# 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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### Finding an example

I am looking for an example to have two distinct decreasing functions $g_1$ and $g_2$ between [0,1], such that their ratio ($\phi=\frac{g_1}{g_2}$) is a periodic function with period one, moreover for ...
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### Sublinearity of Hardy-Littlewood Maximal Function on Sobolev Spaces

Define the (centered) Hardy-Littlewood maximal function by $$\mathcal{M}f(x)=\sup_{r>0}\dfrac{1}{m(B(x,r))}\int_{B(x,r)}\left|f(y)\right|dy,\ f\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{d})$$ We say ...
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### The Cauchy problem for Laplace equation in unit cube

Here is the question: We are given a laplace equation $\Delta u=0$ in $Q:=(0,1)\times(0,1)$. Q1: What are some conditions you can put on this to get uniqueness/existence? Q2:What if you wanted to ...
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### relationship between BMO norm and $L_p$ norm

Are there any relationship between the BMO norm of a function and its $L_p$ norms? For example, one norm is controlled by the other for functions of some special class.
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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 ...
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### Approximating $L^p$ functions using Schwartz functions with compact support on the Fourier side

For $1\leq p<\infty$, how would you show for any $f\in L^p(\mathbb{R})$ and given $\epsilon>0$, there exists $L<\infty$ and $g\in \mathcal{S}(\mathbb{R})$ such that $\|f-g\|_p<\epsilon$ ...
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### An $f\in H^{1/2}$ with self-convolution, showing it is an $C^1$ function.

If $f\in H^{\frac{1}{2}}(\mathbb{R})$ is a Sobelev 1/2 function that $f=f*f$, then how do you show that $f\in C^1$ with a bounded derivative.
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### Showing $f$ is an $L^p$ function if $f$ is "self-convoluted.

If $f$ is $L^2(\mathbb{R})$ and $f=f*f$, show that $f$ is $L^p$ for $2\leq p\leq \infty$.
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### Is “being harmonic conjugate” a symmetric relation?

The question is: Prove or disprove the following: If $u,v:\mathbb{R}^2 \to \mathbb{R}$ are functions and $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (in other words, ...
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### example of maximal operator that is integrable

We know that there are no nonzero functions $f \in L^1(\mathbb R^n)$ such that $Mf \in L^1(\mathbb R^n)$, where $Mf$ is the Hardy Littlewood maximal function. Can we find a maximal operator that is ...
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### How to construct $\operatorname{End}(V_{\pi})$ using a representation $\pi$

Let $(\pi, V)$ be a representation of the group $G$. To make the setting as general as possible, I will not put any restrictions on $\pi, V$, and $G$ from the beginning. By the very definition, for ...
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### Fourier transform of power function

Assume that $$\hat f(x)= (2\pi)^{-n/2} \int_{\mathbb{R}^n} f(y) e^{-i\left<x,y\right>} dy$$ is the Fourier transform of a function $f$. What is $\hat f$ if $f(x)=|x|^{2-n}$?
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### Definition of Zygmund class

I need help with showing that a function f belongs to the Zygmund class. Only helpful suggestions please, no full solution. I am here to learn for myself. This is work for school (we are allowed to ...
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### Integrals of compactly supported functions of positive type

Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest" area $\int f\,dx$ that can be achieved? To be ...
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### About the Fourier transform of the surface measure of the unit sphere

Let $d\sigma$ denote the surface measure on $\mathbb{S}^{n-1}$. To compute its Fourier transform $$\hat{d\sigma}(\xi)=\int e^{-i x\cdot \xi}\, d\sigma(x),$$ a standard technique (cfr. Folland's "...
I'm looking for a proof of the following statement: $$f\in L(\mathbb R^n)\ \text{ radial } \implies f^* \ \text{ radial}$$ where $f^*$ is the Hardy-Littlewood maximal function defined by: f^*(x)= ...