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.
1,589 questions
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Fourier transform of compactly supported functions
Let $F:\mathbb C^n\longrightarrow \mathbb C$, be an entire function of exponential type, that is $F$ is holomorphic on $\mathbb C^n$ and there exist constants $C, A$ such that
$$
\vert F(z)\vert\le C ...
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What are the current best known results for linear dispersive Strichartz estimates on the circle $\mathbb{T}$?
I am currently trying to understand Strichartz estimates for linear disperive equations on the circle:
$$\begin{cases}
i \frac{\partial u}{\partial t}= \Phi(\sqrt{-\partial^2_x})u\:,\: &\text{ in ...
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When are Hecke algebras finitely generated?
Let $G$ be a locally compact totally disconnected group.
Fix a compact open subgroup $K$ and consider the Hecke algebra $H_K$ of all $K$-bi-invariant functions of compact support.
Suppose that $G$ is ...
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Is convolution in $\ell_1(\mathbb Z)$ an open map?
This is a follow-up to this question but in a very specific case that, I must say, drives me crazy so I just want to know the answer.
Is the convolution map $m\colon \ell_1(\mathbb Z)\times\ell_1(\...
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Density of Pisot numbers
I am looking up at the book of Yves Myers "Algebraic Numbers and Harmonic Analysis" where, in a serious treatment of Pisot numbers, they used this particular lemma without proof.
The ...
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Projection onto translation invariant subspaces
I'm currently looking at a subspace of $A \subset \ell^p(\mathbb{Z}^n)$ which is generated by some finitely supported elements and their translations. My question is an old one (but the answer is ...
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Lower bounds for $\|f*g\|_1$ with mean-zero Lipschitz functions on $[0,1]$
Let $f,g \in L^{1}([0,1])$ satisfy
$$
\|f\|_{1}=\|g\|_{1}=1, \qquad \int_{0}^{1} f(x)\,dx=\int_{0}^{1} g(x)\,dx=0,
$$
and assume
$$
f \in \mathrm{Lip}_{L_f}, \qquad g \in \mathrm{Lip}_{L_g}.
$$
...
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Besov spaces under multiplication
Let $ \sigma \geq \frac{n}{2} $. And consider the inhomogeneous Besov space $B^{\sigma}_{2,1}$ with the norm
$$ \Vert f \Vert_{B^{\sigma}_{2,1}}= \sum_{k=0}^{\infty} 2^{\sigma k} \Vert \Delta_{k} f \...
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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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Cotlar type inequality for Riesz transforms
Let $R^s_\mu(x)= \int \frac{y-x}{|y-x|^{s+1}}d\mu(y), x,y \in \mathbb{R}^d, 0<s<d$ be the Riesz transform (of index $s$).
I would like to understand the proof of the following inequality. There ...
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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 ...
1
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1
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252
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Rank of tensor product of irreducible representations over finite symmetric group
Let $G$ be a finite symmetric group, and let $\widehat G$ denote the set of equivalence classes of irreducible unitary representations of $G$.
For any non-trivial $\rho\in \widehat G$, we know that $\...
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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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341
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Sum in tensor product of irreducible representations on $S_n$
Let $G$ be a finite symmetric group, and let $\widehat G$ denote the set of (equivalence classes of) irreducible unitary representations of $G$.
For any non-trivial $\rho\in \widehat G$, we know that $...
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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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On the large sieve inequality involving $GL(2)$ harmonics
I have a question on the large sieve inequality involving $GL(2)$ harmonics. Recall that one has the analog for $GL(1)$ harmonics that, for any complex numbers $\alpha_m,\beta_n$, one has
$$\sum_{q\le ...
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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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140
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Weak L2 norm in proof of Carleson's theorem
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. Under Lemma 2.18 on page 10 it says: "...
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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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Quantitative differentiation via Littlewood-Paley theory
In an article by Cheeger, he presents an elementary method, which he refers to as quantitative differentiation, for obtaining a quantitative version of Rademacher's theorem. While these arguments use ...
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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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Embedding of the Wiener algebra into $C^0_{(0)}$
The Wiener algebra $\mathcal W_d$ is defined as the Banach space $\mathcal F(L^1(\mathbb R^d))$ equipped with the norm
$
\Vert u\Vert_{\mathcal W_d}=\Vert \mathcal Fu\Vert_{L^1(\mathbb R^d)},
$
where $...
- 16.7k
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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 $\pi \prec \rho$ ...
7
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Complex interpolation with a reflexive Banach space yields reflexive Banach spaces
Let $(A,B)$ be a compatible pair of Banach spaces. A result of Calderón states that if at least one of the Banach spaces $A$ or $B$ is reflexive and if $0<s<1$, then the interpolation space $[A,...
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140
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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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Asymptotics of the projection on the kernel of the coboundary operator on infinite cubical lattice
Consider the infinite cubical lattice $\mathbb{Z}^d \subset \mathbb{R}^d$ as a polyhedral cell complex and write $C^k_{(2)}(\mathbb{Z}^d)$ for the Hilbert space of real oriented $\ell_2$ $k$-cochains ...
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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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Boundedness of numerical range of an operator
Let $T$ be a self-adjoint bounded operator on a separable Hilbert space $H$ and $ T=S-K$, where $S$ and $K$ are bounded positive operators with $||S||\leq ||T||$ and $||K||\leq ||T||$.
If $$sup \{ \...
3
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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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Approximation of Lipschitz Domains
Let $\Omega\subseteq \mathbb{R}^n$ be an open set. We say that $\Omega$ is a Lipschitz domain if for each boundary point $p\in \partial\Omega$ there exists an open set $U_p\subseteq\mathbb{R}^n$ ...
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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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Variant of Cordes Inequality
The classical Cordes inequality states the following: suppose that $\|\cdot\|$ is the usual matrix norm, $0 < \alpha \leq 1$, and $A, B$ are $n \times n$ positive semidefinite Hermitian matrices. ...
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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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6
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1
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Restriction of the Hodge decomposition to Kähler submanifolds
Let $(X, \omega)$ be a compact Kähler manifold with kähler form $\omega$, and let $Y\subset X$ be a Kähler submanifolds with the induced Kähler form. The kähler form $\omega$ induces an isomorphism, ...
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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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What is a spectral measure of function on a Pontryagin Dual of LCA group?
At the moment I am reading the book "Mathematics of Ramsey Theory" DOI: https://doi.org/10.1007/978-3-642-72905-8 and I am having difficulties with the understanding of proof of the ...
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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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Definition of the spectrum of a representation
Let $G$ be a locally compact abelian group, $X$ be a Banach space.
Let $(U,X)$ be a representation of $G$ on $X$, $U$ maps $G$ in to the group of investible isometries in $B(X)$.
In the paper, "...
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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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1
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765
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Bourgain and Demeter's Proof of $\ell^2$ decoupling Lemma 3.3
I posted in stackexchange, but then was told that overflow may be a more appropriate place to ask. I'm currently reading Bourgain and Demeter's "The Proof of the $\ell^2$ Decoupling Conjecture”, ...
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125
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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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Discrete Fourier transform converges to torus and real Fourier transforms in what sense?
On the group $\mathbb Z/N\mathbb Z$, we have characters $\chi_j(k) = e^{2\pi i j k/N} \in \text{Hom}_{\mathbf{Grp}}(\mathbb Z/N \mathbb Z, \mathbb T)$, and so equipping with counting measure $\#$, we ...
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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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Sharp $L^{\infty}$ Bernstein inequality for bandlimited functions
I'm interested in finding a proof in the literature for the following result:
Let $f(x)$ let be a smooth function $\mathbb{R} \rightarrow \mathbb{R}$ such that $\hat{f}$ exists and is supported on $[-...
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1
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247
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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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About the doubling condition: Does $U(t) \leq 2^p U(t/2)$ imply $M_k \leq C_p m_{k-1}$? [closed]
📌 Let $U : (1, \infty) \to [0, \infty)$ be a continuous function satisfying the following doubling-type condition:
$$
U(t) \leq 2^p U(t/2) \quad \text{for all } t > 1,
$$
where $p \geq 1$ is fixed....
- 1,185
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0
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69
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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 ...
0
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0
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104
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Necessary and sufficient condition for boundness of positive dyadic operator
My question is, is the following fact well known and, if yes, what is a reference to this.
Let we have positive dyadic operator (not necessary sparse, which is of course important case)
$Tf = \sum_{I \...
0
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1
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230
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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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1
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125
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Positivity of $L^t$ from $L^{-t}$ Integrability on $\mathbb{PR}^d$
Let $\mu$ be a probability measure on $\mathbb{PR}^d$. Let $d$ be a metric on $\mathbb{PR}^d$. Assume that $\mu$ is not supported on any projective space. For $t>0$ and $u \in \mathbb{PR}^d$, ...
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