All Questions
17 questions
3
votes
0
answers
90
views
About BMO space on smooth open bounded domain
Let $\Omega$ be any open domain in $\Bbb R^d$.
Define the $\text{BMO}(\Omega)$ space as
$$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\},
$$
...
- 1,101
0
votes
0
answers
106
views
How to prove that $f(x) := |x|^{\frac{\lambda - n}{p}}(1 - \psi(x))$ satisfies a specific property related to its limit at the origin
Disclaimer. I have asked this question a month ago on MSE (click here to access the original post) and even bountied it. I got an answer on MSE, but unfortunately I don't feel like it has enough ...
- 51
0
votes
0
answers
57
views
Projection measure and an integral formula for Lipschitz functions
Let $n\geq m\geq 0$ be integers and put $k=n-m$. Let $A\subset\mathbb{R}^n$ be Borel measurable, we define the projection measure of $A$ as
$$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
- 75
0
votes
0
answers
126
views
A question about associated operator on continuous functions space equiped with L2 norm
For M a connected compact manifold, $T$ is in $C^{1+\nu}(M,M)$ with $\nu\in(0,1)$, i.e., $DT$ is some Hölder continuous function with Hölder exponent $\nu$. Denote by $m$ the Lebesgue measure on $M$ ...
- 39
2
votes
2
answers
307
views
Preimage of null sets under a monotone increasing function
Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
- 113
0
votes
1
answer
79
views
Convergence in sequential Lebesgue spaces
Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{...
- 1,101
0
votes
1
answer
88
views
An equation in the convolution measure algebra on reals
Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on reals.
Let $\mu$ be a Radon measure in $M(\mathbb{R})$ and $\delta_0$ be the point mass measure concentrated on ...
- 4,058
3
votes
1
answer
322
views
Special version of Tonelli’s theorem
I am trying to prove this theorem. I have not found anything similar to it on the internet.
Special version of Tonelli’s theorem
Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
- 159
1
vote
1
answer
412
views
Support of a measure
Let $T:X\to X$ be a continuous function on a compact manifold $X$ and let $\text{Leb}$ be the Lebesgue measure normalized so that $\text{Leb}(X)= 1$. We denote by $\mathcal{M}(X)$ the space of all ...
- 1,043
2
votes
1
answer
395
views
Existence of integral kernel
I know the following statement ture.
Let $T \in B(L^1(\mathbb{R}^d), L^\infty(\mathbb{R}^d))$, where $B(X, Y)$ denotes all bounded linear operoters from $X$ to $Y$.
Then, $T$ has the integral kernel $...
- 193
1
vote
0
answers
209
views
The translation is continuous in $L^1(\mathbb{R}^n,d\mu)$, $d\mu=\frac{1}{1+|y|^{n+a}}dy$,$ a>0$
For any function $f\colon\mathbb{R}^n\to\mathbb{R}$, set: $\tau_hf(x):=f(x+h)$, $x,h\in\mathbb{R}^n$. Consider the following finite measure on $\mathbb{R}^n$:
$$\mu(A):=\int_A\frac{1}{1+|y|^{n+a}}\,dy$...
- 339
-2
votes
1
answer
1k
views
Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? [closed]
Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...
- 395
2
votes
0
answers
453
views
Is that correct $\mathbb R^2\cong\mathbb R$ as measurable spaces? [closed]
Is that correct $R^2\cong R$ as measurable spaces?
If we consider $R$ and $R^2$ with Borel $\sigma$-algebras, is there measurable map from $R$ to $R^2$ with measurable inverse?
- 29
0
votes
0
answers
299
views
When convolution with exponential kernel is bounded
Let $g(t)=e^{-\omega t}$, $\omega>0$. What is, in terms of well-known function spaces, the space $X$, $L_{loc}^2(0,\infty)\subset X$, of all functions $f:\mathbb{R}^+\to \mathbb{R}^+$, satisfying
$...
- 395
1
vote
2
answers
931
views
A question on the Lebesgue differentiation theorem
In the paper [Jessen, B., Marcinkiewicz, J., and Zygmund, A. Note on the differentiability of multiple integrals. Fundamenta Mathematicae 25.1 (1935): 217-234] it is considered the limit
$$
\lim_{\...
- 2,715
1
vote
0
answers
125
views
Convergence of solutions of the volterra integral equation with convergent kernels
Consider the following Volterra integral equation
$$
g(t) = \int_0^t K_n(t,s)w_n(s) ds
$$
where g(t) and K_n(t,s) are continuous and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ ...
- 111
1
vote
0
answers
331
views
Relationship between weak Lp and strong Lq topologies for q<p
Specificaly:
Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence?
Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If not,...
- 203