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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144 views

Groups with probability measures

Are there algebraic structures that integrate groups with probability measures? For instance, can the closure operation on a group be assigned a probability that says "how much" a member belongs to ...
2
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2answers
308 views

Are the zeroes of the Fourier Transform of compactly supported functions isolated?

I have a continuous function $f$ on a locally compact Abelian group $G$ with compact support, and I would like to say that the zeroes of $f$ are sparse in some sense (isolated would be good, uniformly ...
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1answer
106 views

harmonic balance method for a 2-mass 3-spring system [closed]

I am trying to solve a nonlinear 3spring-2mass system under harmonic loading by using Fourier series expansion of states of the differential equation. The system is just basically two masses, two ...
1
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1answer
57 views

Seeking a specific proof of endpoint boundedness of Riesz potential

The Riesz potential is defined by $$I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.$$ Once $f\in L^{d/\alpha}(\mathbb{R}^n)$, then $I_\alpha f(x)\in BMO$. ...
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0answers
71 views

The decay rate of Hormander lemma is optimal or not?

The Hormander lemma about oscillatory integral operators states that $T_\lambda f(x)=\int e^{i\lambda S(x,y)}a(x,y)f(y)dy$, while the Hessian of $S(x,y)$ is nondegenerate, then $||T_\lambda||_2 \leq ...
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1answer
164 views

$\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$

Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...
3
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1answer
181 views

Is the Fourier-Stieltjes algebra of a locally compact group semi-simple?

Let $G$ be a locally compact group. Is the Fourier-Stieltjes algebra $B(G)$ semi-simple?
4
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3answers
175 views

Show that the Laplacian operator on the Heisenberg group is negative

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ For $z=x+ i y \in \mathbb C$ ...
34
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1answer
2k views

The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?

Yesterday Bourgain, Demeter and Guth released a preprint proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only ...
4
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0answers
78 views

Local factors of Tamagawa measure

This is a reference request to some computations which I hope can be found in the literature somewhere. Let $G\subset GL_n$ be a semisimple linear algebraic group over $\mathbb Q$. The Tamagawa ...
8
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2answers
448 views

Fourier transform of the critical line of zeta?

This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here. Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...
3
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64 views

Almost sure convergence of double nonconventional ergodic averages with respect to $L^p$ function

A famous result of J. Bourgain says that for a probability measure preserving system $(X,\beta,\mu,T)$, with $T_1$ and $T_2$ powers of $T$, we have that for $f_1$, $f_2\in ...
1
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1answer
127 views

Linear Schrödinger equation on $\mathbb{H}^{d}$

Consider the linear Schrödinger equation $i\partial_t u = -\Delta u$, where $\Delta$ is the Laplacian on the hyperbolic space $\mathbb{H}^d$. What are the admissible pairs $(p, q)$ such that we have ...
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1answer
100 views

How to prove the Hölder continuity of a function $u$ by evaluating $\int_{B_{\rho}(x_0)}\frac{|Du(x)|^{2}}{|x-x_0|^{n-2}} dx$?

I'm looking at a video on thin obstacle problem given by Arshak Petrosyan. In his lecture, he uses the following results: Let $0<\alpha<1$, and $B_1$ be the unit ball centered at origin in ...
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1answer
77 views

On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$

There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$ $L^{1}/H^{1}_{0}$ is ...
8
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0answers
148 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 ...
3
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0answers
52 views

$G$-harmonic polynomials, dimension of $\text{Harm}(\mathbb{R}^n, S_n)$? [closed]

Definition. Let $\text{Harm}(\mathbb{R}^n, G)$ be the space of $G$-harmonic polynomials on $\mathbb{R}^n$. My question is, what is the dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?
13
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3answers
427 views

Completeness of nonharmonic Fourier Series

I have the following question: The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$. Thus, certainly the oversampled system ...
12
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1answer
792 views

Where did the term “additive energy” originate?

A fundamental object in modern additive combinatorics and harmonic analysis is additive energy. Given a subset $A$ of (say) an abelian group $G$ the additive energy of $A$ is defined to be the ...
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1answer
115 views

A metric on the set of BV functions, is it mentioned/studied in literature?

I'd like to propose the following metric which operates on the set $M$ of all square integrable functions that are also of bounded variation, of the form $f : (0,1) \to \mathbb{R}$. Given any $x,y \in ...
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0answers
49 views

Left introversion operators associated to function spaces on semigroups

I am stuck on the following question for quite sometime now. Please help, any comment is welcome. Let $S$ be a topological semigroup and $\mathcal{F}$ be a translation invariant, conjugate closed ...
6
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1answer
263 views

Inverse limit of $\Bbb Q/q\Bbb Z$ isomorphic to finite adeles?

Let $\Bbb Q/q\Bbb Z$, for some positive rational $q$, denote the quotient group of the discrete rationals by the subgroup of integers times $q$. For any $q_1, q_2 \in \Bbb Q^+$ and $n \in \Bbb N^+$ ...
2
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0answers
68 views

Combining oscillatory integrals of the first and second kind

Consider an oscillatory integral of the first kind $$ I_\lambda(x)=\intop_{\mathbb{R}^{n}}e^{i\lambda\Phi(x,y)}a(x,y)\,d y,\quad \lambda\geq 0,\; a\in C_c^\infty(\mathbb{R}^{k+n}),\; \Phi\in ...
13
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0answers
262 views

A functional inequality about log-concave functions

Let $f,g$ be smooth even log-concave functions on $\mathbb{R}^{n}$, i.e.,$f=e^{-F(x)}, g=e^{-G(x)}$ for some even convex functions $F(x),G(x)$. Is it true that: $$ \int_{\mathbb{R}^{n}} \langle ...
2
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0answers
92 views

Error term for a Fourier integral

There is a well-known theorem that states that for $f$ continuous and $f,\hat f$ integrable, $$f(0)=\frac{1}{\pi}\lim_{T\to\infty}\int_{-\infty}^\infty f(x)\frac{\sin(Tx)}{x}dx.$$ So it should be that ...
2
votes
1answer
114 views

Expectation equation, harmonic functions, do not understand why equation is true

Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial derivatives are uniformly bounded (or, more generally, have at most polynomial growth as $|x| ...
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50 views

Weak continuity of the Hilbert transform

Is there a simple direct way to prove that the Hilbert transform sends $L^1(\mathbb R)$ into $L^1_w(\mathbb R)$? The Hilbert transform is the convolution by $pv(1/x)$ which is the (distribution) ...
5
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1answer
238 views

harmonic extension of a curve by different parametrization

Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ ...
5
votes
3answers
331 views

Fundamental solution of Discrete Laplace in the plane

We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator Then, it is natural to consider its fundamental solution $u$, i.e. ...
0
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1answer
72 views

Hilbert transform on boundary value of analytic bounded functions

I am considering the boundary values of a bounded holomorphic functions. Suppose $w$ is a bounded holomorphic function in upper half plane, with continuous and bounded boundary value $f$ on real axis. ...
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votes
3answers
555 views

Historical developement of analysis and partial differential equations (especially in the 20th century)

Q: Is there a set of some comprehensive surveys or monographs describing (in technical detail) the historical development of the various subareas of analysis and partial differential equations? ...
14
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102 views

A kaleidoscopic coloring of the plane

Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap ...
9
votes
1answer
307 views

Pontryagin dual of the surreal numbers?

Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown. Alternatively, has this ...
8
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2answers
520 views

Can exponential sums be small on a whole interval?

This is almost certainly routine to an analyst, so forgive me in advance. Let $\alpha_i\in \mathbb{R}$. Consider the functional $$\varphi: L^1[0.9A,A]\to \mathbb{C}$$ via $$f\mapsto \sum_i ...
3
votes
2answers
167 views

Name for an orthogonal decomposition of $L^2 (\mathbb{R}^2; \mathbb{C})$

The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as $$ L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}), $$ where $$ L^2_k (\mathbb{R}^2; ...
20
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1answer
523 views

Bounding Schur symmetric polynomials on the unit circle

Recall the Schur polynomial in $n$ variables, indexed by the partition $\lambda$, with $\ell(\lambda) \leq n$, is given by \begin{equation} s_\lambda(x_1,\ldots, x_n) = a_{\lambda + \delta}(x_1, ...
2
votes
3answers
172 views

Equivalence of Harmonic Maps and Conformal Maps on Genus-0 closed surfaces

By the uniformization theorem, for every genus-0 closed surface $\mathcal{M}\subset\mathbb{R}^3$, there is a conformal map $f:\mathcal{M}\rightarrow \mathbb{S}^2$. Furthermore consider the Dirichlet ...
1
vote
1answer
114 views

Spherical decreasing rearrangement on the sphere

On $\mathbb{R}^n$, we have the concept of spherically decreasing rearragement of a function, which means, given a function $f$, one can design a radial and decreasing function $f^*$ such that $\Vert ...
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0answers
94 views

Looking for some “nontrivial” examples of pseudodifferential operators/symbols

I'm reading up on $\Psi DO$'s and trying to find some examples of symbols that are not quite so trivial. Obviously, the first example of a symbol that most people talk about is just a polynomial in ...
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1answer
63 views

Negativity of a quadratic form on $L^2(M)$

Let $M$ be a compact Riemannian manifold and $V=L^2(M)$. Let $\Delta$ be the negative-definite Laplacian. Let $f \in V$ and $x \in M$ be arbitrary, but fixed. Is it true that ${\rm Re} \ (\Delta f) ...
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197 views

Wavelet-like Schauder basis for standard spaces of test functions?

The Schwartz space of test functions $\mathcal{S}(\mathbb{R})$ is isomorphic to $\mathfrak{s}$ the space of sequences of real numbers with faster than power-like decay. Likewise, the space ...
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53 views

Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane

We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$. What is ...
2
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1answer
177 views

Hardy space, Lebesgue space for $p<1$,

We denote $\mathcal D'(\mathbb R^n)$ the space of distributions, and $\mathcal D(\mathbb R^n)$ the space of smooth, compactly supported functions. Let $\rho\in \mathcal D'(\mathbb R^n)$ such that ...
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40 views

The Best Korn's constant for bounded deformation

I am studying the following version of Korn's inequality. For $u\in BD(\Omega)$, $BD$ denotes the bounded deformation space, we have, there exists a $r(u)\in \operatorname{ker}\mathcal E$ such that ...
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1answer
95 views

One question about the tensor product of $L^1(G)$ and a Banach space $A$

We know that the tensor product of $L^1(G)$ and a Banach space $A$ is isometric to $L^1(G, A)$, the space of all Bochner-integrable $A$-valued functions on a locally compact group $G$. I am looking ...
2
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1answer
79 views

Rajchman measures via strong mixing systems

A Rajchman measure on the unit circle $\mathbb{T}$ is a Borel probability measure $\mu$ with $\lim_{n\to\infty}\hat{\mu}(n)=0$. Where $\hat{\mu}(n)=\mu(z^n)$ for $n\in\mathbb{Z}$ are Fourier ...
2
votes
2answers
123 views

The convolution between weighted $L^1$ space and normal $L^1$ space

Let $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given. We assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant $C>0$, $$ \frac{1}{|B(x,r)|}\int_{B(x,r)}\omega(y)dy\leq C\omega(x) $$ ...
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0answers
48 views

Reference request - Compact embedding of intermediate space

Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$. ...
3
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0answers
93 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 ...
4
votes
0answers
264 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 ...