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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3
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0answers
36 views

A bounded extension operator

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns ...
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93 views

Schrodinger operator with matrix potential

This question is inspired by attempts to extend in some way Shen's 1999 "On fundamental solutions of generalized Schrödinger operators" to Schrödinger operators $- \Delta + V $ with some ...
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94 views

The space of harmonic functions on an open set is infinite dimensional? [closed]

I want to prove that he space of harmonic functions on an open set $\Omega \subset \mathbb{R}^n $ , with $n \geq 2$, is uncountablely infinite-dimensional. I guess that I have to find a linearly ...
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1answer
150 views
+50

Riesz transform of fractional operators

I am interested in Riesz transforms linked to the fractional Laplacian and other fractional operators. I have been hunting down in the literature to find related results but I have not been able to ...
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145 views

Moments of Plücker coordinates on complex Grassmannian and log-concavity

Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...
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61 views

Condition on a function to have a Fourier transform in $L^{2-\varepsilon}$

It is known that in general the Fourier transform of $L^p(\mathbb{R})$ functions for $p>2$ are not even function. However, for regular enough functions, the regularitytransfers into decay for $\hat ...
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1answer
387 views

A characterization of constant functions

In How to recognize constant functions. Connections with Sobolev spaces (Russian Math Surveys 57 (2002); MSN), H. Brezis recalls the following fact: Let $\Omega\subset{\mathbb R}^N$ be connected ...
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172 views

Problem with completeness of an orthogonal system

For $\nu\in (-1, \infty)$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the sequence of positive zeros of the Bessel function $J_{\nu}$. The Fourier-Bessel "Laplacean" is given by \begin{equation}...
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1answer
163 views

Efficient boxing for a mean value in the Bombieri Iwaniec method

One of the nice applications of decoupling is Bourgain’s record towards Lindelöf: https://arxiv.org/pdf/1408.5794.pdf Wooley has developed some techniques known as efficient congruencing which allow ...
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80 views

Proof that Littlewood-Paley vertical square function is NOT bounded on L^infinity

The classical heat semigroup on $\mathbb{R}$ is given by $$ W_t f(x)=\frac{1}{t}\int_{\mathbb{R}}e^{-\pi (\frac{x-y}{t})^2}f(y)dy, \qquad t>0. $$ Then the Littlewood-Paley vertical square ...
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1answer
158 views

the fractional integration method of the proof of Stein-Tomas theorem?

In Schalg's Classical multilinear and Harmonic analysis, he presented two methods of the proof of Stein-Tomas theorem, one of which is called the fractional integration method. As a matter of fact, in ...
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1answer
506 views

History of the notion of irreducible representation

I am looking for the earliest references where the study of irreducible representations appears. There has been many articles and books on the history of representation theory. A fundamental feature ...
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73 views

Is this a positive definite kernel?

Under which conditions on the function : \begin{array}{l|rcl} K : & \mathbb R^+ & \longrightarrow & (0, 1)\\ &t & \longmapsto & K(t) \end{array} is the symmetric ...
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68 views

Strichartz estimate for the Schrödinger equation

Estimates of the extension operator can be seen as estimates of the initial value problem for the evolution Schrödinger equation. If $u(x,t)=e^{it\Delta}u_0$ is the solution to the IVP: $$i\partial_t ...
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70 views

Quantum analogue of certain property of compact groups

Let $\mathcal{A}$ be the category of $C^*$ algebras. For a group $G$ let $\tilde{G}$ be the space of all conjugacy classes of $G$. What is a precise description of a maximal ,or in some sense ...
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2answers
132 views

Measure algebra on the Bohr compactification vs the bidual algebras

The following question probably reduces to some standard abstract harmonic analysis Twister play, but I'd still welcome some comments on it. Let $G$ be a locally compact Abelian group and let $bG$ ...
4
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1answer
171 views

Idea behind Carleson's theorem modern proof “intitial reductions”

I'm having troubles to understand the philosophy behind the modern proof of Carleson's theorem. For convenience, let me state precisely what I am asking for. For any $f \in L^2(\mathbb{R})$, let $\...
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45 views

Space contained in the Interpolation of $L^\infty$ and the Wiener Algebra $\mathcal{F}(L^1)$

Let $\ell^p$ be the space of sequences with power $p$ summable to $\ell^\infty$, $L^p = L^p(\mathbb{R^d})$ be the Lebesgue spaces and $\mathcal{F}$ be the Fourier $d$-dimensional Fourier transform. ...
2
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1answer
68 views

On conditionally negative definiteness of a kernel

We are interested in the kernel $\phi\left( x-y\right) =\frac{1}{2}\log\left( \cosh\left( x-y\right) \right) -\log\left( \cosh\left( \frac{x-y}{2}\right) \right)$ on $\mathbb{R} \times \mathbb{...
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1answer
273 views

Compactly supported probability measure in high dimensions with fast Fourier decay?

For any sufficiently large $d\in\mathbb{N}$, does there exist a probability measure $\Psi$ supported on the Euclidean ball in $\mathbb{R}^d$ for which $|\widehat{\Psi}[\omega]|\le C\cdot \exp(-\|\...
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1answer
94 views

What is the definition of Plancherel density?

I know about the Plancherel measure, but I don't know where the term "Plancherel density" is defined.
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202 views

What is the relationship between Hecke algebras and the enveloping algebra of Lie groups?

Here is the story as I see it. Let $G$ be an abelian locally compact group. Then the (spherical) Hecke algebra for $K=1$ is by definition the endomorphism algebra of $l^2(G)$ as a $G$-module, where ...
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104 views

On Pitt's inequality (weighted Fourier inequality)

One of Pitt's Theorem (from "Theorems on Fourier Series" by H R Pitt, 1937) states that for an integrable periodic function $F$ over $[-\pi,\pi]$, $$ \sum_{n=1}^{\infty} |a_n|^q n^{-q\lambda} \leq K(...
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68 views

Fourier dimension of radial set

In his 1967 article "Sur un theoreme de R. Salem", Gatesoupe proved that if a set $A\subset [0,1]$ has Fourier dimension $\alpha$ then the set $\tilde A:=\{x\in \mathbb{R}^n: |x| \in A\}$ has Fourier ...
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43 views

How does the constant bound on Hormander's $L^2$ estimate for non-degenerate phase depend on the cut-off function?

A classical estimate, due to Hormander, assets that the integral operator $$Tu = \int_{\mathbb{R}^{d}} e^{i\varphi(x,y)/h} a(x,y) u(x) dx, \ \ a \in \mathcal{C}_{c}(\mathbb{R}^{2d}), \ \ \varphi \in \...
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1answer
71 views

Derivation of an uncertainty principle from the symplectic non-squeezing theorem

Is there a derivation of an uncertainty principle or uncertainty-type principle from the symplectic non-squeezing theorem?
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72 views

Classical singular integral operator

I am working on a problem involved the Biot-Savart law in fluid dynamics. I found a theorem of singular integral which is intimately related to my research. Assume that $K(x)$ is a Calderon-Zygmund ...
6
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1answer
79 views

Stationary phase in spherical integral

I'm trying to find asymptotics for an oscillatory integral on $\mathbb{S}^{n-1}$, for which my advisor said I should use stationary phase arguments. The particular, he claims that: If $\lambda\gg 1$...
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72 views

Sobolev convergence of Fourier series

Consider $f\in H^{\sigma}(S^1)=W^{\sigma, 2}$ (the usual Sobolev space on the circle) and let $S_Nf$ be its truncated Fourier series $S_Nf = \sum_{|n|\leq N} \hat{f}(n)e^{2\pi i n x}$. I am looking ...
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202 views

An inequality in harmonic analysis with the BMO flavour

I am asking myself this question (which seems to be a natural generalization of Remark 4.4 of these lecture notes). Question. Let $I_s, s \in \mathcal{S}$ be a collection of intervals included in ...
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96 views

Reference request: harmonic analysis with non-Lebesgue reference measure

The Lebesgue measure on $\mathbb{R}^d$ admits the following polar decomposition: $$ L(dx) = r^{d-1} dr \lambda(dy), $$ where 𝜆 is the uniform measure on the Euclidean unit sphere of $\mathbb{R}^d$ ...
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186 views

Degree of automorphic forms, SL(3,Z), and the elliptic Gamma function

In this article, the authors interpret a certain special function, the elliptic Gamma function, defined as $$ \Gamma(z,\tau,\sigma)=\prod_{j,k=0}^\infty\frac{1-e^{2\pi i((j+1)\tau+(k+1)\sigma-z)}}{1-...
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2answers
96 views

A question about homogeneous distribution

A distribution in $\mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$ is called homogeneous of degree $\gamma \in \mathbb{C}$ if for all $\lambda>0$ and for all $\varphi \in \mathscr{S}\left(\mathbb{...
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222 views

Certain Fourier transforms involving Whittaker function and Bessel functions

I recently meet the following two weird "Fourier transform" questions. (I), Suppose that $F$ is a $p$-adic field (the same question can be asked over any local field, including $\mathbb{R}$ and $\...
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1answer
195 views

Laplace equation on the disk with Robin boundary condition

Consider the following two dimensional Laplace equation on the unit disk $D$ with homogeneous Robin boundary condition: $$\Delta u = 0, ~~\frac{\partial u}{\partial n} = b(x) u(x)~~ \forall x \in \...
2
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2answers
176 views

Asymptotic decay rate of an oscillatory integral

Consider the following oscillatory integral $$ I(n):=\int_{-\pi}^\pi\int_{-\pi}^\pi e^{i n(x+y)}\frac {(1 - \cos(2x)) (1 - \cos(2y))} {2k - (\cos x + \cos y)}\ \mathrm{d}x\,\mathrm{d}y. $$ where $...
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181 views

Discrete superharmonicity

The value at $(n,m)$ of the “Discrete Laplace operator” (see wikipedia) of a function $f$ in $\Bbb Z\times \Bbb Z$ is $\Delta f(n,m)= \frac{1}{4}( f(n+1,m)+f(n,m+1)+f(n-1,m)+f(n,m-1))-f(n,m)$: the ...
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132 views

Extension of subharmonic function: can someone explain the details?

In this paper we have the following situation on page 60. $E$ is a compact subset of $\mathbb{R}^\tau\cup\{\infty\}$ (one point compactification) for $\tau\geq2$, $M_0$ is a point in the boundary of $...
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90 views

Common eigenvector of the Wigner D matrices for the eigenvalue $\lambda = 1$?

Consider the family of spherical harmonics $Y_{n}^m$ on the sphere $\mathbb{S}^2$, with $n \geq 0$ and $-n \leq m \leq n$. The Wigner matrices are matrices $\mathrm{D}_{\mathrm{R},n} \in \mathbb{R}^{(...
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1answer
121 views

References for Neumann eigenfunctions

I am looking for references on eigenfunctions with Neumann boundary condition. In an article, the author wrote in introduction that when a domain is planar polygon, the second eigenfunctions on it ...
3
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0answers
58 views

Logarithmic Sobolev growth of time-space-periodic Schrödinger solutions

Consider the following Schrödinger equation $$i\partial_t \psi (t,x) + \Delta \psi - V(t,x) \psi = 0 \, ,$$ where $x\in \mathbb{T}^d$ and $V(t,\cdot)$ is real, smooth, and periodic (with a diophantine ...
3
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1answer
141 views

Reference request - Weyl's integration formula

Is there a reference discussing in an organized way (with a proof) the Weyl integration formula for a reductive group over a local field (Archimedean or not), expressing the Haar integral on the group ...
10
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1answer
1k views

A question on trig series

Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$ $$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...
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0answers
95 views

Is integration by parts useful for obtaining global cancellation?

If $f$ and $g$ are functions on the real line, and $f$ is oscillatory, then an important technique for bounding the integral $\int fg$ is applying an integration by parts, writing $$ \int_{-\infty}^\...
4
votes
2answers
285 views

The formula for (and computation of) the inverse p-adic mellin transform

So, after scouring the entirety of the internet, I managed to find one (and, so far, only one) source that actually explains how to invert the $p$-adic mellin transform: $$\mathscr{M}_{p}\left\{ f\...
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1answer
86 views

Duality relation of Lorentz space $L^{p,1}$

want to prove the duality relation: $$||f||_{L^{p,1}} =C_{p} \cdot \sup\{\int_X fg d\mu: \text{ for any } ||g||_{L^{p',\infty}}\le 1 \} $$ where $\frac{1}{p}+\frac{1}{p'}=1, p>1, \mu$ is $\sigma$-...
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88 views

How large this subset is to say that it should equal the group?

Let $\alpha$ be a continuous automorphism on a compact group $G$ with normalized Haar measure $m$. We may say $\alpha$ is $n$-splitting, if the set $$\text{Spl}_n(\alpha):=\left\{g\in G: \prod_{k=1}^...
5
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1answer
181 views

Which plane curves can be harmonically parametrized?

In this question, a “(closed oriented plane) curve” $\Gamma$ will mean a continuous map $f \colon \mathbb{U} \to \mathbb{C}$ where $\mathbb{U} := \{z\in\mathbb{C} : |z|=1\}$ is the unit circle, modulo ...
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0answers
66 views

induced maps between group $C^*$-algebras

Suppose $G,H$ are two locally compact groups, if there is a injective homomorphism $\phi:G\to H$, can $\phi$ induce the $*$-homomorphism between group $C^*$-algebras $C^*(G)$ and $C^*(H)$? If it ...
3
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45 views

History and relationship on g-K modules and spherical functions

I am learning Harmonic analysis on real reductive Lie groups. It seems to me that there are two kinds of treatment. One is through $(\mathfrak g, K)$ modules (e.g., Wallach, Real reductive groups), ...

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