Tagged Questions
1
vote
1answer
66 views
Why is logarithm in BMO
I read that $\log|x|$ is supposedly a typical example of a BMO function. How do I see that it is in BMO actually?
1
vote
3answers
36 views
Fourier transform of a compactly supported function
In which space does the Fourier transform of a smooth compactly supported function $\phi$ lie?
I would not say it lies in $\mathcal{S}$, heuristically as one can approximate the step function which is ...
1
vote
1answer
24 views
Measure of the boundary of a union of cubes.
Suppose that we are given a collection of dyadic, mutually disjoint, open cubes in $\mathbf{R}^n$ in which the the union of all the cubes has finite measure. Is it necessary that the boundary of the ...
7
votes
2answers
103 views
Mean Value Property of Harmonic Function on a Square
A friend of mine presented me the following problem a couple days ago:
Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of ...
2
votes
1answer
51 views
Why does this inequality for all characteristic functions imply it for simple functions?
This question is probably obvious, but I'm not seeing how to obtain it.
A simple function is said to be finitely simple if its support is of finite measure. Let $(X_1,\mu_1)$, $(X_2,\mu_2)$, and ...
1
vote
0answers
73 views
Another aspect of Heisenberg uncertainty principle
In fourier transformation theory, we have the Heisenberg uncertainty principle, i.e.
Suppose $\phi$ is a function in [Schwarz space][1] which satisfies the normalizing condition ...
11
votes
2answers
240 views
A series involves harmonic number
How do we get a closed form for
$$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}$$
14
votes
4answers
360 views
About a harmonic series problem : How to prove that $\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$
How to prove that
$$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$$
$H_n$ is the n th harmonic number
1
vote
0answers
99 views
Proof of Euler's general formula for a sum involving harmonic numbers
I have seen this formula, but how to prove this?
$$2\sum\limits_{k=1}^\infty \frac{H_k}{\left( k+1 \right)^m} =m\zeta \left( m+1 \right)-\sum\limits_{k=1}^{m-2}{\zeta \left( m-k \right)\zeta \left( ...
1
vote
1answer
32 views
A Quadratic Maximum?
What does the following mean?
Context: Laplace integrals
Consider the integral $$\int_a^b f(t) \exp(x\phi(t))\,\,\,dt$$ where $\phi(t) $ achieves its maximum at some $t=c\in (a,b)$. Assume that ...
4
votes
1answer
179 views
Hardy-Littlewood-Sobolev inequality for $p=1$
Let $\mu $ be a positive Borel measure on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{d}$ such that $\mu \left( B\left( a,r\right) \right) \leq Cr^{n}$ for some
$n\in (0,d]$ ...
6
votes
1answer
183 views
$1/|x|^n$ is not integrable
Let $\mu $ be a positive Borel measure on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{d}$ such that $\mu \left( B\left( a,r\right) \right) \leq Cr^{n}$ for some
$n\in (0,d]$ ...
4
votes
2answers
102 views
A question from Stein's book, Singular Integral.
A question from Stein's book, Singular Integral. Let $\left\{ f_{m}\right\} $
be a sequence of integrable function such that $$\int_{%
\mathbb{R}^{d}}\left\vert f_{m}\left( y\right) \right\vert dy=1$$ ...
1
vote
1answer
82 views
Is this function a subharmonic function?
Does anyone know, is $h\left(z,w\right):=\frac{\left|zw\right|}{\left|z\right|+\left|w\right|}$
for $z$ and $w$ in the unit disk $\mathbb{D}$ of the complex plane a (pluri)subharmonic function? ...
2
votes
1answer
51 views
Alternate definitions of $C^{1,\alpha}(S^1)$ and $C^{1,\alpha}(\bar{D})$ maps
My question is about the precise definitition regarding the following:
Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
1
vote
2answers
67 views
Estimate on a simple-looking integral arising from harmonic analysis/harmonic extensions
Let $z\in \mathbb{D}, t\in S^1, \beta\in \mathbb{R}$. I was dealing with the following integral arising from some other calculation regarding harmonic extension on $\mathbb{D}$:
...
0
votes
1answer
40 views
how to prove the convolution formular?
let $\overset{\backsim} {g}(x)=g(-x)$;
suppose $u,\phi,\psi$ always make the integral significant,$E_n$ is the n-dimensional euclidean space. Then how to prove
...
0
votes
0answers
70 views
oscillatory integrals in one dimension
Let$$K_{a,b}(x)=\int_{0}^{\infty}{\psi({\xi})\xi^{-b}e^{ix\xi\pm \xi^{a}}d\xi}\quad a,b>0$$
where $\psi\in C^{\infty}$, equals to $0$ when $\xi<\frac{1}{2}$, and equals to $1$ when $\xi>1$.
...
3
votes
2answers
147 views
Shifting a function is continuous
I'm slightly puzzled by the following: if $g(t)$ is a function in $L^q(X)$ then we can show that $g(t-x)$ is continuous function of $t$, i.e. for $\varepsilon > 0$ we can find $\delta$ such that ...
3
votes
0answers
124 views
Series of Maximal Operator
Let $p\in(1,\infty)$. Assume that we have a sequence of functions $\{f_i:i\in\mathbb{N}\}\subset L^p(\mathbb{R}^n)$ such that
$$
\left(\sum\limits_{i=1}^\infty|f_i|^2\right)^{\frac{1}{2}}\in ...
4
votes
1answer
108 views
Fejer kernel applied to a measure
Let $\mu$ be a positive finite measure on $\mathbb R$. Is it true that
$$\int_{\mathbb R} T \text{sinc}^2(Tx) d\mu(x) \sim\frac{\mu([-1/T,1/T])}{1/T}, \text{ as } T\to\infty?$$
Here ...
2
votes
1answer
201 views
Properties of subharmonic functions
A function $f$ is called subharmonic if $f:U\rightarrow\mathbb R$ (with $U\subset\mathbb R^n$) is upper semi-continuous and $$\forall\space \mathbb B_r(x)\subset ...
2
votes
1answer
322 views
properties of a real analytic function
If there are a radius $r>0$ and constants $M,C\in\mathbb R$ for all $y\in U$ with
$$|\partial^if(x)|\leq M\cdot i!\cdot C^{|i|}\space\space\space\space \forall x\in\mathbb B_r(y),i\in\mathbb ...
1
vote
1answer
126 views
Question about support of distributions
I'm reading "Pseudo-differential Operators and the Nash-Moser Theorem" and at the top of on page 8 they write: "Finally, we note that if $u \in C^0(\Omega)$, then the support of $u$ defined above ...
1
vote
1answer
137 views
On covering lemma and Calderón–Zygmund decomposition
I am working on something which needs to understand covering lemmas and Calderón–Zygmund decomposition. These type of lemmas are as in the following link
...
1
vote
1answer
124 views
On a duality Fefferman-Stein's inequality
Let $M$ be the Hardy-Littlewood maximal operator. In the book "Weighted norm inequalities and Related Topics" by Rubio de Francia and J. Cuerva, page 150, theorem 2.1.2 states as the following:
*For ...
1
vote
1answer
76 views
Restriction and completion of Haar measure on $\mathbf{R} \times \mathbf{R_d}$ to Borel $\sigma$-algebra
Let's consider the measure space $(G, \mathfrak{M}, \mu)$, where $\mu$ is the Haar measure on topological group $G:=\mathbf{R} \times \mathbf{R_d}$, ($\mathbf{R}$ is the group of reals with the ...
1
vote
2answers
366 views
Bounded linear operators that commute with translation
I'm trying to read Elias Stein's "Singular Integrals" book, and in the beginning of the second chapter, he states two results classifying bounded linear operators that commute (on $L^1$ and $L^2$ ...
6
votes
1answer
498 views
Theorem of Steinhaus
The Steinhaus theorem says that if a set $A \subset \mathbb R^n$ is of positive inner Lebesgue measure then $\operatorname{int}{(A+A)} \neq \emptyset$. Is it true that also ...
4
votes
1answer
2k views
Criteria for swapping integration and summation order
I have a function (a potential from an electrostatic potential via a Fourier series) in the form of
$$V(x, y, z)=\sum_n\sum_m \ a(x, n, m) b(y, n) c(z, m) \int\int f(u, v) d(u,n) e(v,m)
du\, dv$$
...
2
votes
2answers
105 views
regularity of $d\mu=u dx$
Let $G$ be a abelian, locally compact group with a Haar measure $dx$ on $G$. We know that every Haar measure is inner regular, ie... for any open subset $U$ of $G$, then ...
1
vote
1answer
173 views
Hardy-Littlewood maximal function of a Lipschitz function
In a book, it is said that Hardy-Littlewood maximal function of a Lipschitz function is also Lipschitz. How do we prove this?
+) For Hardy-Littlewood maximal function, see: ...
1
vote
2answers
77 views
Sequence of smooth functions whose image under a maximal operator diverges in $L_p$ norm
For functions $f: \mathbb{R} \rightarrow \mathbb{C}$, define
$$
M f(x) = \sup_{t >0} \frac{1}{2}| f(x+t) + f(x-t) |.
$$
Given $p \geq 1$, I want to construct a sequence of smooth functions ...
4
votes
1answer
82 views
Bound on the relative measure of $\delta$-neighbourhoods of compact sets using the Hardy-Littlewood maximal theorem
Notation: $|A|$ is the Lebesgue measure of $A \subset \mathbb{R}^d$, and $A_\delta = \{ x : \text{dist}(x,A) \leq \delta \} $ is the $\delta$-neighborhood of $A$.
I want to show that there is a ...
