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0
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0answers
112 views

A question about multiplier algebra of $C_0(G)\otimes C_b(G)$ for a locally compact group $G$

Let $G$ be locally compact group. How we can show that $$ M(C_0(G)\otimes C_b(G))=C_b(G,C_b(G)). $$ ($M(C_0(G)\otimes C_b(G))$ is the multiplier algebra of $C_0(G)\otimes C_b(G)$)
3
votes
1answer
150 views

Plancherel formula for non-second-countable (non-unimodular) groups

The Plancherel formula for unimodular, second-countable, type 1 groups can be found in A Course in Abstrict Harmonic Analysis by Gerald Folland (theorem 7.44) or here. It states that we can get a ...
1
vote
0answers
160 views

Laplacian type operator on compact Lie group

Consider the operator $S = \sum_{i,j} X_{ij}^2$ on $L^2(SO(n+1))$, where $X_{ij}$ generates the rotation of the sphere $S^n$ in the $ij$-plane keeping the $(n - 2)$ complementary directions fixed. ...
0
votes
1answer
243 views

Littlewood-Paley theory and norm estimation

In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1. It is claimed that Lemma 2 is ...
6
votes
1answer
296 views

Could we interpolate the compactness of compact operators?

Classical theorems of Marcinkiewicz and Riesz and their extensions to general Banach spaces by Calderón, Lions, Peetre, et al. allow us to interpolate the continuity of two operators, viz., the ...
1
vote
0answers
99 views

Question about a oscillatory integrals on manifold

Let $M$ be a compact oriented Riemannian manifold without boundary. Set $f(x)=a(x)+\sqrt{-1}b(x)$ be a complex-valued function on $M$, where $a(x),b(x)$ are real-valued function on $M$. Then, how to ...
6
votes
2answers
426 views

For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?

(This is essentially a continuation of my previous question, here.) Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...
1
vote
1answer
207 views

Description of Bessel potential spaces

Hi, let $1 < p <\infty $, $ 0 < \alpha < 1$, and $ \mathscr{L}^p_\alpha(R^n) $ be the usual Bessel potential space defined by $$ \mathscr{L}^p_\alpha = (1-\triangle)^{-\alpha/2}L^p(R^n). ...
0
votes
0answers
177 views

Notation for a functional L2 matrix norm

Hi, Let $v(z)$ be a 2x2 matrix depending on a complex variable z, defined on an oriented contour $\Sigma$ in the complex plane. Can anyone tell me the meaning of the notation: ...
8
votes
2answers
758 views

Is there a “right” proof of Riemann's Theta Relation?

Let $\theta$ denote the usual Jacobi Theta function (with auxiliary parameter $\tau = i$, for simplicity), i.e. $$ \theta(z) = \sum_{n \in \mathbb{Z}} \exp(-\pi (a + n)^2 + 2 \pi i n z) \ . $$ I'm ...
3
votes
0answers
240 views

Harmonic analysis on the Heisenberg group

It is well known that: Theorem 1. For $f\in L^{2}(\mathbb H_{n}=\text{The Heisenberg group of dimesion } 2n+1)$ we have the expansion $$f(z, s)= (2\pi)^{-n} \sum_{k=0}^{\infty} \int_{0}^{\infty} f ...
17
votes
2answers
597 views

Image of L^1 under the Fourier Transform

The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is ...
5
votes
0answers
148 views

variant of Haar measure

I found a certain trig identity in a discussion on Lie groups \[ \frac{\prod_{i < j} 2 \sin (\mu_i - \mu_j) \prod_{i< j} 2 \sin (\nu_i - \nu_j) }{\prod_{i, j} 2 \cos (\mu_i - \nu_j) } = ...
4
votes
0answers
249 views

What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$?

Let $G$ be a locally compact group and let $ C_r^\ast(G) $ denote its reduced group $C^\ast$-algebra. Many features of a $G$ can be realized from $L^1(G)$ or $C_r^\ast(G)$. For example, $G$ is ...
3
votes
0answers
147 views

The polynomial Freiman-Ruzsa conjecture for the special case when $f$ is a bijection

The polynomial Freiman-Ruzsa conjecture states that there exists $k > 0$, such that for all $\epsilon > 0$, and for all large $n$ and all functions $ f: \mathbb{F}_2^n \mapsto\mathbb{F}_2^n$, ...
1
vote
0answers
82 views

Dependence between $\langle f(x), g(y) \rangle$ and $\langle x, y \rangle$.

Let $p$ be a large prime and $n \geq 3$ be an integer. Let $f$ and $g$ be two arbitrary bijections from $\mathbb{F}_p^n$ to $\mathbb{F}_p^n$. I want to find a condition on $f$ and $g$ such that the ...
7
votes
0answers
153 views

Inferring asymptotic behaviour from the dominant pole of the Laplace transform

Hi, I am reposting the following question with the hope that a more detailed description will lead to a more descriptive response: dominant pole in the laplace transform I have a vector function ...
1
vote
0answers
189 views

Gauss circle problem and Jacobi-type estimates for higher dimensions

Hello everyone, I was doing some late night random reading and I got to wonder about some stuff about the Gauss circle problem. To begin with, consider a circle in $\mathbb{R}^{2}$ with centre at the ...
2
votes
1answer
228 views

About the boundedness of a multiplication operator.

Let be $f$ a $2\pi-$periodic function and $\hat{f}(k)=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}dx$. Consider the operator: \begin{equation} Tf(x)=\sum_{k\in\mathbb{Z}}sign(k)\ \hat{f}(k)\ e^{ikx}. ...
6
votes
1answer
259 views

For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)?

Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ...
0
votes
1answer
243 views

An interpolation inequality.

For all $s>0$ define for $\epsilon\in(0,1)$ the function: \begin{equation} g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k. \end{equation} Prove that $\exists C>0$ and $\phi(s)$ such ...
3
votes
1answer
253 views

Invariant measures for Cellular automata

An easy question that I have never been able to answer. Suppose we have the CA on $\{ 0,1,2 \}^{\mathbb{N}}$ with local rule given by $f(x,y)=A_{x,y}$ and $A$ the $3\times 3$ matrix ...
1
vote
0answers
197 views

Pencils of circles and Liouville's theorem

Is there any relation (maybe implicit) between the conformal geometry in the space of circles and spheres and the study of harmonic functions? In the original question I was musing whether the ...
2
votes
1answer
129 views

Maximal function associated to convex bodies

Let $B$ be a centrally symmetric convex body in $\mathbb R^n.$ The maximal function associated to $B$ is defined by $$ Mf = \sup_{r>0}(\chi_{B})_{r}*|f|. $$ Bourgain ...
3
votes
0answers
108 views

Growth of inner functions on the disk

Recall that an inner function on the disk $D$ is a bounded analytic function on $D$ having radial limits of modulus one almost everywhere. There has been many works on the growth of the inner ...
3
votes
2answers
277 views

Sufficient condition for $L^p$ multiplier on the torus

Hello, I am searching for a reference of an " easy " sufficient condition insuring that a bounded sequence $(b_{\mathbf{n}\in\mathbb{Z}^d})\in\ell^\infty$ defines a bounded operator from ...
4
votes
2answers
250 views

“geometric” description of the algebra of central functions on a Lie group

I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff ...
2
votes
0answers
239 views

Continuity of multiplicative character

Let $G$ be a discrete group and $\beta (G)$ denote the Stone-Cech compactification of $G$, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication ...
6
votes
2answers
255 views

Orthonormal basis for $L^2(G/H)$.

Let $G$ be a locally compact group and $H$ be a closed subgroup of $G$. Is there any way to define a reasonable orthonormal basis for $L^2(G/H)$? By "reasonable" I mean elements of the orthonormal ...
4
votes
1answer
249 views

Support of functions on compact groups, and their Fourier transforms

I make my question more precisley : Let $f$ be a conjugation invariant function on a compact semi-simple Lie group $G$. So $f$ can be regarded as a function on the maximal tours ( or Lie algebra of ...
2
votes
1answer
251 views

Characters separating points on Maximal Torus modulo Weyl group?

Let G be a compact Lie group, for example, SU(n). Let T be its maximal torus. Let W be its Weyl group. Every finite-dimensional representation of G has a character, which is a function on G, T and ...
1
vote
0answers
106 views

L^p norms of Littlewood-Paley blocks

Is it possible to show that all the $L^p(\mathbb{R}^n)$ norms of a given Littlewood-Paley block are equivalent or else find a counterexample? One inequality follows from the Bernstein inequalities if ...
2
votes
2answers
251 views

Factorization of antisymmetric bounded holomorphic functions

A basic principle in complex function theory is that one can split off zeros of holomorphic functions in a similar way as for polynomials: If $f$ is holomorphic near $0$ and $f(0) = 0$, then $f(z) = ...
2
votes
2answers
328 views

Lebesgue differentiation theorem beyond Euclidean spaces

To formulate Lebesgue differentiation theorem one needs a metric and a measure. Apart from the Euclidean spaces i.e. $\mathbb R^d$, the theorem holds true for homogeneous groups (e.g. Heisenberg ...
4
votes
1answer
144 views

variation of the obstacle in the obstacle problem

Suppose $D \subset \Bbb C$ with smooth boundary. Let $f \in C^{1,1}(D)$. Let $\varphi$ be the supremum of all members in the set $$\lbrace g \in C^{\infty}(\overline{D})| g \ is \ subharmonic \ and ...
3
votes
2answers
383 views

Quotient of a compact Lie group by maximal Torus

I start with a noncompact connected semisimple Lie group with finite center $G$ and fix a maximal compact subgroup $K$ of $G$. I am considering these compact groups $K$. If $\mathbb T$ is the ...
15
votes
0answers
737 views

The Fourier Transform of taking Eigenvalues

The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...
1
vote
2answers
298 views

Harmonic polynomials on complex 2-space

Consider a complex-valued harmonic polynomial $f$ on ${\Bbb C}^2$ and assume that $f(0)=0$. Suppose also that $f$ does not vanish on the unit sphere $S^3\subset{\Bbb C}^2\simeq{\Bbb R}^4$. Does it ...
0
votes
1answer
253 views

Fixed norm problem for analytic functions

Hi there, I have the following problems on my hand: Given arbitrary positive sub-harmonic function $l(x,y)$ on plane, for all analytic functions: $f=a(x,y)+ib(x,y)$ on complex plane, consider the ...
2
votes
0answers
172 views

Using the Mellin transform to invert ill-posed problems of harmonic transforms

I will try to explain the problem using just words. Let's see how far I get! I will use the harmonic transform of the Radon transform for 2 or more dimensions as an example, but there is a large ...
3
votes
2answers
434 views

Corona Theorem in several variables

Hallo, I have read about the Corona Theorem (see link:http://en.wikipedia.org/wiki/Corona_theorem). From this one ca deduce that: Let $f_{1}, ..., f_{n}$ be holomorphic bounded functions on the unit ...
1
vote
2answers
199 views

Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps

My question is about the precise definition regarding the following: Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
6
votes
3answers
763 views

Property/Relations using Fourier series/transform, which give complete information about all the jump singularities of a function.

Consider a function which has only jump singularities of the form of the function itself or one of its derivatives jumping. Now let $\hat{f}(k)$ be its Fourier transform/series. We know the decay of ...
4
votes
3answers
422 views

Meaning of a quote of Doubilet,Rota and Stanley on harmonic analysis and combinatorics

The beginning of the paper On the foundations of combinatorial theory. VI. The idea of generating function (1972) link text says that Since Laplace discovered the remarkable correspondence ...
3
votes
1answer
405 views

Uncertainty principle (really for Mellin, but never mind that!)

Is there a smooth funtion $f:\mathbb{R}\to \mathbb{C}$ such that (a) $f(x)$ decreases faster than $e^{-e^x}$ when $x\to \infty$, (b) $\widehat{f}(t)$ decreases faster than $e^{-|t|}$ when $t\to ...
3
votes
1answer
148 views

Subharmonic envelope

I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me ...
6
votes
5answers
714 views

Background for Hejhal's "The Selberg Trace Formula for $PSL(2, \mathbb{R})$

Reposted from math.stackexchange where my question received only five views and no answers... I'm trying to learn the Selberg trace formula, but have very little background in harmonic analysis. I ...
5
votes
1answer
219 views

What is the Plancherel Measure for $\textrm{SL}_3(\mathbb{Q}_p)$?

I am looking for a description of the Plancherel Measure of $\textrm{SL}_3(\mathbb{Q}_p)$. Has this been calculated yet? I've search many places for it, but I've only found results on real/complex ...
1
vote
2answers
229 views

Approximating a function by fractional powers

Say I have a continuous function $f$ defined on a compact interval $I$ on the real line. As is well-known, I could approximate $f$ arbitrarily well by polynomials. Given $R>0$, how well can we ...
10
votes
1answer
327 views

Locally Compact Quantum Groups application

Recently Mathematicians in harmonic analysis become more and more interested in Locally compact quantum groups and try to transfer concepts from abstract harmonic analysis to the setting of locally ...