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3
votes
1answer
141 views

Generalisation of Lebesgue differentiation theorem to Orlicz spaces

If $f\in L_{p}^{\rm loc}(\mathbb{R}^{n})$ and $1\leq p<\infty$, then a stronger version of Lebesgue differentiation theorem holds: $$\lim\limits_{r\rightarrow ...
6
votes
1answer
203 views

For discrete groups, does the Haagerup property imply the AP of Haagerup-Kraus?

I don't expect to find an explicit counterexample to my question, because any example which was known to have the Haagerup property yet not have AP would have given an exact group without AP, and the ...
4
votes
2answers
68 views

Reference request : Besov spaces on ubounded domains

As I am relatively new to these matters, I would like to know if you could provide me a reference for Besov spaces on unbounded domains, because when I checked the first tome of Triebel's Theory of ...
1
vote
0answers
80 views

Arveson spectrum for a unitary representation of a group on a Hilbert space

Although this is not research, I think the question is a little bit too specific for math.stackexchange Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a ...
7
votes
1answer
102 views

Can the Cesaro limit of a positive definite function be negative?

Let $G$ be a countable amenable group and $\gamma:G\to\mathbb{C}$ a positive (semi)definite function (i.e. such that $\gamma(g^{-1})=\overline{\gamma(g)}$ and $$\sum_{g,h\in ...
8
votes
6answers
1k views

number theory which is close to analysis

I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis. ...
2
votes
1answer
102 views

Commutator with Hilbert transform

Let $H$ be Hilbert transform on $L^p({R})$. It is well known that commutator of $H$ and function $b\in BMO$ is a bounded operator in $L^p({R})$: $$\|[H,b] u\|_p\leq c_p\|b\|_{BMO}\|u\|_p, ...
3
votes
1answer
84 views

Existence of doubling non-Polish metric measure spaces

Let $(X,d,\mu)$ be a metric measure space (i.e. $(X,d)$ is a metric space and $\mu$ is a Borel measure on $X$). Let's say that $X$ is doubling if there exists a constant $C \geq 1$ such that $0 < ...
8
votes
1answer
107 views

Nonconventional ergodic averages for commuting transformations

Let $S$ and $T$ be commuting measure-preserving transformations of a standard probability space $(X,\mu)$, so $S$ and $T$ define an action of $\mathbb{Z}^2$ on $(X,\mu)$. I am wondering about ...
5
votes
1answer
408 views

Green's function of the Ornstein-Uhlenbeck operator

The Ornstein-Uhlenbeck operator $L$ is given by $$ Lu = \Delta u- \frac{1}{2}x\cdot \nabla u. $$ Is there a known closed form expression of the Green's function of $L$ on $\mathbb R^d$ (for $d\geq 2$ ...
1
vote
1answer
48 views

How to prove that the convolution operator associated to a discrete measure on a LCA group has natural spectrum?

Let $\mu$ be a Borel measure with finite variation on a locally compact abelian group $G$, let $\Gamma$ denote the dual group of $G$, and let $\hat \mu: \Gamma \to \mathbb{C}$ be the Fourier-Stieltjes ...
1
vote
0answers
63 views

Find $U \in H^1(\Omega \times (0,\infty))$ such that $\nabla E(u-\bar u)\nabla U \geq 0?$ (PDE harmonic extension)

Let $\Omega$ be a bounded smooth domain. Given $u \in H^{\frac 12}(\Omega)$ with mean value $\bar u = 0$, let $Eu = v \in H^1(\Omega \times (0,\infty))$ solve $$\int_0^\infty\int_\Omega \nabla v\nabla ...
0
votes
1answer
133 views

A $C^{*}$ algebra associated to a group [closed]

Let $G$ be a compact group which act on a Hilbert space $H$. We define a linear map $T$ on the dual space $H^{*}$ with $$T(\phi)(x)=\int_{G} \phi(g.x)$$ The integration is based on the Haar ...
1
vote
2answers
108 views

Growth of a harmonic function on the disc

Here is my question : I have a harmonic function $h$ on the open unit disc in $D \subset \mathbb{C}$, such that $\iint_D e^{2h} d\lambda(z) \leq A < \infty$ ($d\lambda$ is the Lebesgue measure on ...
0
votes
0answers
43 views

Control of Hessian by its trace in a bounded domain

We know if $u:\mathbb{R}^n \to \mathbb{R}$ is a smooth function with compact support, then we can show, via integration by parts, that $$ \|\Delta u\|=\|\nabla^2u\| \tag 1 $$ where $||\cdot||$ is ...
3
votes
0answers
66 views

Is Wiener's Tauberian theorem true in Wiener space?

Let $\gamma$ be the standard product Gaussian measure in $\mathbb{R}^\infty$, and let $\mu$ be a finite variation measure, not necessarily positive, such that $\mu \ll \gamma$. Is the following true? ...
4
votes
1answer
611 views

Cricket and the Hardy-Littlewod maximal function

I'v read somewhere that one motivation for Hardy to define his maximal function is the game of cricket. But I can't see how they are related. Could anyone provide some more information on their ...
1
vote
1answer
55 views

separating two parameters in an oscillatory integral

Consider the following oscillatory integral with two parameters: $$I(a,b)=\int_{\mathbb{R}}e^{i(ax^2+bx)}\psi(x)\,dx$$ where $\psi$ is smooth and supported in $\{x:|x|\in[1/2,2]\}$. Can we write ...
0
votes
1answer
91 views

Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$

My singular integral operator is defined by \begin{align} Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t}, \end{align} that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast ...
3
votes
2answers
127 views

Root of positive function in Fourier algebra

Let $G$ be a locally compact group, let $A(G)$ be the Fourier algebra of $G$. We think of $A(G)$ as a subalgebra of $C_0(G)$. Question 1: Let $f\in A(G)$ be a function that is pointwise positive. ...
3
votes
1answer
360 views

Understanding Bruhat's notion of Schwartz function

I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwartz-Bruhat space ...
1
vote
0answers
172 views

What is the spectrum of $L^1(G:H)$?

Let $H$ be a compact subgroup of a locally compact topological group $G$ and $$ L^1(G:H)=\{f\in L^1(G): R_h f=f\;\text{ a.e. }\; \forall h \in H\}$$ and $\widehat{(G:H)}=\{\xi\in ...
0
votes
0answers
85 views

Explicit formula for Bergman kernel on the unit ball

On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is ...
0
votes
0answers
70 views

Condition for boundedness in stationary phase theorem

I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1. Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and ...
1
vote
0answers
79 views

How do functions operates in a Fourier algebra $A^{q}(\mathbb T)$?

We put , $A^{q}(\mathbb T)= \{ f\in L^{q}(\mathbb T): \hat{f}\in \ell^{q}(\mathbb Z) \}.$ By Helson-Kahane-Katznelson-Rudin Theorem, it follows that, "Let $F$ be a function on $\mathbb C$ and if ...
0
votes
1answer
111 views

Does Trudinger inequality implies this critical Sobolev embedding?

Let $1<p<\infty$, $\mathrm{L}^p_s(\mathbb{R}^n)=J_s(\mathrm{L}^p(\mathbb{R}^n))$, where $J_s=(I-\Delta)^{-\frac{s}{2}}$, or $\mathscr{F}(J_sf)(\xi)=(1+|\xi|^2)^{-\frac{s}{2}}\hat{f}(\xi)$. And ...
2
votes
1answer
95 views

Approximation of subharmonic functions

Let $u$ be an (upper semi-continuous) locally bounded subharmonic function in a domain in $\mathbb{R}^n$. Let $\chi_\epsilon$ be a standard smoothing kernel, namely ...
3
votes
1answer
174 views

Which group algebras in analysis are “true group algebras”?

Let $G$ be a group, $A$ a unital associative algebra over ${\mathbb C}$, and let us call a representation of $G$ in $A$ an arbitrary map $\pi:G\to A$ such that $$ \pi(1)=1,\qquad \pi(a\cdot ...
3
votes
2answers
230 views

Fourier transform of compactly supported distribution is smooth

My advisor made the comment that if $u\in \mathcal{E}'$ is a compactly supported distribution, then $\hat{u}(\xi)\in C^{\infty}(\mathbb{R}^n)$ is actually a smooth function (not merely a distribution ...
12
votes
2answers
398 views

Does the Legendre-Hadamard condition imply a generalized Gårding inequality?

For simplicity, we restrict to constant coefficients. Let $A^{ij}_{ab} \in \mathbb{R}$, $1 \le i, j \le n$ and $1 \le a, b\le m$, satisfy the Legendre-Hadamard condition: $$ ...
0
votes
1answer
107 views

Uniform $L^p-L^{p'}$ bound of a Fourier multiplier

Let $(\tau,\xi)\in\mathbb{R}\times \mathbb{R}^n$, and consider the function $$ m_{\epsilon}(\tau,\xi)=\frac{1}{\tau+|\xi|^4+\epsilon|\xi|^2+i} $$ in $\mathbb{R}^{n+1}$. My first question is that does ...
0
votes
1answer
143 views

What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases?

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an orientation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data ...
0
votes
0answers
86 views

Can we conclude that $\varphi(L_xf_0)\neq0$ for every $x\in G$?

Let $H$ be a compact subgroup of locally compact topological group $G$ and $A=\{f\in L^1(G); R_hf=f(a,e)\forall h\in H\}$ as a subalgebra of $L^1(G)$ by convolution of $L^1(G)$. If $\varphi \in ...
5
votes
1answer
194 views

Can we extend a multiplicative linear functional of a closed left ideal on whole of the algebra?

Let B be a closed left ideal of a Banach algebra A. Also, B has a right approximate identity (in B). If g is a nonzero multiplicative linear functional on B, can we always extend g to a ...
0
votes
0answers
81 views

Is each multiplicative linear functional on $L^1(SL(2,R):SO(2,R))$ is triviall?

We know that if $G=SL(2,R)$ and $H=SO(2,R)$ as a compact subgroup of $G$, then $\{\xi \in \hat{G}; \xi|_H=1\}$is triviall. Can we conclude that each multiplicative linear functional on $\{f\in ...
2
votes
1answer
87 views

Is a matrix element of a norm continuous representation always a trigonometric polynomial?

I asked a similar question for the case of compact groups not long ago in math.stackexchange. Now I understand that the answer was "yes", and I want to modify that question. This is also related to my ...
6
votes
1answer
141 views

Trigonometric polynomials on non-compact and non-abelian groups

I asked this initially in math.stackexchange, but it disappeared almost immediately, so I hope it will be proper to aks this here. Hewitt and Ross define trigonometric polynomial on a locally compact ...
5
votes
2answers
290 views

Conditions for positivity of Fourier transform

Assume you are given a non-negative, continuous, radial function $f\in L^q(\mathbb{R}^3)$ (for any $q\geq 1$). Are there any conditions which would guarantee that the Fourier transform of $f$, that ...
0
votes
0answers
68 views

variation norm of a Fourier transform

Motivated by certain uniform estimate in oscillatory integrals, I am now trying to calculate the Fourier transform of the function ${\large e^{i|t|^{\epsilon}}/t}$ on $\mathbb{R}$, where $\epsilon\in ...
0
votes
1answer
108 views

Heat Equation on LCA Group

I am reading The Laplacian on A Riemannian Manifold. In the book, author defines the heat equation on manifold. In the situation $\mathbb R$ and $\mathbb S$, people often use Fourier transformation ...
6
votes
1answer
218 views

Fractional Laplacian and stereographic projection

The ordinary Laplacian on $\mathbb{R}^N$ behaves nicely under a stereographic projection onto $\mathbb{S}^N\setminus\{P\}$. (Here $P$ is either the north or south pole of the unit sphere ...
1
vote
0answers
53 views

symbol $m\in L^{\infty}$ implies any boundedness of a bilinear operator?

For a linear multiplier operator $T(f)(x)=\int_{\mathbb{R}} m(\xi)\hat{f}(\xi)e^{2\pi ix\xi}d\xi$, we know that $\|m\|_{\infty}$ gives the operator norm of $T$ from $L^2$ to itself immediately. What ...
18
votes
2answers
3k views

Is this statement which relates the Fourier transform of a function to its singularities correct?

I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
54
votes
4answers
3k views

How Does My Radio Work?

Bear with me for a moment while I invoke the real world; the main question at the end is purely mathematical. I live in an area with $n$ AM radio stations and $m$ FM radio stations. AM station ...
-1
votes
1answer
157 views

What is the advantage of the knowledge of jumps for approximating a function with trigonometric polynomials?

Let $f:(a,b)\to\mathbb{R}$ be square integrable, bounded variation and piecewise continuous function. Let the points of jump be $\{x_i/a<x_i<b,i = 1,2,3,...n\}$. The goal is to approximate the ...
1
vote
0answers
70 views

Eigenvalues of the Casimir (Laplace) operator on noncompact semi simple Lie groups

Let $G$ denote a noncompact semi-simple Lie group of real rank $1$ and let $K$ be a maximal compact subgroup. The Casimir (Laplace-Beltrami) operator acts by scalar on every irreducible representation ...
2
votes
1answer
247 views

Connection between the Fourier transform of f and |f|

If $f\in L^p(R)$ with $1\leq p\leq 2$, then Hausdorff-Young inequality implies that the Fourier transform $\widehat{f}\in L^{p'}$, $p'$ is the dual exponent of $p$, and $$ ...
1
vote
0answers
92 views

the (2,2,1) boundedness of a “product” operator

Let $\{E_j\}_{j\in\mathbb{Z}}$ and $\{F_k\}_{k\in\mathbb{Z}}$ be two collections of pairwise disjoint sets in $\mathbb{R}$. Let $C(j,k)$ be a bounded function (e.g. $|C(j,k)|<1$) defined on ...
7
votes
1answer
418 views

Tate's thesis for Artin L-functions

As far as I know, Tate's thesis has been successfully applied in two fronts: Hecke L-functions, by Tate and Iwasawa (and Teichmüller, Witt, Schmid) Automorphic L-functions, by Jacquet, Shalika, ...
2
votes
1answer
327 views

Beurling density and interpolation

Let $\Lambda=\{\lambda_n\}_1^\infty$ a set of points on the real line. We denote by $\bar{n}(r)$ the largest number of points in any interval $[x,x+r]$, $r>0$. Define the upper uniform density ...