Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic ...

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Tensor product of algebra group and banach space

Let G be a locally compact group and A be a banach space. It is known that the tensor product L^1(G)⊙A is isometrically isomorphic to L^1(G,A). I need proof of it.
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601 views

Motivation for BMO

At the moment, I don't have access to the early 1960's paper of John and Nirenberg that (from what I understand) introduced the space BMO (bounded mean oscillation). Why were John and Nirenberg ...
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30 views

Why a cone/parabolic set for the nontangential maximal function?

Suppose $f\in L^p(\mathbb{R}^d)$. Then the Dirichlet BVP for the Laplace equation $\Delta u = 0$ in the upper-half plane $\mathbb{R}^d\times\mathbb{R}_{>0}$ with boundary value $f$ can be solved by ...
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3answers
136 views

Reference request : Besov spaces on ubounded domains

As I am relatively new to these matters, I would like to know if you could provide me a reference for Besov spaces on unbounded domains, because when I checked the first tome of Triebel's Theory of ...
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1answer
99 views

Localization arguments in the paper 'the proof of $l^2$ decoupling conjecture'

I am currently reading Jean Bourgain and Ciprian Demeter's 2015 paper The proof of the $l^2$ decoupling conjecture and would appreciate some help in understanding localization argument used in that ...
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193 views

What is the importance of convergence of variation of Fourier reconstruction to that of variation of the function?

Let $f$ be a periodic function of bounded variation which jumps at a point $x_0\in\mathbb{R}$. Let $S_{N}[f]$ denote the partial Fourier sum of $f$ and let $C_{N}[f]$ denote the Cesaro partial sum. It ...
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1answer
709 views

Hardy spaces: analysis <---> martingales

Let $H^p$ be the Hardy space of analytic functions on the open unit disk $D$: $f \in H^p$ if $f$ is analytic on $D$ and $\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta < \infty$. ...
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1answer
322 views

What's the role of $H^{p}(\mathbb{R}^{n})$ in modern (harmonic) analysis?

The classical theory of $H^p$,due to it's heavy dependence on the complex function theory(such as Blaschke products), seemed to have an insurmountable obstacle barrying its extension to several ...
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6k views

Learning roadmap for harmonic analysis

In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...
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1answer
196 views

The $L^2\times L^2\to L^2$ norm of the bilinear multiplier operator

Consider a general bilinear multiplier operator: $$ T(f,g)(n)=\int_{\Pi}\int_{\Pi}\hat{f}(\xi)\hat{g}(\eta)e^{2\pi i(\xi+\eta)n}m(\xi,\eta)d\xi d\eta, $$ where $\Pi$ is the torus, $n\in\mathbb{Z}$, ...
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1k views

Eliminating Gibbs phenomenon, and approximating with jumping functions in Fourier Analysis : An attempt and a question in this regard

Physical Motivation : Hear to these audio files S_f and P_f. S_f is Fourier partial sum and P_f is the new reconstruction, both use spectrum only in the region (0,4KHz) for reconstructing the ...
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59 views

Ring of SO(n)-invariant differential operators on M_n,m

I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/). There comes a point in the paper (Lemma 2.8) ...
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620 views

Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded?

Let $$f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)} $$ Is it true that $\|f\|_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?
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60 views

What does the Plancherel theorem say about positive-definite distributions?

I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem. The ...
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59 views

Generalization of Pitt's theorem

Pitt's theorem (Pitt 1937), states that the one-dimensional Fourier tranform is well defined and continuous between the weighted spaces $L^p(\mathbb{R},|x|^{bp}dx)$ and $L^q(\mathbb{R},|x|^{\beta ...
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1answer
499 views

Green's function of the Ornstein-Uhlenbeck operator

The Ornstein-Uhlenbeck operator $L$ is given by $$ Lu = \Delta u- \frac{1}{2}x\cdot \nabla u. $$ Is there a known closed form expression of the Green's function of $L$ on $\mathbb R^d$ (for $d\geq 2$ ...
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2answers
164 views

Equivalent Norms on Sobolev Spaces

When $k$ is a positive integer and $1<p<\infty$, we know that there is some $C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$ $$ \left\Vert \left( -I+\Delta\right) ...
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1answer
76 views

Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together

Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$. That is $\mathcal{K}$ is RKHS, ...
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2answers
367 views

Iwaniec's conjecture

Does anyone know whether there is any geometric applications of the Iwaniec's conjecture on $ l^p $ bound of Beurling Alfhors transform (or the complex Hilbert transform). One application could have ...
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2answers
114 views

Are spherical harmonics uniformly bounded?

The spherical harmonics are given by $$Y^m_l(\phi,\theta):=N^m_l e^{im\theta}P^m_l(\cos \phi) $$ where $P^m_l$ are the associated Legendre Polynomials and $N^m_l$ is the normalisation. From ...
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340 views

Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity? $$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$ Here, $|x|$ denotes the pointwise absolute ...
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1answer
331 views

Monge–Ampère with drift

Let $I\subseteq \mathbb{R}$ be an interval. Let smooth $M(x,y):I\times(0,\infty) \to \mathbb{R}$ satisfies PDE: $$ M_{xx}M_{yy}-M_{xy}^{2}+\frac{M_{y}M_{yy}}{y}=0. $$ My question is to ...
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40 views

Relating the R-Transform in Free Probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolution of distribution functions, which plays well with the Fourier Transform. In free (noncommutative) ...
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111 views

Complex sum of squares of vector fields (hypoelliptic operators)

Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$ Now, by ...
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155 views

Does there exist a smooth version of Cohen's factorization theorem?

A famous factorization theorem of Cohen states that for any locally compact group $G$, $$L^1(G)=L^1(G)*L^1(G).$$ I want to know if analogous results exist for the class of smooth functions when $G$ is ...
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1answer
160 views

Spherical harmonics and ellipticity of the Laplacian

Let us consider the sphere $S^n$ and the Laplacian $-\Delta$ on it. Let $L^2(S^n) = \bigoplus_k V_k$, where $V_k$ represents the eigenspace of the Laplacian with eigenvalue $k(k + n - 1)$. We know ...
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53 views

Fourier coefficients of positive polynomials

Let $p(x) \geq 0$ be a positive polynomial on the hypersphere ($x \in S^{n-1}$) satisfying $\int_{S^{n-1}} p(x) = 1$. Writing $p(x) = \sum_{j=0}^s p_j(x)$ where $p_j(x) = \sum_m p_{jm} s_{jm}(x)$ with ...
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1answer
141 views

an analogue of Littlewood-Paley-Rubio de Francia theory

For any function $f$ defined on the set of integer $\mathbb{Z}$, we define its Fourier transform as the following periodic function: $$ \mathbb{F}f(\xi)=\sum_{n\in\mathbb{Z}}f(n)e^{-2\pi i n\xi} $$ ...
2
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1answer
182 views

$BMO$-property via a John-Nirenberg type estimate?

Let $\Omega \subset \mathbb R^d, d\ge 2$, be bounded and denote a ball in $\mathbb R^d$ by $B$. Denote also $$ f_B:= \frac1{|B|}\int_B f \, dx. $$ Suppose $f \in L_{\rm loc}^p(\Omega)$ for all ...
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115 views

the (2,2,1) boundedness of a “product” operator

Let $\{E_j\}_{j\in\mathbb{Z}}$ and $\{F_k\}_{k\in\mathbb{Z}}$ be two collections of pairwise disjoint sets in $\mathbb{R}$. Let $C(j,k)$ be a bounded function (e.g. $|C(j,k)|<1$) defined on ...
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1answer
104 views

The (Hecke) double coset von Neumann algebra

It it well-known in the von Neumann algebra theory that for $\Gamma$ a non-trivial countable group, the von Neumann algebra $L(\Gamma)$ generating by $\Gamma$ acting by left multiplication on ...
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64 views

Does the Total variation of the Fourier partial sum of a bv function with jumps converge to TV of the function as $N\to\infty$

Does the total variation of the Fourier partial sum of a piecewise continuous bv function converge to the total variation of the function as $N\to\infty$. To explain briefly, Let $f$ be a periodic ...
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1answer
177 views

Poincaré inequality for curl-integrable functions

Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and $$ u_B := \frac1{|B|}\int_B u \, dx. $$ The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...
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107 views

Technical question about a Fourier transform

I would like to know if there is an explicit expression for the Fourier transform of the following function: $$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$ or to know where I can find some techniques to ...
5
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1answer
407 views

When does a matrix define a convolution operator on a hypergroup?

Let $H$ be a discrete hypergroup. Suppose I have a matrix $A=(A_{x,y})$ indexed over $H$ with nonnegative entries which defines a bounded operator on $\ell^2(H)$. When does there exist $f\in\ell^1(H)$ ...
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3answers
356 views

Derivatives of radial functions can be bounded by derivatives in terms of radial distance?

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
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61 views

Poisson Kernel and Triangles

The Poisson Kernel is an approximation to the identity, meaning $P_r(\theta) \approx \delta(\theta)$; here is the formula on $\mathbb{D}$: $$ P_r(\theta) = \sum_{n \in \mathbb{Z}} r^{|n|} ...
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43 views

Non interacting complex unit

How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...
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2answers
283 views

Radial limit does not exist almost everywhere

Problem 4 in Chapter 4 of Stein's book "Real Analysis" says $\sum_{n\geqslant 0}z^{2^n}$ doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...
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137 views

application of factorization theorem

Young's inequlity tells us that $L^{1}(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R)$ with norm inequality $$\|f\ast g\|_{L^{p}} \leq \|f\|_{L^1}\|g\|_{L^p};$$ and of course this ...
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1answer
52 views

Limiting absorption principle

I would like to know if there is a book (or a paper) which can give me an introduction to LAP. I tried to read some papers by myself, but I don't feel comfortable. I think that I need the basic ideas ...
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34 views

Is Wiener amalgam spaces $W^{2,1}(\mathbb R)\subset C_0(\mathbb R)$?

I have been learning Wiener amalgam spaces. In Wiener amalgam spaces $W(X, L^2)$, I am taking $X=\mathcal{F}L^{1}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}\},$ and $m(x)=1.$ Take $f(x)= ...
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69 views

mean value estimate $ \frac{1}{T} \int_0^T | \zeta( \tfrac{1}{2} + it)|^2 \, dt = \log \frac{T}{2\pi } + (2\gamma - 1) + O(T^{\delta})$

I was browsing through the recent paper of Bourgain and Watt on mean value estimates of the Riemann zeta function on the critical line. $$ \frac{1}{T} \int_0^T | \zeta( \tfrac{1}{2} + it)|^2 \, dt = ...
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2answers
51 views

extension of the projectivized gradient of a harmonic function

Let $(M,g)$ be a riemannian manifold, $\Delta$ the associated Laplacian, and $\{ f_i \}$ the real-valued eigenfunctions of $\Delta$. Then, $\nabla f_i \in \Gamma ^{\infty } (\mathrm{T} M) $ is defined ...
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122 views

Is the isomorphism between $BMO/\mathbb{R}$ and $(H^1(\mathbb{R}^n))^{\star}$ isometric?

Let $BMO$ the space of bounded mean oscillation functions on $\mathbb{R}^n$ equipped with the Lebesgue measure. If $Q\subset \mathbb{R}^n$ a cube, let $m_Q f$ the average of a function $f\in ...
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28 views

Estimates for derivatives of a positive discrete harmonic function

There is the following estimation (Duffin, Discrete potential theory, Theorem 5): Let $f$ be a discrete harmonic function in a sphere of radius $R$ with the center $p$, all in $\mathbb Z^3$. Then, if ...
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46 views

Stationary averages of harmonic measures

Let $A$ be a countable subset of the open unit disk $\mathbb D$ centered at 0. For a point $x\in\mathbb D$ denote by $\nu_x$ the associated harmonic measure on the boundary circle $\partial\mathbb D$. ...
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45 views

Proof of the chain rule for fractional derivatives in F.M.Christ and M.I.Weinsten's paper

In the paper: Dispersion of small amplitude Solutions of the generalized Korteweg-de Vries equation, JOURNAL OF FUNCTIONAL ANALYSIS 100,87-109 (1991). I find a mistake in the proof of proposition 3.1 ...
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85 views

The Lebesgue measure of the low level sets of the two-dimmension Fourier transform of a compactly supported function

Let $f\in {{L}^{1}}\left( {{\mathbb{R}}^{2}} \right)$ be a density function with the support $\operatorname{supp}\left( f \right)\subset \left[ a,b \right]\times \left[ c,d \right]$. Denoted by ...
5
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116 views

Local version of the Hardy-Littlewood-Sobolev theorem for Riesz potentials: $\|I_\alpha(f)\|_{L^q} \le C \|f\|_{L^p}$?

Recently, I have been studying the properties of the Riesz potential $$ I_\alpha(f)(x) = c_{d,\alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy. $$ The classical Hardy-Littlewood-Sobolev ...