# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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}^...
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### 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 ...
### 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 ...
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), ...