All Questions
5,909 questions
1
vote
0
answers
96
views
System of Poisson equations
Let $(M,g)$ be a closed (compact and without boundary) and oriented Riemannian manifold and let us consider the Poisson equation for a smooth function $\varphi$:
$\Delta \phi = f$,
where $f$ is a ...
- 2,818
4
votes
0
answers
212
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
6
votes
1
answer
803
views
Approximation of a Sobolev function that has vanishing trace on the reduced boundary of a Caccioppoli (i.e. finite perimeter) set
For $\Omega\subset\mathbb{R}^N$ open and bounded, let $W^{1,p}(\Omega)$ denote the usual Sobolev space of $L^p(\Omega)$ functions with weak partial derivatives in $L^p(\Omega)$ and $W_0^{1,p}(\Omega)$ ...
- 512
2
votes
2
answers
947
views
Defining definite integral using indefinite integral
Sometimes definite integral is defined using antiderivatives:
$$\int_{a}^b{f(t)dt}=F(b)-F(a)$$
where $F$ is any continuous function such that:
$$(\forall t\in[a,b]\setminus C)(F'(t)\text{ exists and ...
- 193
2
votes
1
answer
103
views
Matuszewska Index and finite variance
Suppose there is a random variable, $X$, with finite variance, and c.d.f. $F(x)$. Does this imply that the upper Matuszewska index of $\bar F(x)$ exists and is strictly smaller than $-2$?
The upper ...
- 128k
2
votes
1
answer
144
views
Convergence of sequence of images of Schur multipliers
Let $\eta$ be a continuous bounded function on $(0, \infty)^{2}$ so that $\eta(0,0)=1$. Let $A$ be a bounded operator on $\ell^{2}(\mathbb{Z}_{\geq 0})=\ell^{2}$ (by bounded operator I will always ...
- 31
3
votes
1
answer
632
views
Is the sequence $(\log(n!)\mod1)_{n\in\mathbb N}$ dense in the interval $[0,1]$?
This question was raised in the comment by Todd Trimble at how to proof there is a natural number n, the first four digits of n! Is 2018?. I thought the question may be posted separately, as even ...
- 7,193
17
votes
3
answers
2k
views
The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$
Consider a collection of unit vectors $v_1, \ldots, v_n$ in $\mathbb{R}^d$ (we think of $n$ being much larger than $d$). I would like to minimize the sum:
$$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$
...
11
votes
2
answers
2k
views
Operator that commutes with projections
We investigate the Hilbert space $\ell^2(\mathbb{N}_0)$ with standard orthonormal basis vectors $e_n:=(0,...,0,1,0,...).$
Consider the family of self-adjoint rank $1$ projections $P_n\bullet:= \...
- 128k
-1
votes
1
answer
132
views
About a property in a reflexive Banach space
Let $E$ be a reflexive Banach space. Let $\{x_n\}_n$ be a bounded sequence of linearly independent elements of $E$. Does there exist a sequence $\{\phi_n\}_n$ of elements of $E^*$ (the dual of $E$) ...
- 2,118
2
votes
1
answer
964
views
Is the Delta distribution a continuous functional on $H^1(\mathbb{R})$? [closed]
While it is easy to see that $H^1(\mathbb{R})$ are Hölder $1/2$-continuous, I started wondering whether this implies that $\delta_x(\varphi)=\varphi(x)$ is continuous as a functional
$$\delta_x:H^1(\...
- 28k
1
vote
1
answer
394
views
Is fractional Laplacian invariant under rotation?
If $\Delta u=0$, then $\Delta u(Ox)=0$, where $O$ is an orthogonal matrix. From here, do we know whether fractional Laplacian is invariant under rotation? We use the usual definition of fractional ...
- 28k
0
votes
0
answers
58
views
$N-$Green function in $\mathbb R^N$
Let $N \geq 3$. Does there exist solution of the following equation
$$-\Delta_N G + G^{N-1} = \delta_0,$$
where $-\Delta_N = - \text{ div}(|\nabla \cdot |^{N-2} \nabla \cdot )$ denotes $N-$Laplace ...
- 379
1
vote
1
answer
351
views
A Liouville theorem for a uniformly elliptic equation in divergence form
I would like to know if there exists a Liouville theorem for solutions $u : \mathbb{R}^n \to \mathbb{R}$ of uniformly elliptic equations of the kind
$$
D_i \left( a_{ij} D_j u \right) + b_i D_i u = 0.
...
- 17k
1
vote
3
answers
363
views
Estimating L1 functions over the ball with radius 2r
Let $ f $ be in $ L^1(\Omega) $ where $
\Omega $ is an open subset of $ \mathbb{R}^n $. Also, assume that $ B(x_i,r_i) $ is a collection of disjoint open balls in $ \Omega $ such that $ B(x_i,2r_i) \...
- 28k
2
votes
1
answer
149
views
The infinite set of $SBV$ function?
Let $u\in SBV(\Omega)$ where by $SBV$ we denote the special bounded variation function and $\Omega\subset \mathbb R^N$ is open bounded.
Let's identify $u$ by its approximation representative (see ...
- 512
1
vote
1
answer
99
views
Equivalent conditions for a real function to have antiderivates
Lebesgue theorem says that a bounded function $f$ is Riemann Integerable if and only if $f$ continuous almost everywhere.
Unfortunately, we know a function has antiderivative has no relation to ...
- 66.3k
12
votes
5
answers
2k
views
analysis over non-Archimedean ordered fields
Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...
- 10.1k
5
votes
0
answers
349
views
Tietze extension theorem for lower semi continuous functions
On the Tietze extension theorem, if instead of a continuous function "f" we use a lower semi continuous function on a closed subspace of a metric space, is the theorem correct? I mean, can we extend ...
- 29.8k
7
votes
2
answers
999
views
If a measure $\mu$ and Lebesgue measure $\lambda$ are singular, is the derivative of $\mu$ with respect to $\lambda$ $\infty$, $\mu$-a.e.?
If a positive Radon measure $\mu$ and the Lebesgue measure $\lambda$ are singular, can we show that the derivative of $\mu$ with respect to $\lambda$ is $\infty$, $\mu$-a.e.? Namely, can one show that
...
- 734
4
votes
1
answer
259
views
The integrable condition for distance function
Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$. We denote
$d\left(x\right)$ the distance from $x$ to the boundary of $\Omega$,
that is
$$
d\left(x\right):=\inf\left\{ \left\Vert x-y\right\...
- 17.2k
7
votes
0
answers
106
views
The first homotopic Baire class
Let $X$ and $Y$ be topological spaces. A map $f:X\to Y$ belongs to the first Baire class (to the first homotopic Baire class), if there exists a continuous map $H:X\times \omega\to Y$ (a continuous ...
- 367
3
votes
2
answers
2k
views
Is there an example where the error of Gauss-Laguerre quadrature does not vanish?
The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum
$$\sum_{i=1}^n f(x_i) w_i$$
where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...
- 4,034
3
votes
1
answer
262
views
Can the $L^{\infty}\to L^{\infty}$ norm be bounded by the trace norm?
Let $k\in C(\mathbb{R}^2; \mathbb{R})$ be a continuous function. Suppose that the operator $K\colon L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ defined by the formula
$$(Kf)(x)=\int_{\mathbb{R}} k(x,...
- 128k
0
votes
1
answer
52
views
Binarily universal members of $[0,1]$
Let $r\in[0,1]$. We look at the binary represenation of $r$ and say that $r$ is binarily universal if every finite binary string appears in at least one place in the binary representation of $r$. Let $...
- 24.8k
0
votes
2
answers
388
views
Derivative of fractional Laplacian is the fractional Laplacian of the derivative
Is it true that $$\partial_x ((-\Delta)^s u(x)) = ((-\Delta)^s \partial_x
u(x))?$$
- 28k
5
votes
2
answers
2k
views
Chain-rule and change of variables in BV/Sobolev
A lot of results are available for the following chain-rule problem:
(CRP1) Let $f\colon \mathbb R \to \mathbb R$ be a $C^1$/Lipschitz function and let $g \colon \mathbb R^d \to \mathbb R$ be a ...
- 28k
1
vote
2
answers
269
views
Convergence of an iterated sequence
Let $K=[0,1]^2$ be a square and $p\in (0,1)$ be a fixed number. We define a map $F: K^2\to K^2$ as follows.
For $(x_1,y_1), (x_2,y_2)\in K$, it follows by a straightforward computation that there ...
CommunityBot
- 1
3
votes
1
answer
308
views
$f: [0,1]\rightarrow L^1(\Omega)$ as a (measurable?) function from $[0,1]\times \Omega\rightarrow \mathbb{R}$
Given a map from $\big([0,1], \mathcal{B}[0,1], m\big)$ to a Banach space $(X, \|\cdot \|)$. There are strong measurable functions (they are the point wise a.e. limit of simple functions) and weak ...
- 12.9k
4
votes
1
answer
1k
views
Is the Wasserstein-1 metric translation invariant?
Define the Wasserstein-1 metric (or the Earth mover's distance) between two positive measures $\mu_1$, $\mu_2$ by
$$
W(\mu_1, \mu_2)
=
\inf_{\gamma \in \Gamma (\mu_1, \mu_2)} \int \|x_1 - x_2\| \, \...
- 109k
3
votes
1
answer
670
views
A specific mollified functions in the Sobolev space H^1(R)
Let $u>0$ be in $H^{1}(\mathbb{R})=W^{1,2}(\mathbb{R})$, we know that the set of $C^{\infty}$ functions with compact support are dense in the Sobolev space $H^{1}(\mathbb{R})$. Hence, we have a ...
CommunityBot
- 1
1
vote
1
answer
285
views
Recover norm from integral
I am given the following expression where $f \in L^2(\mathbb{R}^2, \mathbb{R}^{2 \times 2})$
$$\int_{\mathbb{R}} \int_{\mathbb{R}} \langle g(x), f(x,y) h(y)\rangle dx dy.$$
The functions $g$ and $h$ ...
- 39.1k
0
votes
1
answer
349
views
Is this function positive?
Could someone tell me if my argument is correct?
Let $\rho_1:[0,1]\to [0,1]$ and $J:\mathbb R\to \mathbb R^+$, I have a system of two coupled PDE's and I proved that its solution $(u_0(t, r), u_1(t, ...
CommunityBot
- 1
1
vote
1
answer
447
views
Dirichlet fractional Laplacian and zero boundary conditions
Does there exists a non-zero function $$f\in C_0([0,1]):=\{f:[0,1]\to \mathbb R:\ f\text{ is continuous and } f(0)=f(1)=0\},$$ such that $(-\Delta)^{\frac\alpha 2}f\in C_0([0,1]) $, where $(-\Delta)^{\...
- 17.2k
2
votes
2
answers
208
views
A problem on the maximal modulus
Let $f$ be a transcendental entire function, we know that
$\log M(r, f)$, with $M(r,f)=\max_{|z|=r}|f(z)|$, is a convex function with respect to $\log r$ and
$\lim\limits_{r\rightarrow\infty}\frac{\...
- 13.6k
1
vote
0
answers
137
views
Is there an analysis theorem analogous to Kuznetsov/Petersson trace formula?
I am thinking about general differential operator acts on a compact manifold. Is there something similar to Kuznetsov trace formula?
For example, let $f_i $ be the eigenfunctions of an operator $D$, ...
- 3,804
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
25
votes
1
answer
8k
views
Convergence of Fourier Series of $L^1$ Functions
I recently learned of the result by Carleson and Hunt (1968) which states that if $f \in L^p$ for $p > 1$, then the Fourier series of $f$ converges to $f$ pointwise-a.e. Also, Wikipedia informs me ...
- 18.7k
1
vote
1
answer
229
views
Which norms on vectors can be consistently decomposed?
I need to know which permutation-invariant norms can be consistently decomposed in the sense that for any vector $v = (a,b,c)$ we have that
$$\|(a,b,c)\| = \|(\|(a,b)\|,c)\|.$$
More precisely, let $v ...
- 702
0
votes
1
answer
138
views
On the essential infimum (over subdomains) of nonnegative measurable functions
let $\Omega:=\left( a,b\right) \subset\mathbb{R}$, and suppose
$f:\Omega\rightarrow\left[ 0,\infty\right) $ is a bounded (Lebesgue)
measurable function, with $f\not \equiv 0$ almost everywhere. It ...
- 29.8k
12
votes
1
answer
191
views
Spectra on different spaces
This is a method request: I am looking for techniques that allow me to investigate problems like this:
Let $T_1: \ell^1 \rightarrow \ell^1$ be a bounded operator with $\Re(\sigma(T_1)) \subset (-\...
- 778
1
vote
1
answer
153
views
Mild solution of 2D surface quasi-geostrophic (SQG) equation
I was reading one of Kato's papers on Navier-stokes equations. A mild solution can be denoted as $u= e^{t\Delta}u_0 + \int_{0}^{t} e^{(t-s)\Delta} \mathbb P\nabla \cdot(u \otimes u)ds$, where $\mathbb ...
CommunityBot
- 1
1
vote
1
answer
723
views
Is the 2D Ladyzhenskaya inequality true for periodic functions?
I have only seen the following version of 2D Ladyzhenskaya inequality in cited references of PDE:
Let $\Omega$ be a Lipschitz domain in ${\bf R}^2$ and let $u: \Omega → {\bf R}$ be a weakly ...
- 16.2k
5
votes
2
answers
541
views
Asymptotic behaviour of $\int f(t)^a\cos(at)dt$
Are there any known necessary or sufficient conditions such that
$$\lim_{a\rightarrow \infty}\int_{-1}^1f(t)^a\cos(at)dt=0$$
where $f:[-1,1]\rightarrow[1,\infty)$ is an even smooth concave real ...
- 1,228
0
votes
1
answer
87
views
Differentiablity of certain composite function
Let $I_1$ and $I_2$ be two closed bounded intervals.
Suppose $W(x,y)$ is a smooth function whose support is contained inside $I_1 \times I_2$.
Suppose I have $\Phi= (\Phi_1(x,y), \Phi_2(x,y)) : \...
- 16.2k
13
votes
2
answers
653
views
The geometry of $\mathbb{R}^n$
Let $X,Y$ be finite-dimensional real normed spaces. Consider the set of linear operators $L(X,Y)$ between the two spaces.
Then we define the set of equivalence classes
$$G(X,Y):=\left\{[T]; T,S \in ...
- 12.6k
3
votes
1
answer
237
views
Asymptotics of the following integral
I am concerned with the asymptotic behavior of this integral
$$
\int_2^{\infty}dx\,\frac{\sin(ax)}{ax}\frac{1}{\log x}\bigg(1+\frac{\log x}{\log(a e^{-5/6})}\bigg)^{-\log(a e^{-5/6})}
$$
I am ...
- 188k
3
votes
0
answers
504
views
Continuity of the conditional expectation
Consider the conditional expectation of $x$ given $y$,
$$
\mathbb{E}(x | y)
$$
where $x \in X$ and $y \in Y$ where $X, Y$ are Hilbert spaces (possibly infinite dimensional).
Question :
I am looking ...
- 3,574
7
votes
1
answer
798
views
Intersection of connected components in $\mathbb{R}^n$
Let $n$ be a positive integer and let $K\subseteq \mathbb{R}^n$ be compact. Pick $x^* \in \mathbb{R}^n\setminus K$.
Let $E$ be the connected component of $\mathbb{R}^n\setminus K$ that contains $x^*$....
- 42k
7
votes
2
answers
626
views
The tangent curve to Bessel functions?
Consider a function from the Bessel family, for concreteness say $f(x) := J_0(x)$, depicted in blue below (the question can be asked for any order of the first or second kind):
I'm interested in the ...
- 2,494