Questions tagged [harmonic-analysis]
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 analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.
1,318
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Use of Stein's maximal principle in Bourgain's paper on Besicovitch sets
I'm trying to understand Bourgain's paper "Besicovitch type maximal operators and applications to Fourier analysis". Let $\xi\in S^2\subset\mathbb{R}^3$ be a unit vector and $\delta>0$, ...
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63
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Value of Plancherel weight on convolutions (Takesaki VII Section 3 Theorem 3.4)
In this post, I follow conventions and notations from Takesaki's second volume "Theory of operator algebras" (chapter VII sections 2 and 3). Let $G$ be a locally compact group with left Haar ...
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1
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125
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Exponential sum with weight in bottom
I am interested in the exponential sum
$$\sum_{n=1}^X \frac{e(c_1n^2+c_2 n)}{1-e(c_1n)}$$
where $c_2$ is irrational and $e(x)=e^{2\pi i x}$. I know if the denominator is not there, this is a Weyl sum ...
- 417
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116
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Iterated exponential sums with cosine
As a continuation of Iterated exponential sums I have a follow up question. Is there anything known about iterated exponential sums of the form
$$\sum_{1\leq n\leq X} e(f(n))\sum_{1\leq m< n} \cos(...
- 417
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1
answer
180
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Iterated exponential sums
Does anyone have any references for iterated exponential sums? That is, sums like
$$\sum_{1\leq n\leq X} e(f(n))\sum_{1\leq m\leq n} e(f(m)),$$
where $e(x)=e^{2\pi i x}$? I am looking for references ...
- 417
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1
answer
99
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The notion of "Admissible" and "Permitted" in the context of convolution with distributions and hypocontinuity
I am reading the paper "On Convolutions" (1958) and have encountered the notion of "Admissible" and "Permitted" spaces.
In p.17-18 of the above paper, it says that an ...
- 1,879
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Are there extremally disconnected groups?
A Hausdorff space is called extremally disconnected or extreme, if for every open set $U$ the closure $\overline U$ is open, too. The question, whether there are extremally disconnected topological ...
- 1,328
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2
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222
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Product of locally Borel sets locally Borel
Let $X$ be a locally compact Hausdorff space with a fixed Radon measure (= Borel measure that is finite on compact subsets, inner regular on open subsets and outer regular on Borel sets) $\mu$ . A ...
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48
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A parametrix construction for heat boundary value problem using Fourier transformation
Let $\Omega$ be a smooth bounded open subset in $\mathcal{R}^{d}$, with $d \geqslant 3
$ and $T>0$. Consider the linear parabolic initial Dirichlet boundary value problem with $f\in H^{-1}(\Omega)$...
- 21
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89
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Mathematical study of dispersive PDEs [closed]
My understanding is that there have been a lot of activities in harmonic analysis and PDEs that use sophisticated tools to study dispersive PDEs like the Schrodinger's equation. E.g. the Strichartz ...
- 225
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0
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176
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Schrödinger representation of the Heisenberg group
Let $\Pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H^n=\Bbb C^n\times\Bbb R$. For $\phi\in L^2(\Bbb R^n)$, we have
$$\Pi_{\lambda} (x,y,t)\phi(\xi)=e^{i\lambda t} e^{...
- 347
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0
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87
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Sobolev estimates on domain with boundary
Could someone point me to a reference for the proof of the following Sobolev estimate
$$
\|u\|_{L^{2 d /(d-2)}(\Omega)} \leqslant C(\|f\|_{L^{2 d /(d+2)}(\Omega)} + \|g\|_{(\partial\Omega)})
$$
for ...
- 21
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0
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Can the best constants in harmonic analysis be approximated in principle?
Consider the trivial example of Holder's inequality $\|f\|_p\,\|g\|_q\geq |fg|_1$ if $\frac{1}{p}+\frac{1}{q}=1, p,q\geq 1$ and $f,g$ are functions on $\mathbb{R}^n$. Let's suppose we don't know how ...
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48
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Existence of solution to prescribed curvature problem with given asymptotic on the punctured unit disc
I have trouble understanding a conclusion in the following paper: Prescribed Curvature and Singularities of Conformal Metrics on Riemann Surfaces by Robert C. McOwen
In the appendix, part B, we are ...
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1
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144
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Reverse estimate on the Riesz potential $I_\alpha : L^{n/\alpha}\to \mathrm{BMO}$
$\newcommand\BMO{\mathrm{BMO}}$Consider the Riesz potential on $\mathbb{R}^n$ given by
$$
I_\alpha f(x) = c_{n,\alpha} \int_{\mathbb{R}^n} \frac{f(y)}{\lvert x-y\rvert^{n-\alpha}} dy.
$$
It is known ...
- 199
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How to give a counterexample of this estimate related to Paley-Littlewood theorem?
I am studying Paley-Littlewood theorem in Harmonic analysis, and I met an exercise. I would like to construct a function $f$ as a counterexample to show that the inequality
\begin{equation}
\| f \|^...
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0
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61
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Question about stationary phase with Hessian close to $0$
Let $\phi$ be a smooth real function in one variable and say $w$ is a smooth function with compact support say $[- 1, 1]$. Let me define
$$
I_{\lambda} = \int_{\mathbb{R}} w(t) e^{i \lambda \phi(t)} ...
- 3,373
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37
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Gehring lemma for fractional maximal functions
Given a function $f\in L^p(\mathbb{R}^n)$, the Gehring lemma states that if there exists $p>1$, a constant $C_0>1$ and a cube $Q \subset \mathbb{R}^n$ such that for almost every $x\in Q$, it ...
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What's the purpose of the operator $(\Delta^{-1}+\lambda)^{-1}$ in Tomita-Takesaki modular theory?
I was reading Tomita-Takesaki modular theory (from all the books, and articles), the goal is to relate a von Neumann algebra $\mathcal{A}$ with its commutant $\mathcal{A}'$ on a Hilbert space $\...
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168
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The convention of Fourier transform on symmetric spaces
I'm from the physics side. When trying to understand the Plancherel formula of reductive symmetric space of Harish-Chandra class, I get confused on the convention of Fourier and related transforms.
$\...
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67
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On Dirichlet eigenfunctions of a domain
Given any bounded domain $\Omega\subset \mathbb R^n$, $n\geq 2$, with a Lipschitz boundary, let $\{(\lambda_k,\phi_k)\}_{k\in \mathbb N}$ be the Dirichlet eigenvalues and eigenfunctions of $-\Delta$ ...
- 3,863
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1
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180
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Compactly supported continuous functions as a Tomita algebra
Let $G$ be a locally compact group with modular function $\delta_G$ and consider $\mathcal{K}(G)$, the set of compactly supported continuous functions $G\to \mathbb{C}$, with the $*$-algebra structure ...
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1
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213
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Why complex conjugate in definition of the Fourier transform?
Let $G$ be a locally compact abelian group and $f:G \to \mathbb{C}$ a function. Its Fourier transform (when it exists) is defined to be
$$\widehat{f}(\chi) = \int_G f(g) \bar{\chi}(g) \mathrm{d} g,$$
...
- 19.4k
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Fixing the duality $L^\infty(X)= L^1(X)^*$ for Radon measure spaces
Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
Let me denote the Borel subsets of $X$ by $\mathscr{B}(X)$. Folland claims that if $\mu$ is a ...
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4
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285
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Every locally compact group gives rise to a locally compact quantum group
A locally compact quantum group (in the sense of Vaes-Kustermans) consists of the data $(M, \Delta, \varphi, \psi)$ with $M$ a von Neumann algebra, $\Delta: M \to M \overline{\otimes} M$ a normal ...
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142
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Haar measures of compact subgroups
Let $G$ be a locally compact group, $K$ a compact subgroup in $G$, and $\mu_K$ the normalized Haar measure on $K$:
$$
\mu_K(K)=1.
$$
Let us denote by $\widetilde{\mu_K}$ the measure on $G$ defined as ...
- 7,091
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0
answers
98
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Norm of the Linear Operator
Let $G$ be a compact group, and $\pi : G \rightarrow \mathcal{U}(H)$ be a continuous unitary representation. Let $f \in L^{1}(G)$ be arbitrary.
By Riesz Representation Theorem we can find a bounded ...
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67
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Positive definiteness with nonnegative weights
Is there a simple criterion to certify if some function $f: \mathbb{R} \to \mathbb{R}$ satisfies that $\sum_{i,j=1}^n c_ic_jf(x_i-x_j) \ge 0$ for all $x_i \in \mathbb{R}$ and $c_i \ge 0$?
Note that if ...
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$ \int_{\mathbb{R}^n} f R_1 f=0$ if $f\in L^2$ Riesz transform
How can I see that? It seems that it has to do with the adjoint of the Riesz transform $R_1^*=-R_1$, but here we do not have the complex $L^2$ scalar product.
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1
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Bound for the product of Sobolev functions in $W^{s,1}$
I would like to bound the product of two functions $f$, $g$ in the space $W^{s,1}$.
$$ \lVert fg\rVert_{W^{s,1}}\leq c\lVert f \rVert \lVert g \rVert. $$
It seems reasonable to want to use Hölder's ...
- 1
2
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0
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168
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Failure of Calderón–Zygmund inequality at the endpoints
$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'd like to prove that the famous Calderón–Zygmund elliptic estimate $$ \norm{ \partial_{ij}u }_{L^p} \leq C \norm{\Delta u }_{L^...
- 250
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0
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97
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Duality of $H^1$ and BMO
While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
- 11
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125
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Inequality for sums of sines with similar frequency
Let $c>0$ be a very small constant and $N \in \mathbb N$ very large.
Assume we have a function $f(x)$ for $x \in S^1$ defined as
$$
f(x) = \sum_{k=\lfloor N/(1+c) \rfloor}^{N} c_k \sin(kx+b_k)
$$
...
- 353
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0
answers
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Does this function belong to local Hardy space?
Let $f\in H^1(\mathbb{R}^n)$, where $H^1(\mathbb{R}^n)$ denotes the Hardy space on $\mathbb{R}^n$. Here, we define a new function
$$
F=\begin{cases}
f&, x_n\ge 0\\
0&, x_n<0.
\end{cases}
$$
...
- 21
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3
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349
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If the Fourier coefficient $\hat{f}(k)$ of $f\in C^1(\mathbb T)$ is zero for all $|k|<N$, then $\|f\|_{L^\infty}\leq \frac CN \|f'\|_{L^1}$?
Let $f\in C^1(\mathbb T)=C^1(\mathbb R/\mathbb Z)$ be a function such that
$$\hat f(k):=\int_{\mathbb T}f(x)e^{-2\pi ikx}\,dx=0,\qquad \forall k\in\{-N+1,\cdots,-1,0,1,\cdots, N-1\}.$$
Do we have $\|f\...
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1
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Littlewood-Paley characterisation of Hölder regularity
I am going through Terence Tao's "Nonlinear Dispersive Equations (Local & Global Analysis)" and trying to work through some of his exercises. However, I find myself being stumped by ...
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1
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125
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The asymptotic behaviour of a singular integral
Given $0<\alpha, \beta<1$, $a,b>0$, $a^2+b^2<1$.
I am trying to determine the asymptotic behaviour of
$$F(a,b):=\int_{\substack{a/2<x<2a\\\\b/\sqrt{2}<\sqrt{1-x^2}<\sqrt{2}b}}\...
- 391
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0
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20
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Bounding $\widehat{G_{m+1}}\ast\widehat{H_m}(0)$ when $\frac{1}{2}\leq\widehat{H_m}(0)\leq\frac{3}{2}$ and $H_m,G_{m+1}$ are smooth over $\mathbb{T}$
To put this question into proper context, what I am asking is related to the construction of smooth function $H_m$ over the torus $\mathbb{T}$ such that
$$\left|\widehat{H_m}(k)\right|\leq C\log(\left|...
- 121
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votes
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117
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Building a smooth function from a rapidly decreasing sequence
Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function?
More precisely:
Let $\lbrace c_k\rbrace _{k \...
- 571
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48
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Eigenfunction expansion theorem for general manifold for smooth functions
The question starts with the well known facts that: if $f$ is a smooth function on $S^1$, then its Fourier series converges to it in smooth topology.
This must be true in more general setting. I have ...
- 31
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3
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Is there a non-constant function on the sphere that diagonalizes all rotations simultaneously?
INTRODUCTION. I am teaching a course in Harmonic Analysis. In class, very often I find myself stressing out the fundamental property that the functions
$$
e_n(x)=\exp(2\pi i n x), \quad \text{where }\...
- 577
0
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0
answers
99
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Using projections to determine equidistribution
Suppose I have a collection of points on $\mathbb{S}^{n-1} \subset \mathbb{R}^n.$ I want to know that they are equidistributed (if you want to be more precise, you have a sequence of such collections, ...
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1
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169
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Uncorrelation of exponential sums generated by irrational rotations over disjoint sets of integers
Assume that $\mathbb{N}=\{0,1,2,\ldots\}$ is partitioned into $k\ge 2$ disjoint sets $J(1),\ldots,J(k)$ such that for every $1\le p \le k$ the set $J(p)$ has an asymptotic density
$$
d(J(p))=\lim_{n\...
- 1,478
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votes
2
answers
258
views
Calculating the Fourier dimension of a real interval $\left[a, b\right]$
(Preliminaries:) 1.) Let $S\subset\mathbb{R}^n$ and define $\mathcal{M}(S) = \{\text{$\mu$ a Borel measure}: \text{$0 < \mu(S) < \infty$ and $\mathrm{support}(\mu)\subset S$}\}$.
2.) Define the ...
- 121
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0
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247
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Question on estimate in one of Jean Bourgain's 1992 papers
The paper in question is A Remark on Schrodinger Operators.
The goal of the argument is to estimate the following integral:
$$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\...
- 655
1
vote
1
answer
99
views
How to prove $ \|u\|_{L^{\infty}}\leq C\|\partial_1\square u\|_{L^1} $ for any $ u\in C_0^{\infty}(\mathbb{R}^{1+2}) $?
It comes from estimates for wave equations.
For any $ u=u(t,x)\in C_{0}^{\infty}(\mathbb{R}^{1+2}) $, which is a smooth compactly supported function, prove that
$$
\|u\|_{L^{\infty}(\mathbb{R}^{1+2}\...
0
votes
0
answers
54
views
Fourier restriction in decoupling inequalities
I'm reading Bourgain and Demeter's paper https://arxiv.org/abs/1403.5335 "The proof of the $l^2$ decoupling conjecture".
On page 1 the paper says, let $P^{n-1}=\{(\xi_1,...,\xi_{n-1},\xi_1^2+...
- 225
0
votes
2
answers
170
views
Well-defined distribution and its singular support
Let $f$ be a smooth function on $X$, an open subset of $\mathbb{R}^n$, with $Im(f) \geq 0$. Let us fix an $\epsilon > 0$.
Let $T_{\epsilon} := \frac{1}{f(x)+i\epsilon} $ in $D’(X)$.
Now if we ...
- 51
1
vote
0
answers
43
views
Localize functions in the Hardy space $\mathcal H^1(\mathbb R^n)$
Let $f$ belong to the Hardy space $\mathcal H^1(\mathbb R^n)$, $B\subset \mathbb R^n$ be the unit ball. Does there exist a $\bar f\in \mathcal H^1(\mathbb R^n)$ with compact support such that $\bar f=...
- 53
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0
answers
53
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Is a discrete harmonic function bounded below on a large portion of $\mathbb{Z}^2$ constant?
In the paper https://doi.org/10.1215/00127094-2021-0037, the main result is if we partition the plane $\mathbb{R}^2$ into unit squares (cells) so that the centers of squares have integer coordinates ...
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