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, ...

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11
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2answers
180 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
46 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
42 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 ...
2
votes
1answer
40 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 ...
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, ...
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 ...
0
votes
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
79 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
votes
0answers
94 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
votes
1answer
188 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
vote
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 $S^2$.
0
votes
2answers
63 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
vote
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
vote
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
votes
1answer
29 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 ...
0
votes
0answers
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
vote
0answers
27 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
43 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
vote
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
44 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 ...
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
votes
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 ...
0
votes
0answers
25 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
98 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 ...
1
vote
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
votes
0answers
22 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 ...
0
votes
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 ...
0
votes
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
64 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
59 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 $B$ ...
1
vote
1answer
28 views

Counterexample to $L^1$-boundedness of the maximal operator $f \mapsto f^\#$ with $f^{\sharp}(x):=\sup_{Q\ni x}\frac{1}{|Q|}\int_{Q}|f-(f)_{Q}|dy$

My question concerns the sharp maximal operator, mapping a locally integrable function $f\colon\mathbb{R}^{n}\to\mathbb{R}$ to $f^{\sharp}\colon \mathbb{R}^{n}\to\mathbb{R}$, where ...
1
vote
1answer
50 views

Proving the Fourier inversion formula in finite abelian groups

The following exercise is exercise 2.2.3 from these lecture notes by Daniel Bump. If $G$ is a finite abelian group, then denote by $G^{\ast}$ the set of its characters, further if $x \in G$ let ...
1
vote
1answer
31 views

Fourier inversion formula on finite abelian groups

The following exercise is exercise 2.2.3 from these lecture notes by Daniel Bump. Let $\mathcal F : L^2(G) \to L^2(G^{\ast})$ be the Fourier transform, defined by $\mathcal{F}f = \hat f$, where ...
1
vote
1answer
65 views

Clarification on something in “Harmonic Analysis - real variable methods, orthogonality and oscillatory integrals” by Elias Stein.

On page 97 in the book, how did Elias inferred that instead of the factor $(1+2^k|y|/t)^N$ we must instead insert the factor: $\frac{t^L (\epsilon+ 2^{-k}t+\epsilon ...
3
votes
0answers
37 views

Fourier analysis on groups, and the isomorphism of characters in the “classical” setting

I am reading these lecture notes By Daniel Bump about Character Theory on Abelian Groups. If $G$ is a group, then $G^*$ denotes its characters, the set of homomorphisms $\pi : G \to \mathbb ...
0
votes
0answers
16 views

Abel transform of $R_z(\Delta)$

The spectrum of the Laplace operator $\Delta$, as self-adjoint unbounded operator on $L^2$ is equal $]-\infty ,0]$. The resolvent $R_z$ is defined for $z \in \mathbb C \setminus ]-\infty ,0] $. Using ...
3
votes
2answers
43 views

Underdamped free vibration proof

I need to prove the solution form of: $$y''+2cwy'+wy=0$$ My book says, after assuming a solution of the form $Ce^pt$, you can show that: $$y=[A\sin(wt)+B\sin(sw)] \cdot e^pt$$ I tried using the ...
0
votes
0answers
34 views

Spherical Fourier Transform of $p_t$

Let $p_t$ denote the heat kernel of the Laplacian $\Delta = \sum_{i=1}^{n}\frac{d^2}{dx_{i}^2}$ on $\mathbb R^n$, I want compute the Spherical Fourier Transform of $p_t$ on $\mathbb R^n$ ? Thank you ...
0
votes
1answer
14 views

Cylindrical boundary condition

There is a cylinder with radius $\rho_o$ and height $h$. The lids are on the planes $z=0$ and $z=h$. $\nabla ^2 \phi = 0$ , $\phi = \phi_o$ on the upper lid, $\phi=0$ every where else on the ...
3
votes
0answers
36 views

Wavelet through the lens of Harmonic analysis

I need some references for starting to study about wavelet. I have enough information about abstract Harmonic analysis. Thanks!
2
votes
1answer
47 views

Harmonic Oscillators: Differential Equations

The book being used for this course is Differential Equations, Dynamical Systems, and an Introduction to Chaos by Morris W. Hirsch. The question is as follows. Suppose there are two masses $m_1$ ...
1
vote
1answer
49 views

Can a lower semi-continuous function blow up to $+\infty$?

I was confused in lecture today that the professor said a l.s.c function can not blow up to $+\infty$... In my point of view, a l.s.c function can blow up to $+\infty$... for example, we take ...
1
vote
1answer
57 views

Is this function bounded below?

Let a bounded open set $\Omega\subset \mathbb R^N$ be given. Let $f$: $\Omega\to (0,+\infty]$ be given and we assume $f$ is locally integrable and there exists a constant $C>0$ so that $$ ...
1
vote
1answer
23 views

Asymptotic behavior of $\sum_{t=1}^T \frac{s_t}{t}$, when $\sum_{i=1}^T s_i = A$

consider the harmonic series: $$ h_T = \sum_{t=1}^T \frac{1}{t} $$ We know that $h_T \in O(\log T)$. Now consider a modified version of the harmonic series: $$ g_T = \sum_{t=1}^T \frac{s_t}{t} $$ ...
1
vote
1answer
41 views

What is the Fourier Transform of an absolute function?

I would like express that the Fourier transform of the function $$ |f(x)| $$ as $$ \widehat{|f|}(\xi) = \text{a function of } \widehat{f}(\xi) $$ In fact, I want to know the relation of ...