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.
1,581 questions
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Another question on large sieve inequality
Recently, I was interested in the large sieve inequalities. A few days ago, I came up with a question on the large sieve inequality involving 𝐺𝐿(2); see On the large sieve inequality involving $GL(2)...
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Mapping properties of the strong maximal function and a question about the article of Cordoba Fefferman
I have been reading the article "A geometric proof of the strong maximal theorem" by A. Cordoba and R. Fefferman which can be found here. Right in the beginning of the article the authors ...
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Harmonic analysis on the non-trivial zeros of the Riemann zeta function?
Suppose I have some function $f(x)$ that satisfies constraints roughly as restrictive as those for Fourier series expansions, and I'm interested in alternative ways of expanding it between some bounds ...
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A common used construction in Harmonic analysis
This may be rather elementary. How to construct such function $\eta$ as shown in the picture?
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Traces mixing tensor products of Fourier coefficients on finite symmetric groups
Let $G$ be a finite (symmetric) group, and let $\widehat G$ denote the set of (equivalence classes of) irreducible unitary representations of $G$. Let $f:G\to \{0,1\}$ be a Boolean function on $G$ ...
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Bounding the largest Fourier coefficient of $f$ minus a class function on symmetric group $S_n$
Let $G=S_n$ be a symmetric group. A class function on $G$ is any $g:G\to\mathbb C$ that is constant on conjugacy classes, i.e. $g(xy)=g(yx)$ for any $x,y\in G$. Let $\mathcal C$ denotes the set of ...
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How to use tube union volume estimates to infer Kakeya conjecture
Wang Hong and Zahl's work "
Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions“ says
Why this is OK ? (We know Kakeya maximum function estimates can ...
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weak containment of representation
The following is the definition of weak containment:
Let $\pi$ and $\rho$ be two unitary representations of the group $G$. Then, we say $\pi$ is weakly contained in $\rho$ denoted as $\pi \prec \rho$ ...
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The dual group of a G-variety
In the paper https://arxiv.org/abs/1702.08264 proposition 5.4, a weak spherical datum for a $G$-variety $X$ is defined, I listened a talk yesterday https://mathematics.jhu.edu/event/number-theory-...
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About bilinear arguments of the Fourier restriction
A bilinear argument of the Fourier restriction (not bilinear Fourier restriction) can be described follows:
This excerpt is from the book of C. Demeter, Fourier restriction, decoupling, applications. ...
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Are these two definitions of an NTA (nontangentially accessible) domain equivalent? If so, is the constant unchanged?
Jerison and Kenig give the following two conditions (alongside an exterior corkscrew condition, which is not relevant to this discussion) in the definition of an $(M,r_0)$ nontangentially accessible ...
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Boundedness of operator in $L^p$ space for $p\neq 2$
This problem arises when I'm reading this paper about Internal modes for quadratic Klein-Gordon equation in $\mathbb R^3$, written by Tristan Léger and Fabio Pusateri: https://arxiv.org/abs/2112.13163....
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About the concept of non-tangential projection
Actually, i'm studying a regularity up to the boundary for the gradient of viscous solutions of an fully non-linear equation with the help of pertubation problem
\begin{equation}
(P_{\epsilon}) \quad ...
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Interpreting $1/f$ as a distribution when $f$ is only smooth
My first question is: does there exist a smooth function $f$ such that $f \neq 0$ on $\mathbb{R}^n \setminus \{0\}$, $f(0) = 0$, and $1/f$, viewed as a distribution on $\mathbb{R}^n \setminus \{0\}$, ...
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Young's Inequality for Hardy spaces
Let $d\ge 1$ be an integer, let $\mathcal H^1(\mathbb R^d)$ be the Hardy space on $\mathbb R^d$ and $L^{d,\infty}(\mathbb R^d)$ be the Lorentz space of index $(d,\infty)$ on $\mathbb R^d$.
Is it true ...
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Equivalent statement of the Wiener-Tauberian theorem?
I would like to know why we have the equivalence between the following statements of the Wiener-Tauberian theorem:
Version 1: Every proper closed ideal in $L^1(\mathbb R)$ is contained in a maximal ...
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Kakeya maximum function inequality and Kakeya inequality
Sorry this is an elementary question. What is the relation between Kakeya maximum function conjecture and Kakeya inequality:
$|\sum_{i=1}^n 1_{T_i} | \le C_{\epsilon} \delta^{-\epsilon} \|\cup_{i=1}^...
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Composition of trace class operator of $L^{2}(X)$ with Multiplication operators
Let $T$ is a positive trace class operator on $L^{2}(X)$ where $X$ is a second countable locally compact hausdorff space with some radon measure $\mu$ and for $\phi \in C_{b}(X)$, define the ...
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A geometric problem in Harmonic analysis
Given scale $\rho > \delta$, can we find a set of finitely overlapping balls with radius $\rho$ or $\rho$-tubes ($\rho$ neighborhood of a segment of line of length one) such that any $\delta$-ball ...
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Importance of Separability of a locally compact group
In the construction of the induced representation of a locally compact group by G.W. Mackey, the group is assumed to be separable. Later, the theory is developed to non-separable cases.
But I could ...
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A question in the proof of multilinear restriction theorem
To prove the multilinear-restriction theorem as follows:
The argument is through multilinear-kekeya inequality and other common tools, as follows
I have a question about local orthogonality. There ...
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Convolution of $L^p(G)$ and $L^2(G)$ into $L^2(G)$ implies compactness of an amenable group
Problem. Let $G$ be a locally compact amenable group. If for some $p>1$ we have $L^p(G) * L^2(G) \subset L^2(G)$ for the Haar measure, then the group $G$ must be compact.
This statement can be ...
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How soon does an energy-critical NLS with a finite $L^1$-deficit blow up?
For spatial dimension $d \geq 3$ consider the energy-critical, focusing nonlinear Schrödinger equation
$$i\partial_t u + \Delta u + c(t)|u|^{\frac{4}{d-2}}u = 0, \qquad (t,x) \in (0,\infty) \times \...
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What does a square-function estimate tell us in harmonic analysis?
I’ve noticed in harmonic analysis that many researchers are interested in the so-called square-function estimate (see arXiv:1906.05877 or arXiv:1909.10693). However, what I’m not clear about is what ...
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Name for integral representation of Riesz potential
Let $\phi:\mathbb{R}^d \rightarrow \mathbb{R}$ be a sufficiently nice (e.g. Schwartz) radial function . Then it is classical by scaling that the Riesz potential $|x|^{-s}$, for $s>0$, may be ...
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