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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21 views

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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10 views

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 ...
0
votes
0answers
17 views

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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9 views

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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0answers
31 views

Do we have correspondance of all of $L^{p}(T)$ and $h^{p}(D)$ $p\in (1,\infty)$?

Let $h^{p}(D)=\{u ;u \ $harmonic in D and bounded means of order p$ \}$ Its well known that the Hardy space "sits nice" in $L^{2}$ by considering Fourier series. Do we have analagous result for $h^{...
1
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2answers
34 views

On an inequality for harmonic functions

I'm trying to understand the following; $| u(z)|^{p}= |\int P_{z}u \ d\lambda |^{p} \le (\int P_{z} |u |^{p} \ d\lambda) (\int P_{z})^{p-1} $ where $P_z$ is the Poisson kernel and $u$ is harmonic in ...
6
votes
2answers
94 views

Why this abuse of notation correctly solves the heat equation

Here's a stupid method I observed to solve the heat equation in $\mathbb R^d$, \begin{align*} \partial_tu=\Delta u,\quad u|_{t=0}=u_0. \end{align*} Pretend that $\Delta$ is a constant so this just ...
1
vote
1answer
27 views

Can a schwartz class function be dominated by an exponential?

Given a function $f$ from Schwartz class, does there exist a constant $C$ such that $|f(x)|<Ce^{-|x|}$. For me its seems true, if it is not true, any counterexample would be very illustrative to me
2
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0answers
37 views

The eigenfunction of modified 1-laplace equation

Let $\Omega\subset \mathbb R^2$ be a bounded, smooth boundary domain. I am interested in the following operator $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=0. $$ ...
0
votes
0answers
43 views

Getting the DFT of irregularly spaced points

I am trying to estimate the discrete Fourier transform of a discrete surface, $x:\{1,\dots,N\}\times \{1,\dots,N\} \to\mathbf{R}$, given a sparse set of samples on the grid. If we had all the ...
27
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2answers
399 views

If $f$ is a smooth real valued function on real line such that $f'(0)=1$ and $|f^{(n)} (x)|$ is uniformly bounded by $1$ , then $f(x)=\sin x$?

Let $f : \mathbb R \to \mathbb R$ be a smooth ( infinitely differentiable everywhere ) function such that $f '(0)=1$ and $|f^{(n)} (x)| \le 1 , \forall x \in \mathbb R , \forall n \ge 0$ ( as usual ...
0
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0answers
27 views

Why is this true about MRA-wavelets?

In the article Characterization of wavelets and MRA wavelets on local fields of positive characteristic, in page 11, As seen in the picture, The author refers to 16(Multiresolution analysis on local ...
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22 views

How theorem 3.1 is a consequence of lemmas 3.2-3.4?

In the article "Tight wavelet frames on local fields", the author states that "Theorem 3.1 is an easy consequence of lemmas 3.2-3.4". How?
2
votes
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17 views

Complex interpolation between $H^1$ and $L^1$

We have the complex interpolation $[H^1,L^p]_\theta=L^q$ for $1<p<\infty$ where $H^1$ is the Hardy space and $\frac{1}{q}=1-\theta+\frac{\theta}{p}$. This raises the obvious question as to ...
2
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0answers
46 views

Elementary proof of $C^\infty_c$ is dense in $L^p (L^q)$ mixed space

As it well known, $C^\infty_0 (\mathbb{R}^n)$(the space of infinitely differentiable functions with compact support) is dense in $L^p (\mathbb{R)^n}$ Here I want to consider the same result with ...
0
votes
1answer
24 views

Fourier coefficents of harmoinc $L^{1}$ functions in the disk

I just did an exercise in a some lecture notes, my result implied that the Fourier coefficients of an harmonic $L^{1}$ function $u$ was the integer values of the borel measure in the Poisson integral ...
1
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27 views

convolution of two decaying polynomial

Show that if the functions $f,g$ defined on $\mathbb{R}^n$ satisfying $|f(x)| ≤ A(1+|x|)^{−M} $and $|g(x)| ≤ B(1+|x|)^{−N}$ for some $M,N > n$, then $|(f ∗g)(x)| ≤ ABC(1+|x|)^{−L} $, where $L = ...
0
votes
1answer
19 views

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 $\int_{\mathbb{R}}\phi(x)dx=0$. ...
1
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1answer
22 views

$G$ 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 ...
1
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1answer
43 views

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.) ...
2
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0answers
32 views

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 ...
3
votes
2answers
22 views

BMO and truncation

I'm trying to solve Exercise 7.1.4 of Grafakos' book $\textit{Modern Fourier Analysis}$: consider two real numbers $K<L$ and a function $f \in BMO(\mathbb{R^n})$. Define the truncation of $f$ $$f_{...
2
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1answer
16 views

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 \...
2
votes
2answers
60 views

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 ...
0
votes
2answers
44 views

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 ...
4
votes
1answer
32 views

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 $f\in L^2(\mathbb{T}^2)$. ...
1
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25 views

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 $L^p$ ...
3
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26 views

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 ...
3
votes
2answers
35 views

Question on Schwarz space why is it $\sup_{x\in \mathbb R^n}|(1+|x|^N)\partial _x^\alpha f(x)|<\infty $

Why Schwarz space is given by the set of $f\in \mathcal C^\infty (\mathbb R^n)$ s.t. $$\sup_{x\in \mathbb R^n}|(1+|x|^N)\partial _x^\alpha f(x)|<\infty ,$$ where $\alpha \in \mathbb N^n$ and $N\in\...
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26 views

Why $(2\pi x)^\alpha f(x)(2\pi i\xi)^\beta e^{-2i\pi x\cdot \xi}=(\partial _x)^\beta [(2\pi x)^\alpha f(x)]e^{-2i\pi x\cdot \xi}$?

Why $$(2\pi x)^\alpha f(x)(2\pi i\xi)^\beta e^{-2i\pi x\cdot \xi}=(\partial _x)^\beta [(2\pi x)^\alpha f(x)]e^{-2i\pi x\cdot \xi} ?$$ Indeed, to prove that the fourier transform is in the Schwarz ...
2
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2answers
59 views

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 ...
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13 views

Irreducible representations of locally compact semigroups

What class of semigroups have finite dimensional irreducible representations. For example, If we have a compact groups $G$ then every continuous irreducible representation of $G$ is finite ...
5
votes
1answer
68 views

Is $\Delta C_c^\infty$ a dense subset of $L^p(\mathbb{R}^d)$?

I'm struggling to obtain some density result. It is well known that $C^\infty_c(\mathbb{R}^d)$ is dense in $L^p (\mathbb{R}^d)$ for $1\leq p<\infty$. It is well known that for $\lambda>0$, $(\...
2
votes
0answers
18 views

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 $m\in \mathbb{Z}^+$, ...
5
votes
1answer
75 views

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 !
0
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28 views

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 ...
4
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28 views

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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38 views

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. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
0
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1answer
31 views

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 ...
0
votes
1answer
65 views

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 $\mathbb{K}^+$ ...
1
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0answers
21 views

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 ...
5
votes
1answer
1k views

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 ...
1
vote
0answers
17 views

Finite dimensional irreducible representations of Sp(2).

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 ...
2
votes
2answers
62 views

Morrey's inequality for Sobolev spaces of fractional order

Let $H^s(\mathbb T)$, where $s\in\mathbb R$, be the space of $2\pi$-periodic functions, $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, such that $$ \|u\|_{H^s}^2=\sum_{k\in\mathbb Z}(1+k^2)^{...
6
votes
1answer
91 views

A version of Hörmander multiplier theorem

Let $m>n/2$ be an integer. Let $h\in H^m_{loc}(\mathbb{R}^n)$ satisfy that $\displaystyle \exists M>0,\forall R>0,\sum_{|\alpha|\le m}\int_{\frac R2\le|w|\le2R}R^{2|\alpha|}|\partial^\alpha ...
0
votes
0answers
30 views

Semigroups and Finite Dimensional Representaions

It is well known that if a group $G$ is compact then every irreducible continuous representation of $G$ is finite dimensional. As far as I know the semigroup equivalent of this statement is not true. ...
3
votes
0answers
33 views

Classification of irreducible (g,K)-modules for other g than sl2

Harish Chandra showed how to associate to an admissible representation $(\pi,V)$ of a real semisimple Lie group $G$ the so-called Harish-Chandra module $V_K$ of $K$-finite vectors in $V$. This is a $(\...
2
votes
1answer
54 views

Problem on the pointwise boundedness of the partial sums of the j-series in Tuomas Hytonen's paper

Recently, I have read Tuomas Hytonen's paper On Petermichl's Dyadic Shift And The Hilbert Transform and got into trouble in a certain part of his article. In the first place, we should have some ...
4
votes
2answers
109 views

Why does the coordinate transformation from Cartesian coordinates leads to an additional term in the biharmonic operator in spherical coordinates

I am trying to solve a problem in physics where the biharmonic operator is involved. I think that the bihahmonic operator can be obtained by taking twice the Laplace operator, such that $\nabla^4 f = \...
0
votes
1answer
18 views

Estimation with Hölder condition

Do you have any hints about how to prove (or find a counterexample) that, given $f \in \mathcal{C}^1 ( \mathbb{R}^n \smallsetminus \{ 0 \}) $ such that $$\int_{|x|=r} f(x) \, dS(x) = 0$$ for all $r&...