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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1answer
109 views

Convolution algebra $L^1(G)$ for non sigma-finite $G$

Let's assume that $G$ is locally compact and Hausdorff topological group, hence it carries a Haar measure, $\mu$. We can than consider space of integrable functions $L^1(G)$ (class of functions to be ...
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44 views

Is there a stable probability distribution on the rational numbers?

Does there exist a (non-trivial) probability distribution on the rational numbers $$\sum_{r\in\mathbb{Q}}p_r=1$$ with $0\leq p_r$, which is stable, meaning that the sum of two i.i.d. random variables ...
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0answers
21 views

Spherical resolvent kernel on $H^n(\mathbb R)$

Is there an explicit formula in the literature for the spherical resolvent kernel $R_{\lambda}(r)$ of the Laplacian $\Delta_{H^n}$ on $H^n(\mathbb R)$ the real hyperbolic space ? Such that: $rad(\...
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70 views

Fractional Sobolev Space Trace Inequality

Let $\phi\in\mathcal{S}(\mathbb{R}^{n})$ be a Schwartz function, such that ${\phi}\equiv 1$ on the unit ball $|\xi|\leq 1$ and $\text{supp}({\phi})\subset B_{2}(0)$. Set $\phi_{0}=\phi$ and $\phi_{j}=\...
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11 views

Resolvent kernel of Hyperbolic space

The expression of the spherical functions $\varphi_\lambda(x)$ and the resolvent kernel $r_\lambda(x)$ on Hyperbolic space are known. The spherical functions are the Jacobi functions $\varphi_\lambda(...
2
votes
1answer
42 views

Hardy-Littlewood theorem about the Poisson integral for $p=1$

(Hardy-Littlewood Theorem) : Let ‎$ u(r,‎\theta)‎$‎ be the Poisson integral of ‎$ ‎\varphi ‎\in L‎^{p}‎‎$‎, ‎$ 1<P ‎\leqslant‎ ‎\infty‎$‎ , and let $ U(‎\theta)=\sup‎_{r<1}‎|u(r,‎\theta)|‎ $‎. ...
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63 views

$k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set $\...
4
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1answer
113 views

Is Rudin correct here? Fubini's theorem and product measures

Let $X, Y$ be locally compact Hausdorff spaces with nonnegative regular measures $\mu, \lambda$. By definition (in the book I'm reading) a regular measure is a Borel measure for which every Borel set ...
2
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2answers
95 views

Product of two sinusoidal functions model

I'm trying to make a model of the rise and fall of sea levels. According to this explanation and image in the textbook, the product of two sinusoidal functions should look something like this: (...
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36 views

Fourier Transform of a kernel

Let $\omega \in \mathcal{S}(\mathbb{R}^{2})$ and define $u(x) = \int_{\mathbb{R}^{2}}\frac{(x-y)^{\perp}}{|x - y|^{2}}\omega(y)dy$, where $(x-y)^{\perp} = \begin{bmatrix}x_{2} - y_{2}\\-x_{1} + y_{1}...
4
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86 views

Continuous Littlewood-Paley Inequality

I am trying to prove Q13 from Terence Tao's Fourier Analysis notes (number 4): For every $t>0$, let $\psi_{t}:\mathbb{R}^{d}\rightarrow\mathbb{C}$ be a function obeying the estimates $$|\...
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0answers
18 views

Exactness of harmonic sum using Mellin transformation

I'm trying to learn how you can use the Mellin transformation to obtain closed expressions of harmonic sums. There are demonstrations om MSE how show this technique. eg Proving $\sum_{n =1,3,5..}^{\...
3
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1answer
45 views

Real vs Complex Interpolation

The two major classical interpolation theorems in analysis are Riesz-Thorin Theorem (complex method) and Marcinkiewicz Theorem (real method). One can see the statements of the theorems and realize ...
0
votes
1answer
33 views

What condition on $f$ makes the formula $(−\Delta)^sf(x)=c_{n,s}\int_{\mathbb{R}^n}\frac{f(x)−f(y)}{|x−y|^{n+2s}}dy$ true?

I'm trying to understand the concept of fractional Laplacian, and I found the page https://www.ma.utexas.edu/mediawiki/index.php/Fractional_Laplacian,and the formula $$(−\Delta)^sf(x)=c_{n,s}\int_{\...
2
votes
2answers
40 views

Integration of hypergeometric functions?

I would calculate the following integral \begin{equation} I_x = \int_{0}^{1} y^{b+\mu-1} (1-y)^{\nu-1}\, _2F_1(a,b+\nu +\mu;c; xy) \, dy. \end{equation} Such that $\quad \Re a,\Re b,\Re \mu, \Re \nu &...
11
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2answers
181 views

average of maximal function is less than its infimum?

Let M be the dyadic Hardy-Littlewood maximal operator. Prove the following: there is a constant $C$ such that for any $f$, $$ \inf_{x\in I}Mf(x)\le C 2^k\inf_{x\in J} Mf(x) $$ where $I$ and $J$ are ...
4
votes
1answer
47 views

Fundamental solution of a shifted operator

what is the fundamental solution of the shifted operator $ \Delta + \lambda^2 $, i.e, what the function $f$ satisfying the following equation $$ (\Delta + \lambda^2 )f(x) = \delta(x),$$ where $ \Delta ...
1
vote
1answer
44 views

Convolution with imaginary Gaussian cannot be a $(p, p)$ operator unless $p=2$

Let $g=e^{-ix^2}, x\in \mathbb{R}$. Let $T$ be an operator defined as $T(f)=f*g$. Show that $T$ cannot satisfy a $(p,p)$ inequality unless $p=2$. Note: We say an operator $T$ satisfies a $(p,p)$ ...
2
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1answer
45 views

A stronger form of the weak $(1,1)$ inequality for the Hardy-Littlewood maximal function

I am trying to show that for $f \in L^1(\mathbb R^d)$, if $f^*(x)$ is the Hardy Littlewood Maximal function, then the following inequality is satisfied:$$|\{x : f^*(x)> \alpha\}|\leq \dfrac{c}{\...
6
votes
2answers
104 views

Dense Subspace of $L_{0}^{1}(\mathbb{R}^{n})$

Let $L_{0}^{1}(\mathbb{R}^{n})$ denote the the closed subspace of $L^{1}$ functions whose Fourier transform vanishes at the origin (equivalently, $\int f=0$). At the top of pg. 231 in E.M. Stein, ...
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0answers
26 views

Riemann Lebesgue Lemma for locally compact ableian groups

I'm looking for a reference (or proof here) of the generalized RL Lemma for LCAGs. One result is that if $G$ is a LCAG then $$\{ \hat{f} : f \in L^1(G)\} \subset C_0(\Gamma)$$ where $\Gamma$ is the ...
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0answers
16 views

Number of solution of $ (\Delta - \lambda) f = \delta $

How they are solutions of the equation $$ (\Delta - \lambda) f (x) = \delta (x)$$ Where $\Delta$ is the Laplacian operator and $\delta(x) = \begin{cases} 0, \quad x\neq 0;\\ +\infty, \quad x= 0 \...
2
votes
2answers
80 views

$L^{1}$ Boundedness of Hilbert Transform on $\left\{f\in L^{1}(\mathbb{R}) : \int_{\mathbb{R}}f=0\right\}$

It is well-known that the Hilbert transform $H(f)$ of a bounded, compactly supported function $f:\mathbb{R}\rightarrow\mathbb{C}$ belongs to $L^{1}(\mathbb{R})$ precisely when $\int f=0$. One can ...
5
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0answers
96 views

Criterion for Membership in Hardy Space $H^{1}(\mathbb{R}^{n})$

Let $H^{1}(\mathbb{R}^{n})$ denote the real Hardy space (I am agnostic about the choice of characterization). It is known that if $f:\mathbb{R}^{n}\rightarrow\mathbb{C}$ is a compactly supported ...
8
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1answer
208 views

Error in Stein Shakarchi Exercise on $H^{1}(\mathbb{R})$ and $L\log L$

In Stein and Shakarchi's Functional Analysis (Princeton Lectures in Analysis Vol. 4), the authors claim in Section 2 Exercise 17 that the function $$f(x):=\dfrac{\chi_{|x|\leq 1/2}}{x(\log|x|)^{2}}...
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0answers
31 views

Reference for the spectrum of the Bochner Laplacian on the 2-sphere.

I am looking for a reference for the spectrum of the Bochner Laplacian on $S^2$.
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2answers
64 views

Harmonic function and Neumann Compatibility Condition

______________________________________________________________ Conversely, in a different approach using Green's 1st identity, I showed that by choosing v ≡ 1, that the compatibility condition is ...
1
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1answer
47 views

If $\|g - g_x\|_{\infty}$ is small, then so is $\|g - g_x\|_p$

Let $G$ be a locally compact (additive) abelian group with Haar measure $\mu$. Let $g \in C_c(G)$ with support $K$, and $1 \leq p < \infty$. Then $g$ is uniformly continuous on $G$, so there ...
1
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1answer
24 views

Use the Holder inequality to show that $f \ast g \in C_c(G)$

Let $G$ be a locally compact abelian group, and let $f \in L^p(G), g \in L^q(G)$. I'm trying to prove that $f \ast g \in C_0(G)$. The book I'm reading (Rudin, Analysis on Groups) gives the following ...
0
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1answer
30 views

How to show that the metric $\frac{|dz|^2}{y^2}$ and the 2-form $\frac{dz \wedge d\overline{z}}{(-2i)y^2}$ are $SL_2$-invariant measures?

I am reading the lecture notes. On page 19, let $G=SL_2$. It is said that the metric $\frac{|dz|^2}{y^2}$ and the 2-form $\frac{dz \wedge d\overline{z}}{(-2i)y^2}$ are $SL_2$-invariant measures on the ...
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19 views

$\hat{H} \cong G/H^{\perp}$?

Let $G$ be a locally compact Hausdorff abelian group, and $H$ a closed subgroup of $G$. Let $\hat{G}$ denote the Pontraygin dual of $G$, i.e. the group of coninuous homomorphisms $G \rightarrow S^1$ ...
1
vote
1answer
60 views

Is a bounded function always the Hilbert transform of some other function?

Given $f \in L^\infty(\mathbb R)$, does there always exist a $g$ (in some space) such that \begin{equation*} Hg=f, \end{equation*} where $Hg$ is the Hilbert transform of $g$ ? In other words, is the ...
0
votes
1answer
41 views

Unbounded Hilbert Transform of $C_{c}(\mathbb{R})$ Function

I want to give an example of a continuous, compactly supported function (denoted $C_{c}(\mathbb{R})$) $f$, such that the distribution $H(f)$, where $H$ is the Hilbert transform, does not coincide with ...
1
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0answers
28 views

Littlewood-Paley theorem at endpoints

Littlewood-Paley theorem says that the $L^p$ norm of the square function associated with $f$ is equivalent to the $L^p$ of $f$ when $p\in (1,\infty)$. I'm interested in the endpoint case. Why it ...
5
votes
1answer
45 views

Example of a compactly supported Lipschitz function with non-Lipschitz Hilbert transform

Suppose $f$ is a compactly supported measurable function (say in the interval $[-1,1]$) which is Hölder continuous of order $\alpha\in (0,1)$. I have read that the Hilbert transform $Hf$ of $f$ is ...
1
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1answer
23 views

How can large u(1/2)?

Actually it is rudin's exercise 11.13. Let $u$ is positive harmonic on $U$, unit disc and $u(0)=1$. How can large $u(1/2)$ be? What I tried is this; By theorem 11.30 that every positive harmonic ...
2
votes
1answer
48 views

Calderon-Zygmund Operator Associated to Zero Kernel

We say that a Calderon-Zygmund operator (CZO) $T:L^{2}(\mathbb{R}^{n})\rightarrow L^{2}(\mathbb{R}^{n})$ (i.e. a bounded linear operator) is associated to a CZ kernel $K:\mathbb{R}^{n}\times\mathbb{R}^...
1
vote
1answer
16 views

Solving for values of a Fourier transform

Let $f(x)$ be an integrable function on $\bf R$. Given constants $A,B>0$, does there always exist an $x$ satisfying the following equation? $$A=Bx+\hat{f}(x)=Bx+\int^\infty_{-\infty}f(t)e^{ixt}dt$$ ...
4
votes
1answer
58 views

Expected value formula holds for family of functions.

Does Dynkin's formula hold for any function $f: \mathbb{R}^d \to \mathbb{R}$ such that $f$ and all of its partial derivatives of order $\le 2$ have at most polynomial growth at $\infty$?
2
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1answer
39 views

The distribution solution to $L^{+} u=0$ is u=0, where $L^{+}=-\frac{d}{dx}+x$?

Consider the creation operator $L^{+}=-\frac{d}{dx}+x$. If $u\in L^2(\mathbb{R})$ and is a distribution solution to the equation $L^{+}u=0$, then for any $\phi\in C_0^{\infty}(\mathbb{R})$ we have $\...
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0answers
26 views

Computing Haar Measure

How do we compute both (Left and Right) Haar measure on the following group $$G = \left\{ \begin{bmatrix}x & y \\ 0 & 1 \end{bmatrix} \bigg| \ x\in \mathbb R \ \ \text{and }\ \ y \in \...
0
votes
2answers
106 views

Reference for Harmonic Analysis?

I'm looking primarily for references for Harmonic Analysis. I'm mostly considering Doran&Fell or Deitmar, but I have access to lectures using Stein as well. The important thing is covering ...
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0answers
18 views

Fourier Transform of Distribution Equal to the Distribution Itself

We define $T(t)=a\delta^{(n)}+bt^n$ for $a, b$ nonzero complex constants and $n$ a nonnegative integer. I want to find the combinations of $a, b, n$ such that the fourier transform of $T$ is equal to ...
2
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0answers
23 views

*-representations, unitary representations, and adjunctions

I've been reading Folland's Abstract Harmonic Analysis, and I am currently in the section on the correspondence between unitary representations of a locally compact group $G$ and $*$-representations ...
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0answers
29 views

Haar measure on $\mathbb{C} \setminus 0$.

I want to construct a Haar measure on $\mathbb{C} \setminus 0$. That is, a Borel measure $\mu$ on $\mathbb{C} \setminus 0$ such that $\mu(zS) = \mu(S)$ for all $z \in \mathbb{C} \setminus 0$ and all ...
1
vote
1answer
30 views

A question on maximal operator in Stein's real variable methods, orthogonality and oscillatory integrals.

On page 111 of Stein the maximal function defined as $\mathcal M_0 f(x) = \sup_{t>0} |f*\Phi_t(x)|$ (on page 106), where $\Phi$ is a smooth function supported in the unit ball about the origin with ...
0
votes
1answer
22 views

Some properties of the homogeneous spaces on $\mathbb{T}$

I am reading the first chapter of Y. Katznelson "An introduction to harmonic analysis". There is a definition of homogeneous space $B$ on $\mathbb{T}$, and then it is proved that the trigonometric ...
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0answers
15 views

Invariant subspaces of $L^{\infty}(G)$

I am reading a book named "Lectures on Amenability". The author Dr. Volker Runde defines the following : Let $G$ be a locally compact group, and let $E$ be a subspace of $L^{\infty}(G)$ containing ...
1
vote
1answer
67 views

BMO space and log-lipschitz regularity

Let $\Omega$ an open bounded domain in $R^n$ with smooth boundary and consider a function $u \in W^{1,p}(\Omega) (2<p < +\infty)$ . Suppose that for every ball $B \subset \subset \Omega$ exists ...
4
votes
0answers
60 views

(Exponential) Growth of Operator Norm of Uncentered Maximal Function

Define the uncentered Hardy-Littlewood maximal operator $M$ by $$Mf(x):=\sup_{x\in B}\dfrac{1}{\left|B\right|}\int_{B}\left|f\right|,$$ where we the supremum is taken over all (open) balls $B$ ...