# Tagged Questions

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### Why are function spaces typically defined on open sets?

I never really bothered to ask this question and now it seems silly...but why do we always seem to define function spaces $X(U)$ (e.g. $L^2(U), BV(U)$ for open sets $U\subset\Bbb{R}^n$? What breaks if ...
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### Real version of the Jensen's formula.

Prove the Jensen's formula $$\int_{T}f(z+re^{2\pi i\theta})d\theta-f(z)=\iint_{D(z,r)}\log{\frac{r}{|w-z|}}\Delta f(w)dm(w)$$ where $w$ is in $D(z,r)$ and $f$ is a two-dimensional $C^2$ ...
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### Hardy Space Cancellation Condition

I have been reading the chapter on Hardy spaces in Stein's Harmonic Analysis book, and I am having a lot of trouble figuring something out. The setting here is $\mathbb{R}^n.$ Let $f \in L^q$ be ...
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### On convergence rate of kernel approximation

Let $\{K_\epsilon\}$ be a sequence of mollifiers, (or often take heat kernels). We have known from classical analysis that if $f$ is uniformly continuous then the error ...
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### Integrability of function and its Fourier transform implies differentiabilty

Is the the following true: "Assume $f\in L^1[0,1]$ and $\hat{f}\in L^1(\mathbb{R})$, then $f$ is differentiable a.e"
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### A function $f:\mathbb{R} \to \mathbb{R}$ with infinite norm but finite weak seminorm

Let $0<q\leq p<\infty$. For $f:\mathbb{R}\to \mathbb{R}$, we define the norm \|f\|_{p,q}=\sup_{a\in \mathbb{R},r>0} r^{\frac{1}{p}} \left(\frac{1}{r} \int_{a-r}^{a+r} ...
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### Zero convolution of a function with a measure

Suppose $0\not=f\in L^1_{loc}(\mathbb{R}^2)$ and $\mu$ is a positive Borel measure with compact support. Given $f\ast\mu=0,$ what can be said about $\mu$?
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### The property of positive fourier series. [duplicate]

This is the problem in the book 'Classical and multilinear harmonic analysis, volume 1' Let $f(x)=\sum_{n=0}^{N}[a_{n}\cos{2\pi nx}+b_{n}\sin{2\pi nx}]$ be a nonnegative function defiend on $[0,1]$. ...
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### Harmonic inside with zero average

Assume $\Omega\in\mathbb{C}$ is a domain with nice enough boundary,say smooth boundary. What can be said about $f\in C(\bar\Omega)$, harmonic in $\Omega$ and $\int_{\partial\Omega}f(z)|dz|=0$, where ...
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### A Curious sum of sines and cosines with angles in arithmetic progression

I am stuck with the following problem: \begin{align} \max_{\theta\in\mathbb{R}}\sum_{i=1}^{N}(a_i\sin(i\theta)+b_i\cos(i\theta)), \end{align} where for $i(1\leq i\leq N$)$, a_i$ and $b_i$ are real ...
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### Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, ...
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### Proof of Riemann-Lebesgue lemma

I read a book, and this mention to the following lemma of Rieman-Lebesgue type. Lemma. Let $-\infty<a<b<\infty$ and $f(x,y):[a,b]^2\to\mathbb R$ be an integrable and nonnegative function. ...
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### Convention in Riesz representation theorem vs. tempered distribution theory

We are working over the complex field here. Sometimes analysis textbooks say that every continuous linear functional on $L^p$ is integration against some $f \in L^{p'}$ for $p\in (1, \infty)$, rather ...
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### Sobolev spaces vs. Hardy Spaces

I have seen sobolev spaces (the ones with the p norms of the derivatives of a multivariable function) and Hardy spaces (the objects investigated in harmonic analysis when one asks about tangential and ...
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### How to prove $\widehat{(fg)}(n) = \hat{f} (n) \ast \hat{g} (n); (n \in \mathbb Z)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
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### Are the two definitions of the complementary Young function equivalent?

The description of the problem: For a Young function I would refer the reader to the book "Function spaces" by Pick Luboš, Kufner Alois, John Oldrich and Fucík Svatopluk, and published by de Gruyter. ...
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### Determine polynomials with $n$-variables

Here is a funny problem arise from harmonic analysis: Let $E$ be a measurable subset of $\mathbb R^n$ with $m(E)>0$, where $m$ is the usual Lebesgue measure on $\mathbb R^n$. In practice, $E$ ...
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### Abstract Fourier Analysis

I am trying to prove that given a locally compact abelian (Hausdorf) topological group, the characters on it parameterize the multiplicative linear functionals on the banach * algebra $L^1(G, d\mu)$ ...
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### A bound for the product of two functions in BMO

The question: Let's consider $f\in L^\infty(\mathbb{T})$ and $g\in BMO(\mathbb{T})$. I'm trying to figure out if the following inequality is true $$\|fg\|_{BMO}\leq C\|f\|_{L^\infty}\|g\|_{BMO}.$$ ...
### Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?
Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
### Show that $\int_{S^{n-1}}(\Delta _{S}f) gd\sigma=\int_{S^{n-1}}f( \Delta _{S}g)d\sigma$
Elias M. Stein said that by an application of Green's theorem the following equality holds $\int_{S^{n-1}}(\Delta _{S}f) gd\sigma=\int_{S^{n-1}}f( \Delta _{S}g)d\sigma$ where $\Delta _{S}$ is a ...