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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$k[x_1, \dots, x_n]$ 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 ...
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36 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 ...
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1answer
100 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 ...
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82 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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29 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} + ...
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1answer
182 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 ...
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2answers
329 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 ...
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1answer
80 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 ...
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214 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 $$\int_{% \mathbb{R}^{d}}\left\vert f_{m}\left( y\right) \right\vert dy=1$$ ...
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150 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 ...
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247 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 ...
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134 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?
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36 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 ...
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17 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 ...
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2answers
102 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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37 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 ...
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41 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 ...
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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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1answer
146 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 ...
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75 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 ...
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15 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 ...
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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 ...
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58 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 ...
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Lower bound for the Hardy-Littlewood maximal function implies it is not integrable

I am working on the following problem from Stein and Shakarchi: Let $f$ be an integral function on $\mathbb{R}^d$ such that $\|f\|_{L^1} = 1$ and let $f^*$ by the Hardy-Littlewood maximal function ...
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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 ...
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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 $\infty$?
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28 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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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 ...
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1answer
59 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 ...
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$\hat{H} \cong G/H^{\perp}$?

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 $G \rightarrow S^1$ ...
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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 ...
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1answer
40 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 ...
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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 ...
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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 ...
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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? $$A=Bx+\hat{f}(x)=Bx+\int^\infty_{-\infty}f(t)e^{ixt}dt$$ ...
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1answer
87 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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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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23 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 ...
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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 ...
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*-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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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 ...
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1answer
29 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 ...
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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 ...
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1answer
53 views

Distributivity of projective tensor product over direct sum

Let $I$ is a non-empty set and $\{A_i\}_{i\in I}$ is a family of Banach algebras and $B$ is a Banach algebra. Define $$\ell^1-\oplus_{i\in I}A_i=\{a=\{a_i\}_{i\in I}: \|a\|_1=\sum_{i\in ...
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Unital amenable Banach algebras which is a proper two sided ideal in its second dual

I need some examples of "unital amenable Banach algebras which is a proper two sided ideal in its second dual".
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1answer
59 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 ...
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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 ...
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1answer
21 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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(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 $B$ ...