking for the best known constant in the boundedness result of the fractional Riesz integral. In particular, I am interested in the dependence on the dimension $d$ and on the parameter $\alpha&...
- 778
1
vote
0
answers
122
views
Metric transforms that preserve $\ell^1$ embeddability
Consider a function $f$ from reals to reals such that $f$, when applied to pairwise Manhattan distances between $n$ points, always results in a set of Manhattan distances.
Work by Schoenberg and ...
- 93
1
vote
0
answers
109
views
Is this a positive definite kernel?
Under which conditions on the function :
\begin{array}{l|rcl}
K : & \mathbb R^+ & \longrightarrow & (0, 1)\\
&t & \longmapsto & K(t) \end{array}
is the symmetric ...
1
vote
0
answers
62
views
Reference request - Compact embedding of intermediate space
Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.
...
- 679
1
vote
0
answers
269
views
$L^{p}(\mathbb R)\subset L^{1}(\mathbb R) \ast L^{p}(\mathbb R), (1< p< \infty)$?
Let $\mathbb T$ be a circle group.
In 1939, Salem, has shown that, every member of $L^{1}(\mathbb T)$ can written as a product(convolution) some other two members of $L^{1}(\mathbb T),$ that is, $L^...
- 1,051
0
votes
0
answers
34
views
Locally compact groupoid with range map restricted to isotropy groupoid is open
Suppose the action groupoid 𝐺=𝐻⋉𝑋, where 𝐻 is a locally compact group and 𝑋
a locally compact space is such that isotropy subgroups of H are isomorphic to each other.
Can this be an example of a ...
0
votes
0
answers
89
views
Maximal function on mixed $L^{p}$
Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is
$$
\Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
- 31