Questions tagged [harmonic-analysis]

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 analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.

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Average size of the Fourier--Stieltjes transform of the fractal measures

For $0<\theta<1/2$ define $\mu_\theta$ to be the uniform (self-similar) measure on the Cantor set obtained from the dissection pattern $(1-2\theta,\theta)$. For example, when $\theta=1/3$ the $\...
Subhajit Jana's user avatar
1 vote
1 answer
118 views

estimate involving Gaussian data

Let $x\in \mathbb{R}^{d}$, $d\geq 1$ and $p>\frac{2}{d}$. For any $R>0$ and any $0<s<\frac{dp}{2}-1$ \begin{align} &\int_{\mathbb{R}}\int_{|x|>R}\left(\frac{1}{1+t^2} \right)^{\frac{...
Julian Bejarano's user avatar
2 votes
0 answers
58 views

A lemma in the application of Lions's concentration compactness pricnciple in Hardy-Littlewood-Sobolev inequality

I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2". The Hardy-Littlewood-Sobolev (HLS) ...
IMOS's user avatar
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5 votes
1 answer
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Is the local maximal function bounded from $W^{1, 1}$ to $L^1$?

Let $f \in W^{1, 1} (\mathbb R^d)$. For every $\varepsilon > 0$, we consider the local maximal function $M_\varepsilon f: \mathbb R^d \to \mathbb R$, defined by $$M f_{\varepsilon} (x) = \sup_{r \...
Nate River's user avatar
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Characterising wavelet frames using Fourier transform

As usual, for $f\in L^2(\mathbb R)$ $$ D_j(f)(x) = 2^{j/2} f(2^jx), T_k(f)(x) = f(x-k) $$ and let $\{D_j T_k \varphi\}$ be a system of wavelets on $\mathbb R.$ A simple result is that, $T_k \varphi$ ...
Ma Joad's user avatar
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4 votes
1 answer
450 views

Showing that decay results of Fourier coefficients are sharp

I originally post the question here but the it seems too advanced for undergraduate level. At least, an example is very hard to find, so I am putting it here. It is well-known that if $f$ is ...
Ma Joad's user avatar
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2 votes
1 answer
221 views

Examples of non-discrete, cocompact subgroups

I am looking for non-trivial examples of the following: $G$ is a locally compact group $H\subset G$ a closed subgroup Both are unimodular and non-discrete The quotient space $G/H$ is compact, but $G$ ...
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4 votes
1 answer
738 views

How to get $\int_{\mathbb R^d} |\partial_i\partial_j(1-\Delta)^{-\frac{\delta}{2}}p_t(\cdot-y)(x)| \, \mathrm d x \lesssim t^{\frac{\delta}{2}-1}$?

We consider the Gaussian heat kernel $p_t$ on $\mathbb R^d$, i.e., $$ p_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}}, \quad t>0, x \in \mathbb R^d, $$ and define the operator $P_t$ by $$ ...
Akira's user avatar
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1 answer
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Heat kernel of left-invariant metric on 3-sphere

This post concerns the heat equation on $S^3 (\simeq \mathrm{SU}(2))$ endowed with an arbitrary left-invariant metric. We think of $S^3$ as the space of unit quaternions, and its Lie algebra $\...
o0BlueBeast0o's user avatar
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1 answer
431 views

If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ is in the Hardy space $\mathcal H^1(\mathbb R^n)$?

Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's "Harmonic Analysis", Chapter 3). It is well known that $\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$ and its ...
Lorenzo Pompili's user avatar
1 vote
0 answers
64 views

Eigenvalue problem of The Dirichlet problem

Consider the nonlocal problem \begin{aligned} &u_t(x,t)=\int_{\mathbb{R^n}}J(x-y)u(y,t)dy - u(x,t),&x\in\Omega,t>0,\\ &u(x,t)=0&x\notin\Omega,t>0,\\ &u(x,0)=u_0(x).&x\in\...
Phan Trung Hiếu's user avatar
5 votes
3 answers
300 views

The integrability of $\widehat{e^{-|x|^a}}$, $a>0$

Let $f_a(x)=e^{-|x|^a}$, $x\in \mathbb{R}^n$. Then $f \in L^p(\mathbb{R}^n)$ for every $1\leq p\leq \infty$. It is also smooth away from the origin and decays faster than any polynomial as $|x|\...
Medo's user avatar
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1 answer
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Does this dyadic sum converge?

Let $a\in (0,1)$ and define $$J(j):=\int_{0}^{\infty} e^{- 2^{j} s} \frac{s^{a}}{1+s^{2a}} ds,\quad j\in \mathbb{Z}.$$ Note that rescaling $2^{j} s\mapsto s$ shows that $$J(j)\leq 2^{-j(1+a)}\int_{0}^...
Medo's user avatar
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2 votes
1 answer
152 views

Continuity of an upper semi-continuous function over periodic points

Let $f: X \to \mathbb{R}$ be an upper semi-continuous function on $X$, which is a compact subspace of a vector space. Let sequence $x_n, n \in \mathbb{N}$, with positive elements - periodic: there ...
Adam's user avatar
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1 vote
0 answers
62 views

A kind of weak convergence for Sobolev spaces with zero on boundary

Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$. Is it true that if $(\varphi_n)_{n\geq 1}\subset C^{\infty}_c(\Omega)$ with $\varphi_n\to \...
Bogdan's user avatar
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2 votes
0 answers
409 views

Generalized conjugacy classes in (topological) groups

Let $G$ be a topological group. We define an equivalence relation on $G$ as follows: For $a,b\in G$ we set $a\sim b$ if the following two maps are topologically conjugate: $$x\mapsto ax,\qquad x\...
Ali Taghavi's user avatar
2 votes
0 answers
161 views

Approximating $L^p$ functions by eigenfunctions of Laplacian

I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932. In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
ze min jiang's user avatar
1 vote
0 answers
147 views

The space of ergodic elements of a topological or Lie group

Let $G$ be a compact topological group with normalized Haar measure $\mu$. An element $g\in G$ is an ergodic element if the mapping $L_g:G \to G $ with $x\mapsto gx$ is an ergodic map. The ...
Ali Taghavi's user avatar
1 vote
2 answers
177 views

Solution of $\Delta f -\frac{1}{2}hf = 0$ behaves asymptotically as $f(x) = 1 - C/|x|$

Let $f: \mathbb{R}^{3} \to \mathbb{R}$ be the solution of the following PDE: $$\Delta f -\frac{1}{2}h f = 0$$ where $h \in C_{c}^{\infty}(\mathbb{R}^{3})$ (compactly supported an smooth) and $f$ ...
IamWill's user avatar
  • 3,151
4 votes
1 answer
170 views

When is $W^{1,p}(\Omega)$ a Banach algebra?

Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$. My question: knowing that $f,g\in W^{1,p}(\Omega)$ for what $p$ can we conclude that $f\...
Bogdan's user avatar
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3 votes
0 answers
115 views

$L^\infty-L^1$ norm of an oscillatory integral operator

Let $\lambda>1$, $x,y\in [0,1]$, and $K_\lambda(x,y)=(1+\lambda|x-y|)^{-2}e^{i\lambda|x-y|}$. Consider the operator $$T_\lambda f(x)=\int_0^1 K_\lambda(x,y)f(y)dy.$$ We want to precisely estimate ...
capitalone's user avatar
3 votes
0 answers
48 views

Matrix argument K Bessel functions at half integral orguments

As a working definition I will define: $$ K_\nu^{(n)} (z) = \frac{1}{2^n}\int_{\mathcal{P}} e^{- \operatorname{tr}( y + y^{-1}) z/2} \det y^\nu d \mu(y) $$ where $\mathcal{P}$ represents the space of ...
Max K's user avatar
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15 votes
4 answers
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Steinhaus theorem and Hausdorff dimension

Assume for simplicity that sets $A_i\subset\mathbb{R}$ are compact. If $A_1$ and $A_2$ have positive length, then $A_1+A_2$ contains an interval. That is a variant of the classical Steinhaus theorem ...
Piotr Hajlasz's user avatar
4 votes
1 answer
189 views

Fourier coefficients of Selberg polynomials

In Montgomery's "Ten Lectures on the Interface Between Number Theory and Harmonic Analysis" a bound for the Fourier coefficients of the Selberg polynomial $S^+_K$ is obtained by using what ...
Johnny T.'s user avatar
  • 3,547
-1 votes
1 answer
121 views

Riesz energy for open sets in dimension $1$

This is a continuation of the question Calculation of Riesz energy for balls . As there are three questions,;I am posting a new question here. Riesz energy for a ball $B(x_0,r)$ is given by $$I_s(B(...
Sarthak's user avatar
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2 votes
0 answers
276 views

Recent progress restriction conjecture - Problem 2.7 (Terence Tao lecture notes)

I've been tackling the following problem for some time, Problem 2.7. (a) Let $S:=\left\{(x, y) \in \mathbf{R}_{+} \times \mathbf{R}_{+}: x^2+y^2=1\right\}$ be a quarter-circle. Let $R \geq 1$, and ...
Daniel Fonseca's user avatar
8 votes
2 answers
604 views

Uniqueness of the uniform distribution on hypersphere

I'm looking for a uniqueness-type result for the following problem, which is related to the uniform distribution in the hypersphere $\mathbb{S}^{p-1}$. Suppose $f$ is a sufficiently smooth function on ...
pat2211's user avatar
  • 81
1 vote
0 answers
90 views

Showing Vaaler polynomial is a good approximation to saw tooth function

Vaaler's polynomial is defined $$ V_K(x) = \frac{1}{K+1}\sum_{k=1}^K\left(\frac{k}{K+1} - \frac12\right) \Delta_{K+1}\left(x - \frac{k}{K+1}\right) + \frac{1}{2 \pi (K+1)}\sin 2 \pi (K+1) x - \frac{1}{...
Johnny T.'s user avatar
  • 3,547
2 votes
1 answer
130 views

Orthonormal bases in RKHSs via interpolating sequences

Definitions and setting Let $\mathcal{H}$ be a separable, infinite-dimensional, reproducing kernel Hilbert space on a nonempty set $X$. As usual, denote the reproducing kernel on $\mathcal{H}$ by $K$ ...
ABIM's user avatar
  • 5,019
6 votes
1 answer
191 views

Is the Hardy Littlewood “minimal function” comparable to the original function in $L^1$ norm?

Given $f \in L^1 (\mathbb R^d)$, and $\varepsilon > 0$, define the minimal function $m_\varepsilon f$ by $$m_\varepsilon f(x) := \inf_B \frac1{|B|} \int_B |f| ,$$ where the infimum is taken over ...
Nate River's user avatar
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3 votes
1 answer
189 views

Eigenforms of the Laplacian on Lie groups

I am a bit rusty in my differential geometry and I would like to confirm that my reasoning below holds, and I have some related questions (and all references to related concepts are of interest to me)....
Daniel Robert-Nicoud's user avatar
1 vote
0 answers
112 views

Idempotent conjecture and non-abelian solenoid

Is there a discrete non-abelian group whose dual in a reasonable sense is isomorphic to the solenoid constructed via a sequence of quaternions $S^3$ instead of a sequence of circles? The motivation ...
Ali Taghavi's user avatar
0 votes
0 answers
94 views

Idempotent conjecture and (weak) connectivity of (a reasonable) dual group

What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space? The Motivation: The motivation comes from the idempotent conjecture of ...
Ali Taghavi's user avatar
1 vote
0 answers
58 views

$L^p$ norm of Fourier transform of function composed with a diffeomorphism

Suppose $f$ is a compactly supported smooth function from $\mathbb{R}^n$ to $\mathbb{C}$ and $A$ is a diffeomorphism on $\mathbb{R}^n$, do we have any theorems relating the $L^p$ norm of $\hat{f}$ and ...
Simplyorange's user avatar
2 votes
1 answer
232 views

Maximal function on small cubes

I am reading "The Uncertainty Principle" by Fefferman (Bull. AMS, 1983) and have some issues following the arguments. In Lemma $C$ we have the following setting: Let $Q^0\subseteq \mathbb{R}...
Severin Schraven's user avatar
0 votes
0 answers
143 views

Why is this function in $L^1$?

I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
Bobo's user avatar
  • 101
2 votes
1 answer
194 views

Asymptotics for oscillatory integral

Consider the following integral for $f \in C_c^{\infty}(\mathbb R^n)$, $x_0$ fixed (possibly zero), and $n \ge 3$ $$F(\lambda) = \int_{\mathbb R^n} e^{i\lambda \vert x-x_0 \vert^2} \frac{f(x)}{\vert x ...
António Borges Santos's user avatar
1 vote
0 answers
66 views

Estimating commutator of Fourier integral

Let $f(x)= \log(\vert x\vert)$ on $\mathbb R^2$ and define $s_n:H^2 \to L^2$ where $H^2$ is the second Sobolev space by $$ s_n(g)(x) = \frac{nf(x)}{4\pi i} \int_{\mathbb R^2} e^{\frac{in\vert x-y\...
António Borges Santos's user avatar
1 vote
0 answers
49 views

Are integration over restricted direct products only useful for specific functions?

So I've been reading Tate's thesis currently. In that we have defined integration of functions on $G$, which are basically formed from restricted direct products of locally compact groups $G_{\...
Rits's user avatar
  • 133
3 votes
2 answers
272 views

Calculation of Riesz energy for balls

I was reading stuffs about Riesz energy which is defined for an open subset $U\in\mathbb{R}^d$ by $I_s(U)=\int_U\int_U|x-y|^{-s}\ dx\ dy$ where $dx$ and $dy$ are Lebesgue measure in $U$. Now if I take ...
Sarthak's user avatar
  • 73
4 votes
1 answer
154 views

Uniqueness of critical points for Lipschitz perturbations of uniformly convex Hamiltonians

Consider a macroscopic free energy functional of the form $$\mathcal{F}_\beta(\mu):= \frac{1}{\beta}\int_{\mathbb{R}^d}\log(\mu)\mu dx + \int_{\mathbb{R}^d}V(x)\mu(x)dx + \iint_{(\mathbb{R}^d)^2}g(x-y)...
Matt Rosenzweig's user avatar
2 votes
1 answer
85 views

Controlling the tensor product of functions in $H^1$ with lower derivatives

Given some function $\phi:D\to\mathbb{R}$, with $D\subseteq \mathbb{R}^d$. I was wondering whether we can find an estimate of the form $$ \|\phi\otimes\phi\|_{\dot{H}^1(D\times D)} \lesssim \|\phi\|_{\...
Víctor's user avatar
  • 123
8 votes
0 answers
168 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 ...
Jared White's user avatar
0 votes
0 answers
68 views

Oscillation of a polynomial

Recently I came across a statement in a paper that I am unable to verify. Namely, it roughly says that the oscillation of a polynomial on a cube can be controlled by the oscillation of the polynomial ...
Severin Schraven's user avatar
3 votes
1 answer
330 views

Is there an operation in topology analogous to the operation of averaging over a compact subgroup in harmonic analysis?

Let me start with the following Illustration: Let $G$ be a compact group, and let $\pi:G\to H$ be its (surjective) continuous homomorphism onto a (compact) group $H$. So we can think that $H$ is the ...
Sergei Akbarov's user avatar
2 votes
0 answers
53 views

Wave equation time decay

I am trying to deduce the dispersive estimate for the free wave equation in $\mathbb{R}^{d+1}\equiv\{(x,t) : x\in\Bbb R^d \wedge t\in\Bbb R\}$ $$u_ {tt}-\Delta_xu=0$$ The fundamental solutions of this ...
user509139's user avatar
3 votes
1 answer
76 views

Spectral disjointness of unitary representations of Type I groups and orthogonality

Background: If $\mathcal{H}$ is a Hilbert space and $U:\mathcal{H}\rightarrow\mathcal{H}$ is a unitary operator, then for each $f \in \mathcal{H}$ the sequence $(\langle U^nf,f\rangle)_{n = 1}^\infty$ ...
Sohail Farhangi's user avatar
5 votes
1 answer
264 views

Maximal operator estimates for the Schrödinger equation

Let $a>0$ and consider the operator $$Tf(t,x)= \int_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$ When $a=2$, the function $Tf$ solves the Cauchy problem ...
Medo's user avatar
  • 698
4 votes
0 answers
121 views

Distinguishing the Besov and Triebel-Lizorkin spaces

Theorem 2.3.9. in Triebel's Theory of Function Spaces states that the Besov space $B^{s_1, p_1}_{q_1} (\mathbb R^d)$ coincides with the Triebel-Lizorkin space $F^{s_2, p_2}_{q_2} (\mathbb R^d)$ if and ...
Jason Zhao's user avatar
3 votes
1 answer
91 views

$L^\infty$ bound of $x^m \psi_n(x)$ where $\psi_n$ is a Hermite function and $m,n \in \mathbb{N}$ - extension from Cramer's inequality

For each $n \in \mathbb{N}$, the Hermite function $\psi_n : \mathbb{R} \to \mathbb{R}$ is a Schwartz function defined by \begin{equation} \psi_n(x):=(-1)^n(2^n n!\sqrt{\pi})^{-1/2} e^{x^2/2} \frac{d^n}...
Isaac's user avatar
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