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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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 ...
1
vote
2answers
36 views

Showing $\hat{\tilde{f}}=\tilde{\hat{f}}$ where $\hat{f}$ is the Fourier transform, and $\tilde{f}(x) = f(-x)$

I'm trying to prove that $\hat{\tilde{f}}=\tilde{\hat{f}}$ for any integrable function $f$, where $\hat{f}$ denotes the Fourier transform of $f$ and $\tilde{f}$ denotes the mapping $f(x)\to f(-x)$, ...
1
vote
1answer
68 views

Question regarding constructing a function via its Fourier transform

Let $\varepsilon>0$. I was interested in understanding the justification of defining the following function $\phi$ via its Fourier transform, satisfying the following properties: (1) $\widehat{\...
1
vote
0answers
34 views

Proof of an inequality with using maximal operator

I want to prove an inequality such that $$ \int_{B}|f(y)|dy\leq |B|^{1-\frac{1}{p}}\|f\|_{L^p(B)}, $$ where $B\subset\mathbb{R}^n$ is a ball, $p>1$ and $\|f\|_{L^p(B)}=(\int_{B}|f(y)|^pdy)^{\frac{1}...
0
votes
0answers
34 views

Question related to decay of Fourier transform and smoothness

Suppose $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies $$ |\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}. $$ Let $$g(x) = \frac 1{...
10
votes
2answers
85 views

$W^{s,p}(\mathbb{R}^{n})$ Is Not Closed Under Multiplication when $s\leq n/p$

For $s\in\mathbb{R}$, $1<p<\infty$, and $n\geq 1$, define the Sobolev space $W^{s,p}(\mathbb{R}^{n})$ by $$W^{s,p}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}(\mathbb{R}^{n}) : \|(\langle{\xi}\...
1
vote
0answers
38 views

Additive combinatorics in Hamming weights of addition of numbers modulo $2^n$ with prescribed Hamming weight

I wonder if anyone could point me to a reference about the following type of combinatorics problem: Fix $n $. For an integer $k \in [n] = \{1, \ldots, n\}$, let $A(k)$ be the set of integers in $[0, ...
0
votes
1answer
17 views

Proving Holders inequality for the sequence space $l_p (\mathbb(N)$

We first look at when $p=1$ and $q=\infty$ And we look at the non trivial case when the sequences $x=(x_k)_{k \in \mathbb{N}}$ and $y=(y_k)_{k \in \mathbb{N}}$ are both not equal to zero. We first ...
0
votes
0answers
44 views

How is $\lim_{\rho \to 0} \rho^{1 - n} \int_{\partial B_\rho} u ds = n \omega_n u(y)$?

I'm looking at some material on harmonic functions and it says $$\lim_{\rho \to 0} \rho^{1 - n} \int_{\partial B_\rho} u ds = n \omega_n u(y)$$ where $n$ is the number of dimensions and $\omega_n$ ...
1
vote
1answer
50 views

Convolution of indicator functions with values in a finite field

This is something I haven't seen online yet, indicator functions with values in a finite field. Probably for a good reason, but I would like to know why, and if there are still things that can be said....
4
votes
1answer
39 views

Can the system of shifts of an $L^2(\mathbb{R})$ function be an ONB?

In Wavelet theory, one constructs wavelet bases via translations a dialations of an $L^2$ function... Is it possible for some set of translations alone to form an Orthonormal Basis? That is: Does ...
2
votes
1answer
56 views

Critical Homogeneous Sobolev Embedding

For $s\in\mathbb{R}$, $1<p<\infty$, define the homogeneous Sobolev space $\dot{W}^{s,p}(\mathbb{R}^{n})$ as follows: For $f\in\mathcal{S}_{0}(\mathbb{R}^{n})$ (Schwartz functions with Fourier ...
1
vote
0answers
37 views

Real zeros of a function holomorphic in the upper half plane

Suppose that $f(z)$ is holomorphic in the upper half-plane and $\lim_{\, |z| \to \infty} f(z) = 1$ along any ray in the open upper half-plane. Suppose also that $f(z)$ has a continuous extension to ...
1
vote
1answer
51 views

Finding a continuous $f\in C(\mathbb{T})$ with $\{\hat{f}(n)\}\notin \ell^p$

Denote the circle by $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Find an example for a continuous function $f\in C(\mathbb{T})$ with coefficients of the fourier series, $\{\hat{f}(n)\}\notin\ell^p,\forall p&...
0
votes
0answers
25 views

Are exponential functions a complete system on $L^2\left(\mathbb{T},\mu\right)$

Let $\mathbb{T}$ denote the unit circle in $\mathbb{C}$, and let $\mu$ be a complex Borel measure on $\mathbb{T}$. Consider the space of square integrable functions $L^2\left(\mathbb{T},\mu\right)$. ...
1
vote
1answer
27 views

Convergence of measures on $\mathbb{T}$

Denote by $M(\mathbb T)$ the set of complex-values measures on the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$.Prove that $D(T)$, the set of discrete measures on $\mathbb{T}$ is: closed in $M(\...
2
votes
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 ...
1
vote
1answer
22 views

Convergence in $H^1(\Omega)$ and $L^2(\Omega)$

Let $\Omega$ be a bounded domain (maybe that doesn't matter), if $f_n\rightarrow f$ in $H^1(\Omega)$, does it follow $f_n\rightarrow f$ in $L^2(\Omega)$ since $H^1$ is dense in $L^2$? Is it true that $...
5
votes
1answer
44 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 ...
0
votes
0answers
21 views

Spectrum of Laplacian on divergenceless vector on $T^3$

I encountered the following calculation: $\int dA A_\mu\Delta A^\mu$, where $A_\mu$ are divergenceless vectors and the theory is on flat $T^3$. What is the determinant we get by integrating over $...
1
vote
1answer
25 views

Why such a net will exist?

G- locally compact group & $\lambda(x)f(y) = f(x^{-1}y)\ \forall \ y \in G$. The following condition is called Reiter's finite condition. $P(G) := \{f \in L^1(G): f \geq 0, \|{f}\|_1 = 1\}.$ ...
0
votes
0answers
35 views

Boundedness of singular integral operators on $L^{p}$ spaces

Let $\Omega \in L^{1}(S^{d-1})$ have mean zero. Prove that, if the operator $T_{\Omega}: L^{p} \rightarrow L^{q}$ given by $T_{\Omega}f(x) $:= p.v. $\int_{\mathbb{R}^{d}} \frac{\Omega \left(\frac{y}{|...
0
votes
1answer
44 views

Why a left-invariant Haar measure is $\mu(A)=\int_A \frac{1}{a^2}da\,db$ and a right-invariant Haar measure is $\mu'(A)=\int_A\frac{1}{a}da\,db$?

Let $G$ be the group of affine transformations of $\mathbb R$, $x\mapsto ax+b$, $a>0$. $G$ is the half-plane $(a,b);a>0$. A left-invariant Haar measure is $\mu(A)=\int_A \frac{1}{a^2}da\,db$, ...
1
vote
0answers
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 ...
1
vote
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(\...
4
votes
0answers
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}=\...
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 ...
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_{\...
0
votes
0answers
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)|‎ $‎. ...
2
votes
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: (...
4
votes
0answers
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 $\...
2
votes
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}{\...
4
votes
1answer
112 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 ...
4
votes
0answers
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 $$|\...
0
votes
0answers
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}...
1
vote
1answer
210 views

On the Hilbert Transform of a Bounded Function

Let $f: \mathbb R \rightarrow \mathbb R$ be a bounded function that is smooth and also in $L^1(\mathbb R) \cap L^2(\mathbb R)$. I want to prove that the Cauchy transform of this function $Kf$ is in $H^...
1
vote
2answers
338 views

(Obvious?) Half-Space Poisson Kernel Estimate

In Stein's Singular Integrals and Differentiability Properties of Functions, Theorem 1 (a) of Chapter 8 (pg. 197) he makes the claim without proof that the Poisson Kernel for the half-space $\mathbb{R}...
1
vote
1answer
93 views

$L^p$ boundedness of Riesz potential.

Why studying, I repeatedly see people use the following result. That is there exists $C > 0$ such that $$\|\nabla \Delta^{-1}\nabla \times u\|_p \le C \|u\|_p$$ for every $u \in C^1_c(\mathbb{R},\...
4
votes
2answers
219 views

A question from Stein's book, Singular Integral.

A question from Stein's book, Singular Integral. Let $\left\{ f_{m}\right\} $ be a sequence of integrable function such that $$\int_{% \mathbb{R}^{d}}\left\vert f_{m}\left( y\right) \right\vert dy=1$$ ...
2
votes
0answers
157 views

What are the sequels to Rudin's Functional Analysis?

Briefly speaking my purpose, I'm looking for the sequels to Rudin's Functional Analysis. How about the following books by Stein? Are there any other nice ones? Harmonic Analysis: Real-Variable ...
0
votes
1answer
255 views

Question from Stein's Singular Integrals and Differentiability Properties of Functions.

My question is in regards of Stein's proof that Hilbert transform is of weak $(1,1)$ property, on page 30 of the textbook I mentioned in my title. On page 32 he writes that because $|\nabla K| \leq B ...
2
votes
0answers
135 views

Intuition for Calderon-Zygmund operator?

What is the best intuition for Calderon-Zygmund operators? Why are they so important in singular integrals, and complementary, which singular integrals don't they cover?
3
votes
1answer
43 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
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..}^{\...
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, ...
2
votes
2answers
38 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 &...
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 $(p,p)$ ...
1
vote
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 ...
8
votes
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}}...