**1**

vote

**1**answer

122 views

### Does the product function $fg$, where $f$ is in $L^2$ and $g$ is in $C^{\infty}_0$ belong to hardy space $H^1$?

I am struggling to know whether the product function $fg$, where $f$ is in $L^2$ and $g$ is in $C^{\infty}_0$ belong to hardy space $H^1$.
$fg$ has compact support but I can't figure out how I can try ...

**0**

votes

**0**answers

32 views

### prove that K is bounded operator [closed]

Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space, $1\leq p <\infty$, and suppose that $k:X\times X\rightarrow \mathbb{F}$ is an $\Omega \times \Omega$ measurable function such that for $f$ ...

**4**

votes

**0**answers

112 views

### Distributions on the Jacquet module $C_c^\infty(X)_{H,\chi}$ and functions

Let $X$ be an $\ell$ space (in the sense of Bernstein-Zelevinski), $H$ be an $\ell$ group which acts on $X$ and $\chi$ be a character of $H$. Denote $C^\infty(X)^{H,\chi}$ the space of locally ...

**5**

votes

**1**answer

174 views

### A sufficient condition (or not) for positive semidefiniteness of a matrix?

Let $A\in M_{n}(\mathbb{C})$ be a Hermitian matrix. If for all $z_1,...,z_n\in\mathbb{C}$, $$\sum_{i,j=1}^{n}A_{ij}z_{i}\overline{z_{j}}\ge 0$$ then A is positive semidefinite.
I do not think the ...

**2**

votes

**1**answer

224 views

### upper bound on derivatives of a function defined on an arc

This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here.
You can skip examples below and read from "General setting" at ...

**0**

votes

**1**answer

66 views

### Question on Littlewood-Paley trichotomy

In proving the product estimate, we need the Littlewood-Paley trichotomy. See http://www.math.ucla.edu/~tao/254a.1.01w/notes3.ps.
In the decomposition
$$P_k (fg)=\sum_{k',k''\in Z} P_k (P_{k'} f ...

**1**

vote

**1**answer

109 views

### The reproducing kernel for harmonics on compact manifolds

Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") ...

**12**

votes

**1**answer

286 views

### Definition of discrete spectrum and continuous and basic properties

I apologize if this is too basic for MO.
I have an embarrassing admission to make: I don't know the actual definition of the discrete/continuous spectrum of a reductive group $G/\mathbb{Q}$ (in the ...

**2**

votes

**0**answers

82 views

### Primitive ideal space and unitary dual of a [SIN] group - when are they Hausdorff?

Recall that a locally compact group $G$ is said to be
an $[FC]^-$ group, if each conjugacy class in $G$ has a compact closure;
an $[SIN]$ group, if each neighborhood of the identity includes a ...

**1**

vote

**0**answers

37 views

### The decay rate of the spectrum of the Gaussian kernel on compact manifolds

It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...

**4**

votes

**1**answer

223 views

### Weil's Haar measure construction from below

Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function.
I would need to know something similar for an ...

**0**

votes

**0**answers

146 views

### Johnson's Theorem - Proof (Runde) Clarification

I am reading Runde's Lectures on Amenability. In the proof of Johnson's theorem where he proves "$L^{1}(G)$ is amenable Banach algebra implies $G$ is amenable" : he defines a $L^1(G)$ bimodule action ...

**0**

votes

**0**answers

48 views

### Bounds on the the spherical harmonics on $S^{p-1}$

The only reference I could find in this regard is for upper bounding the n-homogeneous spherical harmonics on $S^{p-1}$ as in equation 4.29 here, ...

**0**

votes

**0**answers

8 views

### Does Weil-Brezin transform provide Fourier basis in C^0 on Heisenberg manifold?

I know that $L^2$ functions of the Heisenberg nilmanifold are spanned by images of the Weil-Brezin transform. Is it true that they are also dense in C^0? Is there a reference for this?

**2**

votes

**1**answer

201 views

### Does Fourier Algebra of locally compact group separate compact sets of the group?

Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of ...

**3**

votes

**1**answer

724 views

### Two questions on Elias Stein paper (1976)

I am working on some results related with a paper of Elias Stein (on the almost every where convergence of wavelet summation methods), and I have the following questions:
The maximal function ...

**3**

votes

**0**answers

161 views

### Non-compact analogue of Peter-Weyl

I have the following situation: $G$ is a real unimodular locally compact semisimple Lie group. Then it is known that the regular representation $H:=L^2(G,\mu_H)$ decomposes as
\begin{equation}
...

**5**

votes

**1**answer

612 views

### Is the following integral nonzero?

Recently I met an integral as follow:
$$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq ...

**2**

votes

**1**answer

104 views

### condition for $f \in H^{1/2} \cap C^0$

I need to show that for $f \in H^{1/2}(S^1) \cap C^0(S^1)$ we have :
$$ \iint\limits_{S^1 \times S^1}{ \frac{|f(x)-f(y)|^2}{\sin^2(\pi(x-y))}dx dy}< + \infty $$
and that, conversely, if $f$ is a ...

**4**

votes

**0**answers

90 views

### Well-definedness on $C_{0}^{\infty}(\mathbb{R}^{n})$

Let $T$ be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel $K\in CZK_{\alpha}$ of order $\alpha>0$ and $b\in BMO(\mathbb{R}^{n})$. Then for $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ ...

**2**

votes

**1**answer

87 views

### A singular integral of several functions

While playing with some PDE I came across a singular integral that looks something like
...

**4**

votes

**0**answers

56 views

### L^1 maximal inequalities for the Ornstein-Uhlenbeck semigroup in infinite dimension

For an infinite-dimensional Gaussian random vector $X$ consider the Ornstein-Uhlenbeck maximal operator:
$M f(X) := \sup_{\rho \in [0,1]} \mathsf{E} [f(\rho X + (1-\rho^2)^{1/2} X^\prime) \mid X]$
...

**6**

votes

**1**answer

318 views

### Liouville property - a very basic question

Let $\mathbb{F}_2$ be the free group on two generators. By a result of Kaimanovich and Vershik, for each measure $\mu$ on $\mathbb{F}_2$ such that the support of $\mu$ generates $\mathbb{F}_2$, we ...

**2**

votes

**0**answers

39 views

### Restricted weak type bound at the endpoint

We know that if we have an operator that is (restricted) weak type $(p,p)$ and (strong) type $(\infty,\infty)$ with norm 1, then it's also of strong type $(q,q)$ for all $p<q<\infty$ by the real ...

**3**

votes

**2**answers

73 views

### Sequence of subharmonic functions on shrinking domains

Set $G_\eta:=\{(x,y)\in \mathbb{R}^2|-\eta<x<\eta, 0<y<1\}$. If $u_\eta\geq 0$ is a sequence of subharmonic functions defined on $G_\eta$ such that
$$
\int_{G_\eta}|u_\eta|^2dx\wedge ...

**1**

vote

**1**answer

118 views

### Harmonic/Subharmonic lifting of functions on an annulus

Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h\in ...

**4**

votes

**1**answer

207 views

### Application of Factorization Theory to Oscillatory Integral Estimates

In the article "Some New Estimates on Oscillatory Integrals" by Bourgain in the book Essays in Honor of Elias M. Stein, Bourgain considers operators of the form
...

**3**

votes

**1**answer

140 views

### Hardy-Littlewood-Sobolev inequality in Lorentz spaces

Hardy-Littlewood-Sobolev inequality states that if $1<p<q<\infty$, $1/r=1-1/p+1/q$, then we have
$$\left\|\frac{1}{|x|^{n/r}}\ast f\right\|_{L^q(\mathbb R^n)}\le\|f\|_{L^p(\mathbb R^n).}$$
...

**8**

votes

**1**answer

257 views

### Have the explicit Poisson-type formulas of Guinand and Meyer been observed before?

In a recent paper of Meyer Measures with locally finite support and spectrum PNAS vol. 113 no. 12:3152–3158 (behind a paywall, but see also these seminar notes) some new explicit Poisson-type formulas ...

**7**

votes

**1**answer

203 views

### When does a $C^*$-algebra have no nonzero projection?

Let $A$ be a $C^*$-algebra and $\hat{A}$ its spectrum of $A$,the set of classes of non-zero irreducible representation of $A$ endowed with hull-kernel topology. suppose $\hat{A}$ is a non-compact ...

**1**

vote

**0**answers

63 views

### Differentiability criterion in the Zygmund class

Let $ f: \mathbf{R}^{m} \rightarrow \mathbf{R} $ be a continuous
function, $ \omega $ be a modulus of continuity and assume
$$ | f(x+h) +f(x-h) -2f(x) | \leq \omega(|h|)|h| $$
whenever $ x,h \in ...

**2**

votes

**0**answers

68 views

### Bilinear Approach to Bochner-Riesz Conjecture in Two Dimensions

In some old lecture notes on the Restriction and Kakeya conjectures (Notes 5, specifically), Terence Tao gives a proof of the restriction conjecture (for the sphere) in two dimensions via a bilinear ...

**0**

votes

**0**answers

63 views

### Matrix representation of the Heisenberg quaternionic group

Equiped with the law $(a,b,c)\circ (a',b',c') = (a + a', b + b', c + c' + ab'),$ the matrix representation of the Heisenberg group $H^3= \mathbb C\times \mathbb R$ is given by
$$
\begin{pmatrix}
1 ...

**3**

votes

**2**answers

172 views

### Sampling Theorem for non-bandlimited Functions

The classical Shannon sampling theorem states that
a bandlimited function with $\mbox{supp } \hat f\subset [-1/2,1/2]$ can be uniquely determined by its samples $(f(i))_{i\in \mathbb{Z}}$ (The symbol ...

**7**

votes

**1**answer

306 views

### The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$?

I'm going through the last steps of Bourgain and Demeter's proof of the $l^2$ decoupling conjecture, but I'm unable to see how the first inequality in (43) goes through. I'll water down the question a ...

**7**

votes

**1**answer

240 views

### What does the unique mean on weakly almost periodic functions look like?

There is a unique invariant mean $m$ on WAP functions on any discrete group (see definitions below, theorem of ?). However, the proofs I found are fairly non-explicit on how to obtain this invariant ...

**2**

votes

**0**answers

32 views

### Global Euclidean Carleman Estimate with a linear phase

I am interested in deriving the following global Carleman estimate which I think should hold :
$ \| e^{\tau \phi} \triangle u \|_{L_{\delta}^2({\mathbb{R^3})}}> C \tau \| e^{\tau \phi} u ...

**13**

votes

**0**answers

439 views

### A possible mistake in Walter Rudin, “Fourier analysis on groups”

I have the following lemma 4.2.4 on page 80 in the book (we have locally compact abelian topological groups $G_1, G_2$ and their duals $\Gamma_1, \Gamma_2$):
Suppose $E$ is a coset in $\Gamma_2$ ...

**0**

votes

**2**answers

98 views

### $L^p$ estimates for elliptic equation of divergence form

Consider the scalar elliptic equation of divergence form
$$div((1+a)\nabla\pi)=div F\ \ in\ \ R^3,$$
where $a$ is a Schwartz function with $1+a\geq c=const>0$, $F=(F_1,F_2,F_3)$ is a vector-valued ...

**4**

votes

**0**answers

50 views

### Sobolev spaces defined on non-compact Lie groups

In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and ...

**1**

vote

**0**answers

103 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 ...

**13**

votes

**1**answer

411 views

### Does there exist some $C$ independent of $n$ and $f$ such that $ \|f''\|_p \geq Cn^2 \| f \|_p$, where $1 \leq p\leq \infty$?

Let $f$ be a trigonometric polynomial on the circle $\mathbb{T}$ with $\hat{f}(j) = 0$ for all $j \in \mathbb{Z}$ with $\lvert j \rvert < n$. Does there exist some $C$ independent of $n$ and $f$ ...

**8**

votes

**1**answer

163 views

### Real rank zero of group $C^*$-algebras

The concept of real rank zero of a $C^*$-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if $X$ is a ...

**1**

vote

**1**answer

56 views

### Multivariate analogue of Jackson's inequality's modulus of continuity form

According to Jackson's inequality, there is $c > 0$ s.t. for any continuous function $f: S^1 \rightarrow \mathbb{R}$ (where $S^1 := \mathbb{R} / \mathbb{Z}$ is the circle) and integer $n$ there is ...

**4**

votes

**0**answers

196 views

### A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$
where $a,b$ are two given points in the plane and $\lambda$ is a constant.
Now we consider the ...

**2**

votes

**1**answer

135 views

### Rational valued functions on the Cantor set with $\int_{C} f^{3}d\mu=1 $

Let $C$ be the Cantor set as a compact Abelian topological group, isomorphic to countable product of $\mathbb{Z}/2\mathbb{Z}$.
Its normalized Haar measure is denoted by $\mu$.
Is there a ...

**6**

votes

**0**answers

98 views

### How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law
$$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$
For $z=x+ i y \in \mathbb C$ ...

**2**

votes

**0**answers

74 views

### Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$,
$\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$
where the scalar complex function ...

**1**

vote

**0**answers

76 views

### Subgroups of finite non-zero Haar measure of abelian locally compact groups

Is it true that every subgroup of finite non-zero Haar measure of an abelian locally compact group should be open and compact? This is obviously true for the case of discrete abelian groups. Thanks.

**2**

votes

**1**answer

281 views

### Decoupling in mixed norm spaces

Bourgain and Demeter's proof of the $L^2$ decoupling conjecture decouples $\|f\|_{L^p}$ into an $L^2$ sum of $\|f_\theta\|_{L^p}$, where $\hat f$ is supported on a curved hypersurface $S$, where ...