Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic ...

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58 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
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1answer
330 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
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0answers
45 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
2answers
74 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 dy\...
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1answer
126 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 H^1(\...
5
votes
1answer
296 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 $$S_{N}g(x):=\int_{\mathbb{R}^{n}}g(y)e^...
3
votes
1answer
154 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
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1answer
282 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
1answer
209 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 ...
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71 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
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77 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
0answers
67 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
2answers
173 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
1answer
321 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
1answer
264 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 ...
3
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69 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 e^{-\tau \phi} u \|_{L_{\delta+1}^2({\mathbb{R^3})}}> C \tau \| u \|_{L^...
13
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447 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
2answers
106 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
0answers
53 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 ...
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0answers
105 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
1answer
429 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
1answer
172 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
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1answer
57 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
0answers
216 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
1answer
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
0answers
104 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
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0answers
75 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
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0answers
79 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
1answer
289 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 $\...
0
votes
0answers
69 views

Necessary condition for decouplings for surfaces in $\mathbb{R}^4$

I'm currently studying the paper Decouplings for surfaces in $\mathbb{R}^4$ written by Bourgain and Demeter. This paper is available in here. As an example of nondegenerate $2$-dimensional surfaces ...
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44 views

Kernel for projection operators onto the spaces of piecewise linear loops

For every $m\in\mathbb{N}_+$, $k=0,\dots,m-1$, denote $I_{m,k}:=(\frac{k}{m},\frac{k+1}{m})$. Denote $W = H^1(S^1,\mathbb R^{2n})$ = The Hilbert space of $C^1$ maps maps $v(t)$ from the circle to $\...
2
votes
2answers
463 views

Pointwise convergence of Fourier series, Fefferman's article

This is the first time I ask a question here, so sorry if I make any mistake in the way I ask it. I'm studing Fefferman's article Pointwise Convergence of Fourier Series, and I have two questions: 1)...
8
votes
2answers
533 views

Interpret Fourier transform as limit of Fourier series

Let $V=\mathbb{R}^n$, $\Lambda_r=2\pi r \mathbb{Z}^n \subset V (r>0)$ a lattice; $V^*\cong\mathbb{R}^n$ the dual vector space of $V$, and $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z}^n =\text{Hom}(\...
0
votes
0answers
54 views

Definiteness and infinite divisibility of kernels including heat semigroup

Let $P_{t}$ be the usual heat semigroup. Can one show (preferably) or disprove that for arbitrary $k \in \mathbb{R}_{>0}$ and $n \in \mathbb{N}$ we have \begin{equation} \sum_{i,j=1}^{n}a_{i}a_{j}\...
2
votes
0answers
150 views

Groups with probability measures

Are there algebraic structures that integrate groups with probability measures? For instance, can the closure operation on a group be assigned a probability that says "how much" a member belongs to ...
2
votes
2answers
356 views

Are the zeroes of the Fourier Transform of compactly supported functions isolated?

I have a continuous function $f$ on a locally compact Abelian group $G$ with compact support, and I would like to say that the zeroes of $f$ are sparse in some sense (isolated would be good, uniformly ...
1
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1answer
130 views

harmonic balance method for a 2-mass 3-spring system [closed]

I am trying to solve a nonlinear 3spring-2mass system under harmonic loading by using Fourier series expansion of states of the differential equation. The system is just basically two masses, two ...
1
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1answer
60 views

Seeking a specific proof of endpoint boundedness of Riesz potential

The Riesz potential is defined by $$I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.$$ Once $f\in L^{d/\alpha}(\mathbb{R}^n)$, then $I_\alpha f(x)\in BMO$. ...
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The decay rate of Hormander lemma is optimal or not?

The Hormander lemma about oscillatory integral operators states that $T_\lambda f(x)=\int e^{i\lambda S(x,y)}a(x,y)f(y)dy$, while the Hessian of $S(x,y)$ is nondegenerate, then $||T_\lambda||_2 \leq C\...
0
votes
1answer
164 views

$\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$

Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...
3
votes
1answer
185 views

Is the Fourier-Stieltjes algebra of a locally compact group semi-simple?

Let $G$ be a locally compact group. Is the Fourier-Stieltjes algebra $B(G)$ semi-simple?
4
votes
3answers
203 views

Show that the Laplacian operator on the Heisenberg group is negative

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 $z=x+ i y \in \mathbb C$ ...
34
votes
1answer
2k views

The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?

Yesterday Bourgain, Demeter and Guth released a preprint proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only ...
4
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85 views

Local factors of Tamagawa measure

This is a reference request to some computations which I hope can be found in the literature somewhere. Let $G\subset GL_n$ be a semisimple linear algebraic group over $\mathbb Q$. The Tamagawa ...
9
votes
2answers
550 views

Fourier transform of the critical line of zeta?

This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here. Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...
3
votes
0answers
66 views

Almost sure convergence of double nonconventional ergodic averages with respect to $L^p$ function

A famous result of J. Bourgain says that for a probability measure preserving system $(X,\beta,\mu,T)$, with $T_1$ and $T_2$ powers of $T$, we have that for $f_1$, $f_2\in L^{\infty}(\mu)$,$$\frac{1}{...
1
vote
1answer
128 views

Linear Schrödinger equation on $\mathbb{H}^{d}$

Consider the linear Schrödinger equation $i\partial_t u = -\Delta u$, where $\Delta$ is the Laplacian on the hyperbolic space $\mathbb{H}^d$. What are the admissible pairs $(p, q)$ such that we have ...
1
vote
1answer
103 views

How to prove the Hölder continuity of a function $u$ by evaluating $\int_{B_{\rho}(x_0)}\frac{|Du(x)|^{2}}{|x-x_0|^{n-2}} dx$?

I'm looking at a video on thin obstacle problem given by Arshak Petrosyan. In his lecture, he uses the following results: Let $0<\alpha<1$, and $B_1$ be the unit ball centered at origin in $\...
0
votes
1answer
80 views

On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$

There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$ $L^{1}/H^{1}_{0}$ is ...
8
votes
1answer
263 views

$k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set $\...