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Questions tagged [harmonic-analysis]

Harmonic analysis is the generalisation of Fourier analysis. Use this tag for analysis on locally compact groups (e.g. Pontryagin duality), eigenvalues of the Laplacian on compact manifolds or graphs, and the abstract study of Fourier transform on Euclidean spaces (singular integrals, Littlewood-Paley theory, etc). Use the (wavelets) tag for questions on wavelets, and the (fourier-analysis) for more elementary topics in Fourier theory.

Harmonic analysis can be considered a generalisation of Fourier analysis that has interactions with fields as diverse as isoperimetric inequalities, manifolds and number theory.

Abstract harmonic analysis is concerned with the study of harmonic analysis on topological groups, whereas Euclidean harmonic analysis deals with the study of the properties of the Fourier transform on $\mathbb{R}^n$.

2797 questions
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Powers of a BMO function

We know that $f(x) = |\log{|x|}|$ is in BMO (bounded mean oscillation). Using this, how can we prove that $f(x) = |\log{|x|}|^p$ for $0 < p < 1$ is also in BMO? I tried proving this using really cumbersome computational arguments but I feel like I…
Iguana
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Hardy-Littlewood maximal function: the relation between $\mu_{\lambda}^{Mf}$ and $\mu_{\lambda}^f$

Denote $M_f$ as uncentered Hardy-Littlewood maximal function. It is well known that $M_f\in L^{1,\infty}$, i.e. the distribution $\mu_{\lambda}^{Mf}$ is closely related to $||f||_{1}$, thus closely related to $\mu_{\lambda}^f$. Actually, from…
89085731
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property of Calderon-Zygmund kernel

I have a kind of vague question about the property of Calderon-Zygmund kernel. If given a k dimensional Calderon-Zygmund kernel $K$, can we say immediately that it is Lipschitz continuous except at the origin, $K(rx)=r^{-k}K(x)$ for all $r>0$ and…
cali
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Is multiplication by bounded, continuous functions bounded on $BMO$?

I know that if $f\in L^\infty$ then the multiplication operator $\varphi\mapsto\varphi f$ is not bounded on $BMO$, see eg. here. The obvious counterexample is something like $f=1_{x\geq0}$ and $\varphi=\log|x|$, since the $BMO$ness of $\log|x|$…
Funktorality
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Why $|K(x-y)-K(x)|\leq B|y|\left(\frac{|x|}{2}\right)^{-n-1}$?

Let $K:\mathbb R^n\backslash \{0\}\longrightarrow \mathbb C$ differentiable s.t. $|K(x)|\leq B|x|^{-n}$, and $|\nabla K(x)|\leq B|x|^{-n-1}$. Suppose $|x|>2|y|$. By IVT, $$K(x-y)-K(x)=-\int_0^1 \nabla _x K(x-ty)\cdot yd…
user330587
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Extend the Fourier transform over $L^2(\mathbb R^n)$

Using Plancherel theorem, we have that the Fourier transform is an isometry over $L^2(\mathbb R^n)$. But anyway. In my course it's written that Plancherel theorem is extremely important since it allow us to prolonge the fourier transform from…
user349449
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Fourier series calculation

I had an exam question today that stated something along the lines of the following: "Let $f$ be an even function given by $f(x)=x$ on $[0,\pi]$ and extend $f$ to $\mathbb{R}$ by $2\pi$-periodicity. Find the Fourier series of $f$" Now for this…
Dan Burrows
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positive definite character

‎‎We know each character on dual group of ‎$‎Z‎; ‎‎\widehat{Z}‎$‎‎, is positive definite and if‎ ‎$‎‎\chi‎‎ ‎\in‎ \widehat{Z}‎$ then ‎$‎‎\left\| ‎‎‎\chi‎‎ ‎\right\|‎_{‎\infty‎}‎‎=‎‎\chi(1)‎$‎. But I can not prove the problem that:‎ ‎ Let ‎$‎‎\chi‎‎…
user244115
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Topology on dual of an abelian discrete topological group.

We define the compact-open topology on the dual of an abelian topological group. Please describe compact open topology more explicitly in the case where G is equipped with the discrete topology, for example G=Z. Are the sup-norm topology and…
user244115
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Unitary representation on L_1(G)

If $u$ is a unitary representation of a locally compact group $G$ on a Hilbert Space $\mathcal{H}$ then $\pi_u :L^1(G)\rightarrow \mathcal{B}(\mathcal{H})$ given as $$\langle \pi_u(f)\xi\vert\eta\rangle = \int f(x) \langle u(x)\xi\vert\eta\rangle…
Parish
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Have a question about $L^p$ multiplier.

I'm studying $L^p$ multiplier. While reading the book, it says "The characteristic function of the unit disk is not an $L^p$ multiplier on $\mathbb{R}^n$ when $n\ge2$ unless $p=2$." How can I verify this? It says it will be discussed in later…
Xaviere
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"Tube domain over symmetric cone" thesis references?

My professor has told me to read "tube domain over symmetric cone". While saying so he said something related to complexificatin of real Lie algebra and representation theory. What is connection between all these concepts? What mathematical…
mathonlymath
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Proof that $\frac{dx\ dy}{x^2+y^2}$ is a Haar measure on the multiplicative group $\mathbb C\setminus\{0\}$

How can it be proven that for every Borel subset of $\mathbb{C}\setminus\{0\}$ as A we have $\mu(cA)=\mu(A)$? $$ ∬_{cA} \frac{dx\ dy}{x^2+y^2}=\iint_{A} \frac{dx\ dy}{x^2+y^2} $$ I'm confused...
fahimeh choobtarash
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Existance of Haar Measure

While showing the existence of Haar measure We consider all finite sequences of positive numbers $(c_i)_{i < n}$ and all finite sets $\{x_i\mid i < n\}⊂G$ Such that $f(x)\le \sum_i c_i g(x_i^{-1}x)$ for all $x∈G$ we define…
Milan Amrut Joshi
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A question about the proof of refined Young's inequality in Bahouri, chemin and Danchin's book.

The question may be very trivial. I am reading the proof of refined Young's inequality in page 8 of Bahouri,chemin and Danchin's famous book named Fourier Analysis and non-linear PDEs. The authors used the method of atomic decomposition (see page 7…
vent de la paix
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