# Tagged Questions

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-...

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### Integral kernel of resolvent of Laplacian

Consider the Laplacian $-\Delta: C^\infty (S^1) \to C^\infty(S^1)$ where $\mathbb R/\mathbb Z=S^1$ and for a periodic function $f:\mathbb R\to\mathbb R$ we have $-\Delta f=-f''$. For the orthonormal ...
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### Proving Navier-Stokes bilinear operator is bounded on $H^s$

I'm trying to understand the proof that the bilinear operator arising in the Navier-Stokes equation, $$B(u,v)=\int_0^te^{(t-s)\Delta}\mathbb P\nabla\cdot(u\otimes v)ds$$ where $\mathbb P$ is the Leray ...
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### Show that $\lim_{|y|\to\infty}|y|^{n-2}u(y)>0$

For $n\ge 3$,Let $u\in C^2(R^n)$, $\Delta u\le 0, u>0$ in $R^n$ ,show that $\lim_{|y|\to\infty}|y|^{n-2}u(y)>0$. I consider the maximum principle,but I don't know how to deal with.
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### Do conformal maps preserve subsolutions of elliptic PDE?

The fact is well-known for the Laplace equation for regions in $\mathbb R^n$ but I'm wondering if it extends to general elliptic PDE.
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### Image under Hilbert Transform in $L^1$

I have a question concerning the proof of following proposition: Proposition: Let $\phi\in S(\mathbb{R})$ be given. Then $H\phi \in L^1(\mathbb{R})$ if and only if $\int_{\mathbb{R}}\phi(x)dx=0$. ...
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### $G$ is dense in $X^*$ in weak* sense if and only if $G$ is total set

I have some question on functional analysis. Recently, I'm reading an article of Coifman and Weiss, "Extensions of hardy spaces and their use in analysis". They proved some important theorem to me by ...
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### Null space of the Laplacian operator?

(I guess the answer to my question is well-known in harmonic analysis, but I consider it in the framework of Schwartz distributions and in any dimension, and could not find a satisfactory answer.) ...
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### Determining a function is harmonic from mean value property for just three(?) radii.

This theorem is well-known (maybe it can be called Morera's theorem): A continuous function satisfying the mean value property on balls is harmonic. I was recently surprised to hear in a talk ...
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### Riesz basis of Paley-Wiener space.

Let us consider the Paley-Wiener space: $$PW_\pi:=\{f\in L^2(\mathbb R)\cap C(\mathbb R),\ \operatorname{supp}\hat f\subset [-\pi,\pi] \}.$$ Let $\{\lambda_n\}_{n\in\mathbb Z}$ be a sequence of real ...
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### What does $f$ continue on $\mathbb S^1$ mean?

Let $\mathbb S^1:=\mathbb R/\mathbb Z$. What does $f$ continuous on $\mathbb S^1$ mean ? That it's continuous over $[0,1)$ or $[0,1]$ ? I would say $[0,1)$ but I have doubt since we sometimes take the ...
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### Pointwise convergence of Fourier series in two dimensions

By Carleson's Theorem, we know that for every $f\in L^2(\mathbb{T})$ $$f(x)=\lim_{N\rightarrow\infty}\sum_{k=-N}^N\hat{f}(k)e^{2\pi ikx}\;\text{ a.e.}$$ Suppose now that $f\in L^2(\mathbb{T}^2)$. ...
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### Are holder space dense in $L^p$ ? or in Schwarz space?

I recall that $f\in \mathcal C^\alpha ([0,1[)$ where $\alpha \in (0,1)$ if $$[f]_\alpha :=\sup_{x,y\in\mathbb [0,1[}\frac{|f(x)-f(y)|}{|x-y|^\alpha }<\infty .$$ Does those space are dense in $L^p$ ...
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### The equivalence of the two definitions of fractional Laplacian

Using the Fourier transform we can easily define the fractional Laplacian by $$(-\Delta)^{s/2}f(x)=(|\xi|^s\hat f(\xi))^\vee(x), \ \ f\in C_0^\infty.$$ However, I learned that there is another ...
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### Singular integral operator on a decay Hölder space

Denote $$E^{k,m}=\left\{f\in C^m(\mathbb{R}^3)\mid\sup_{x\in\mathbb{R}^3}(1+|x|)^{k+|\alpha|}|\partial^\alpha_xf(x)|<\infty\right\}.$$ Now given $f\in E^{2,m}$ for any fixed $m\in \mathbb{Z}^+$, ...
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### Finding $f \in L^p(\mathbb{R}^n)$ such that $\hat{f} \notin L^p(\mathbb{R}^n)$

I heard there were functions $f \in L^p(\mathbb{R}^n)$ such that $\hat{f} \notin L^p(\mathbb{R}^n)$. Is there a concrete example of such functions ? Thanks in advance !
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### The discrete Laplacian

I am working on the $d$-dimensional integer lattice. Let $S$ be a random walk with increment distribution $p$. Given the distribution $p$ we can define the discrete Laplacian just as in Wikipedia is ...
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### On sharp bounds of some dyadic operators

I am now having interest in finding the sharp bounds of some kinds of dyadic operators which map $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$. For example, the martingale transform $T_{\sigma}$ which ...
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### Proof of iff condition of harmonic map

How to compute the equation above the red line in the picture below ? Below picture is from the Harmonic maps and their heat flows. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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### A specific maximal function of of a potential function

Let $$f(x)=\frac 1{(1+|x|)^2},$$ Then what's the maximal function of $f$ ? By definition $$Mf(x)=\sup_{r>0}\frac 1{|B_r(x)|}\int_{B_r(x)}\frac 1{(1+|y|)^2}dy,$$ If one can prove that the average ...
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### where can I find the proof of this theorem of wavelet frames?

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
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### What are the applications of quantum shuffle multiplication?

I am writing a research paper on quantum shuffle multiplication, and there is just one piece that I'm missing: I want to give one or several examples of its applications, and so far I have not been ...
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### Neumann problem for Laplace equation on Balls by using Green function

It is well known that for Dirichlet problem for Laplace equation on balls or half-space, we could use the green function to construct a solution based on the boundary data. For instance, one could ...
I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$. I can ...