# 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.

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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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### 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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### 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 ...
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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 ...
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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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### 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 ...
1 vote
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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 ...
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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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### 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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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 $\lbrace c_k\rbrace _{k \... 3 votes 0 answers 48 views ### Eigenfunction expansion theorem for general manifold for smooth functions The question starts with the well known facts that: if$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 ... 15 votes 3 answers 1k views ### 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 $$e_n(x)=\exp(2\pi i n x), \quad \text{where }\... 0 votes 0 answers 99 views ### Using projections to determine equidistribution Suppose I have a collection of points on \mathbb{S}^{n-1} \subset \mathbb{R}^n. I want to know that they are equidistributed (if you want to be more precise, you have a sequence of such collections, ... 0 votes 1 answer 169 views ### Uncorrelation of exponential sums generated by irrational rotations over disjoint sets of integers Assume that \mathbb{N}=\{0,1,2,\ldots\} is partitioned into k\ge 2 disjoint sets J(1),\ldots,J(k) such that for every 1\le p \le k the set J(p) has an asymptotic density$$ d(J(p))=\lim_{n\... 0 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 ... 3 votes 0 answers 247 views ### 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: $$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\... 1 vote 1 answer 99 views ### How to prove \|u\|_{L^{\infty}}\leq C\|\partial_1\square u\|_{L^1} for any u\in C_0^{\infty}(\mathbb{R}^{1+2}) ? It comes from estimates for wave equations. For any u=u(t,x)\in C_{0}^{\infty}(\mathbb{R}^{1+2}) , which is a smooth compactly supported function, prove that$$ \|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+...
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 ...