All Questions
12,935 questions
-1
votes
1
answer
149
views
Necessity of Lebegue's convergence criterion? [closed]
the well-known Lebesgue’s dominated convergence theorem states that pointwise convergence of a sequence of functions implies convergence of the sequence of integrals if an integrable function ...
-1
votes
1
answer
191
views
Nearly elliptic equations [closed]
If you have a second order elliptic equation but the coefficients of the second order terms only form a nonnegative (instead of positive definite) matrix, then, do you know if there is any literature ...
- 1
-1
votes
1
answer
139
views
$L^1$ convergence
Setting
For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
- 125
-1
votes
2
answers
407
views
Conditional expectation: commuting integration and supremum
Let $X$ and $A$ be compact Polish spaces endowed with Borel $\sigma$-algebras. Let $\mathcal{A} = X\times \mathcal{B}(A)$ be the $\sigma$-algebra consisting of cylinders whose projections on $A$ are ...
- 109
-1
votes
1
answer
200
views
Is the unordered sum of measurable functions measurable?
Let $E$ be a normed $\mathbb R$-vector space and $I$ be a nonempty set. Remember that $(x_i)_{i\in I}\subseteq E$ is called summable if there is a $x\in E$ such that for all $\varepsilon>0$, there ...
- 167
-1
votes
1
answer
108
views
Are local diffeomorphisms Fredholm maps with index zero? [closed]
Does this statement correct? if it does how we can prove it. In Banach spaces
a map is local diffeomorphism if and only if it is a Fredholm map of index zero with no critical points?
- 71
-1
votes
2
answers
642
views
Invariance of spectrum under conjugation
Let $T$ be a self-adjoint invertible operator on $\mathcal{H}$ with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators $V$( with continuous ...
- 677
-1
votes
2
answers
466
views
What conditions imply that a function over $\mathbb{Z}$ is a polynomial? [closed]
How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can satisfy....
- 1,129
-1
votes
1
answer
110
views
Proving that $\max_{w \in B(z)} e^{f(w)} \leq Ce^{f(z)}$
Let $f : \mathbb R^2 \to \mathbb R $ be a smooth function statisfying
$$
0 < \alpha \leq \Delta f(w) \leq \beta < \infty, \ \ \forall w \in \mathbb R^2
$$
where $\Delta$ denotes the Laplace ...
- 437
-1
votes
1
answer
396
views
Interpolation Inequality's Proof
Let $\Omega \subseteq R^{n}$ bounded domain. I need to prove that for $u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$:
\begin{equation}
\|\nabla u\|_{L^{2}(\Omega)}^{2}\leq \|u\|_{L^{2}(\Omega)}\|\Delta u\...
-1
votes
1
answer
153
views
$\ell^q$ analog of square function
It is a classical result in harmonic analysis that
$$ \|\|P_kf\|_{\ell^2_k}\|_{L^p_x}\approx\|f\|_{L^p} $$
for $p\in(1,\infty)$, where $P_k$ is the Littlewood-Paley decomposition onto frquency $\...
- 5,169
-1
votes
2
answers
263
views
$L^{n/2}$ norm of scalar curvature
In the Wikipedia article on scalar curvature, it is noticed that the Hölder inequality implies
$$Y(g) \geq - \left(\int_M |R(g)|^{n/2} \mathrm{d}V_g\right)^{n/2}$$
for the Yamabe functional $Y(g)$ and ...
- 9,853
-1
votes
1
answer
357
views
carleman inequality
Is there a connection between carleman inequality discovred by T. Carleman in 1922 if I am not mistaken in his research on quasianalytic functions and what is called Carleman estimates used in the PDE ...
- 308
-1
votes
2
answers
407
views
Almost isometric subspaces of $\ell_p$
1) Given $p\in (1,\infty)$.
2) Let us fix two, non-isometric subspaces $X,Y\subseteq \ell_p$ isomorphic to $\ell_p$.
3) Are there an $\varepsilon\in (0,1)$ and an isomorphism $S\colon X\to Y$ such ...
- 537
-1
votes
1
answer
98
views
Spectrum of sum of positive and negative operators
Let $(\mathscr{H}, \langle \cdot, \cdot \rangle)$ be a separable complex Hilbert space, and let $\mathscr{D}$ be a dense subset of $\mathscr{H}$. Let $P: \mathscr{D} \to \mathscr{H}$ and $N: \mathscr{...
- 21
-1
votes
1
answer
168
views
Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 349
-1
votes
1
answer
168
views
A question in functional analysis about selfadjoint operator [closed]
In Hilbert space $u$, Let $T_1$,$T_2$ is selfadjoint operator, if exit $c>0$ such that $cI\le T_1\le T_2$, prove $T_1$,$T_2$ have a bounded inverse operator and $c^{-1}I\ge T_1^{-1}\ge T_2^{-1}$.
I ...
- 1
-1
votes
1
answer
246
views
Density of normal elements in a C*- algebra [closed]
Let $A$ be a unital C*-algebra.
I wanted to know if there is a necessary and sufficient condition for normal elements to be dense in $A$?
- 149
-1
votes
1
answer
127
views
Solving a fully nonlinear first order PDE
given a symmetric matrix of Holder continuous functions $A(x)$ such that
$$
\frac{1}{C} |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq C |\xi|^2
$$
find a vector field $\Phi$ such that
$$
D \Phi(x)^t D ...
- 261
-1
votes
1
answer
102
views
Is it true that $\nabla_x \int_0^\infty f(t,0) dt = 0 \implies \nabla_x f(t,0) = 0 \ \forall t>0$? [closed]
Let $f:\mathbb R_+ \times \mathbb R^N \to \mathbb R$ and $$F(x) = \int_0^\infty f(t,x) dt.$$ If $\nabla_x F(0) = 0$ do we have that $\nabla_x f(t,0) = 0$ for all $t \in \mathbb R_+$? If not, which ...
- 131
-1
votes
2
answers
129
views
Is it possible for all of the smooth/continuous curves in $R^3$ to form a Hilbert space? [closed]
Under which condition can it form a Hilbert space? Or what space can it form?
You can write down certain condition to make it to be a Hilbert space, e.g., Let $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...
- 11
-1
votes
1
answer
215
views
Dense linear span implies closed convex hull has non-empty interior
Let $X$ be a Fréchet space and let $Y\subseteq X$ such that $\overline{\operatorname{span}(Y)}=X$. It seems intuitive to me that $\operatorname{int}\big(\overline{\operatorname{co}(Y)}\big)$ is a non-...
- 5,405
-1
votes
1
answer
122
views
Approximation of function in general measure space
Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with
$$
\int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
- 411
-1
votes
1
answer
135
views
Prove that Cartesian composition $c_0 \times c_0$ is not isometric isomorphic [closed]
Prove that Cartesian composition $c_0 \times c_0$ with rate $ \Vert (x_1 ,x_2)\Vert = \Vert x_1 \Vert_{c_0} + \Vert x_2 \Vert_{c_0} $ is not isometric isomorphic to space $c_0$.
- 21
-1
votes
1
answer
346
views
Riesz representation theorem for Hilbert-to-Hilbert mappings [closed]
Assume $\phi:\mathbb{H}_1\rightarrow \mathbb{H}_2$ is a continuous linear mapping between two real Hilbert spaces $\mathbb{H}_1$ and $\mathbb{H}_2$. If $\mathbb{H}_2=\mathbb{R}$, then the Riesz ...
- 313
-1
votes
1
answer
237
views
Theorem with an example [closed]
i have this theorem
in the paper they gives an example:
but here $H_1$ is not satisfied !
How to correct it please?
- 277
-1
votes
1
answer
280
views
Showing there is a unique spectral measure
All the books I have seen have proved that, for a normal bounded operator $T$, there is a unique spectral measure $E$ such that $\int_{\sigma(T)}^{}\lambda\,dE=T$ by first proving in it for a general ...
- 127
-1
votes
1
answer
457
views
Why can't I get global existence to linear PDE in this way? [closed]
For any $n > 0$, standard theory implies there is a unique $u_n \in L^2(0,n;V)$ with $u_n' \in L^2(0,n;V^*)$ such that
$$u_n' + Au_n = f\quad\text{as an equality in $L^2(0,n;V^*)$}$$
$$u_n(0) = u_0$...
-1
votes
1
answer
77
views
Applications and motivations of resolvent for elliptic operator
Let $ A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ be a matrix satisfying ellipticity condition, that is
\begin{align}
\mu^{-1}|\xi|^2\geq \sum_{i,j=1}^da_{ij}(x)\xi_i\xi_j\geq\mu|\xi|^2
\...
- 1,219
-1
votes
1
answer
83
views
"Large" compact sets in separable normed space
Let $(X, \lVert \cdot \rVert)$ be a separable normed space. Can we always guarantee that there is a nonempty compact set $K \subseteq B_X$, where $B_X$ is a closed unit ball in $X$ such that:
$$\...
- 820
-1
votes
1
answer
116
views
Continuous surjection of $\mathbb{R}^{n-1}$ onto the interior of the $n$-simplex with continuous right inverse
Let $n$ be a positive integer. Clearly $\mathbb{R}^{n-1}$ and the interior of the $n$-simplex $
\delta_n := \{x \in [0,1]^n:\,\Sigma_k x_n =1, (\forall i)\,x_i>0\}
$ are homeomorphic. What I'm ...
- 5,405
-1
votes
1
answer
120
views
Definition of a $\psi$-Banach space [closed]
Let $X$ be a Banach space. Let $\mathcal{F}$ be the family of all the bounded subsets of $X$. If $I$ is the identity map on $X$, we shall denote by $\operatorname{span}\{I\}$ the vector space ...
- 291
-1
votes
1
answer
176
views
How do we approximate the pressure in the Boussinesq equations of hydrodynamics? [closed]
How do we approximate the pressure or the gradient of it in the Boussinesq equations of hydrodynamics ?
Is the pressure limited or can it be any amount?
- 11
-1
votes
1
answer
70
views
Is this kind of interpolation correct?
Let $f=\sum f_j$ be a finite sum. Assume that
$$ \|f\|_2\le(\sum\|f_j\|_2^2)^\frac12$$
$$\|f\|_\infty\le C\max_j\|f_j\|_\infty$$
Then can we conclude that for $2<p<\infty$
$$\|f\|_p\le C^{1-\...
- 1
-1
votes
1
answer
328
views
About the critical points of quasi-convex functions
What do we know about the structure of critical points of quasi-convex functions?
I am looking for statements like "the critical points of a quasi-convex function are always either a global minima ...
- 2,246
-1
votes
1
answer
346
views
An infinite set in a compact space
Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...
-1
votes
1
answer
360
views
Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]
Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...
- 7
-1
votes
1
answer
148
views
Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]
Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...
-1
votes
1
answer
104
views
Question about measure lemma?
"Let (u_j) be a bounded sequence from $W^{1,p}(\Omega)$ how to prove that there exists a subsequence such that $u_j\rightharpoonup u$ in $W^{1,p}_0(\Omega)$ and $|\nabla u_j|\rightharpoonup d\mu,$ $|...
- 277
-1
votes
1
answer
516
views
Equivalence of two definitions of Sobolev spaces
Good morning,
I am looking for a reference about the following fact that seems to be folklore. Define the Sobolev (Beppo Levi?) space
$$
D^{1,p}(\mathbb{R}^N) = \left\{ u \in L^{p^*}(\mathbb{R}^N) \...
- 9
-1
votes
1
answer
128
views
Proving convergence of an integral-differential equation [closed]
I have a second order nonlinear ordinary differential equation which I transformed into an integral-differential equation by multiplying the ODE by $y'$ and integrating.
My question is where can I ...
- 1,594
-1
votes
1
answer
187
views
Limit of a function in a weighted Sobolev space
I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense
$$\lim_{|x-y|\to 0} f(x)$$
? ($y$ is a fixed point)
If i have $f$ in $H^2$ I can say that
$$\lim_{|x-y|\to 0} f(x)=...
- 25
-1
votes
1
answer
1k
views
relation between inclusion and embedding [closed]
Assume that $X$ and $Y$ are two Banach spaces, now we have that $X$ is included in $Y$, in the sense that $\forall a\in X$, we have $a\in Y$. Then can we get that $X$ is embedded in $Y$, namely, $\...
- 331
-1
votes
1
answer
286
views
Check an equation on the Heisenberg group $H_1$
The Heisenberg group $H_1$ is the set $\mathbb C\times \mathbb R$ endowed with the group law
$$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right); \quad \forall z,w \in \mathbb C\,...
- 650
-1
votes
1
answer
153
views
Sobolev estimates $\|\nabla\phi\|_{\infty}\leq C\|\phi\|_{H^2}$
This is a cross post in continuation to this question on Mathematics Stack Exchange. I wanted to know if this inequality holds true in two or three dimensions,
$\|\nabla\phi\|_{L^{\infty}(\Omega)}\leq ...
- 101
-1
votes
1
answer
79
views
A question about the commutator $[J^s,u]\partial_x u$
I am studying the use of the commutator for finding the estimate of energy. During my looking through many papers I found that this paper contains a possible typo. Here is the archive version which ...
- 159
-1
votes
1
answer
87
views
Is integration by parts allowed for the $J^s$ derivative, where $s \in \mathbb R$
I am having the following integral:
$$I = \int u\, J^s(\partial_x \overline{u})- \overline{u}\, J^s(\partial_x u))dxdy$$
where $J^S= (I-\Delta)^\frac{2}{2}$, $\mathbb{R} \ni s \geq 1$ and $u=u(x,y)$,
$...
- 159
-1
votes
1
answer
164
views
Closure of the point spectrum of an unbounded diagonalizable operator
Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
- 969
-1
votes
1
answer
78
views
Fundamental of a signal
Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$
or any reasonable Euclidean norm such that bounded ...
- 1,902
-1
votes
1
answer
75
views
name of coordinates and reference (elliptic pde)
Consider working on a domain $\Omega$ in $ R^N$ and we assume that $r=|x|$ and $ \theta$ is the angle between the $x_N$ axis and the $ R^{N-1}$ plane. I am looking at functions and domains that ...
- 1,385