# Tagged Questions

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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### 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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### 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 ...
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### $k[x_1, \dots, x_n]$ 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 ...
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### $G$-harmonic polynomials, dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?

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 $\text{Harm}(\mathbb{R}^n, S_n)$?
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### 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 ...
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### 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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### 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) ...
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### 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 $h:\mathbb{D}\rightarrow \mathbb{R}^3$ ...
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### 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. ...
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### 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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### 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? ...
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### 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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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 ${\rm Re} \ (\Delta f) ... 0answers 174 views ### Wavelet-like Schauder basis for standard spaces of test functions? The Schwartz space of test functions$\mathcal{S}(\mathbb{R})$is isomorphic to$\mathfrak{s}$the space of sequences of real numbers with faster than power-like decay. Likewise, the space ... 0answers 44 views ### Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane We consider the hyperbolic plane and the harmonic function there. Pick any point$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$. What is ... 1answer 143 views ### Hardy space, Lebesgue space for$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)$such that ... 0answers 32 views ### The Best Korn's constant for bounded deformation I am studying the following version of Korn's inequality. For$u\in BD(\Omega)$,$BD$denotes the bounded deformation space, we have, there exists a$r(u)\in \operatorname{ker}\mathcal E$such that ... 1answer 89 views ### One question about the tensor product of$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$. I am looking ... 1answer 68 views ### Rajchman measures via strong mixing systems A Rajchman measure on the unit circle$\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}$are Fourier ... 2answers 113 views ### The convolution between weighted$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$, $$\frac{1}{|B(x,r)|}\int_{B(x,r)}\omega(y)dy\leq C\omega(x)$$ ... 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$X_1$. ... 0answers 84 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 ... 0answers 199 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 ... 1answer 223 views ### What is the importance of convergence of variation of Fourier reconstruction to that of variation of the function? Let$f$be a periodic function of bounded variation which jumps at a point$x_0\in\mathbb{R}$. Let$S_{N}[f]$denote the partial Fourier sum of$f$and let$C_{N}[f]$denote the Cesaro partial sum. It ... 0answers 76 views ### Ring of SO(n)-invariant differential operators on M_n,m I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/). There comes a point in the paper (Lemma 2.8) ... 1answer 220 views ### The$L^2\times L^2\to L^2$norm of the bilinear multiplier operator Consider a general bilinear multiplier operator: $$T(f,g)(n)=\int_{\Pi}\int_{\Pi}\hat{f}(\xi)\hat{g}(\eta)e^{2\pi i(\xi+\eta)n}m(\xi,\eta)d\xi d\eta,$$ where$\Pi$is the torus,$n\in\mathbb{Z}$, ... 0answers 83 views ### What does the Plancherel theorem say about positive-definite distributions? I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem. The ... 0answers 77 views ### Generalization of Pitt's theorem Pitt's theorem (Pitt 1937), states that the one-dimensional Fourier tranform is well defined and continuous between the weighted spaces$L^p(\mathbb{R},|x|^{bp}dx)$and$L^q(\mathbb{R},|x|^{\beta ...
Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$. That is $\mathcal{K}$ is RKHS, ...