All Questions
950 questions
2
votes
0
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184
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Properties of the optimal decomposition for the $K$-functional between $\ell_1$ and $\ell_2$
Background: For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$:
$$
\lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\...
- 1,372
2
votes
2
answers
634
views
Continuous upper envelope of upper semicontinuous function
Let $u$ be a upper semicontinuous function on a compact set $K$ in $\mathbb R^d$. Define a space of continuous function dominating $u$ by
$$A = \{\phi \in C(K): \phi \ge u\}.$$
[Q.] Is the following ...
- 1,399
2
votes
0
answers
71
views
Gluing together mixed normed vector spaces with mixed topologies
This is a variant of this question.
Definitions/Facts
$Ball_1(X)$ denotes the unit ball (about $0$) in a normed vector space $X$.
MixTop of triples of pairs $(X,\tau)$ of normed vector spaces $X$ ...
- 5,405
2
votes
0
answers
77
views
Homomorphism of composition to additive structure
Consider the following topological groups
$\operatorname{Homeo}(\mathbb{R}^d)$ be the topological group of all homeomorphism from $\mathbb{R}^d$ onto itself; equipped with the compact-open topology (...
- 5,405
2
votes
2
answers
2k
views
Separable quotients of non-separable Banach spaces?
I am reading the Functional Analysis book of Conway, one question from the book is find a closed subspace M of $l^{\infty}=l^{\infty}(\mathbb{N})$ with the property that $l^{\infty}/M$ is separable. I ...
- 281
2
votes
1
answer
416
views
Questions about Maharam's classification theorem
I am studying von Neumann algebras. In the wiki article abelian von Neumann algebras, it mentions that every abelian von Neumann algebras acting on a separable Hilbert space is *-isomorphic to $L^{\...
- 523
2
votes
0
answers
83
views
Reference request for (co-)free constructions
Following a comment of user131781, posted to an answer of this question on MO, I am looking for references to the construction of (co)-free functors from categories into the category of Banach spaces ...
- 5,405
2
votes
2
answers
257
views
Reference request on Min-Max theorem
Consider the following min-max problem
$$\inf_{x\in M} \sup_{y\in N} F(x,y),$$
where $F: M\times N\to\mathbb R$ is Lipschitz and $y\mapsto F(x,y)$ is concave for all $x\in M$. Could we derive $\...
user128095
2
votes
0
answers
131
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Can we conclude $\sup_g\int f_1g\le\sup_g\int f_2g$ from $\int f_1\le\int f_2$ in this situation?
Disclaimer: Please bear with me, the question isn't as complicated as it looks like, but I wasn't able to find any simplification for which no counterexample comes to my find.
Let $(E,\mathcal E,\...
- 167
2
votes
2
answers
255
views
Do we have a name for this space?
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Consider the class
$$
\mathcal{F}=\{f\in L^{1}(\Omega):\exists C>0 \text{ s.t. } \int_{U}|f|\leq C\sqrt{|U|},\text{ for any }U\subset \Omega.\...
- 21
2
votes
1
answer
702
views
Correction term in the relation between the Itō and Stratonovich integrals in Hilbert spaces
I'm reading the paper On the relation between the Itō and Stratonovich integrals in Hilbert spaces and there is something I don't understand.
In the notation of the paper, let
$H,H_1$ be separable $\...
- 167
2
votes
1
answer
315
views
Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum
How does one show directly that the solution following parabolic partial differential equation (PDE) of $p(t,v)$ approaches its stationary solution which is a solution of an elliptic partial ...
- 2,239
2
votes
1
answer
338
views
Why is this test function admissible? [Paper explanation]
Reading Non-linear Elliptic and Parabolic Equations Involving Measure Data by Boccardo$\&$Gallouet , I had trouble understanding the following:
Why is $\psi(u_n)\chi_{(0,t)}$ admissible as a ...
- 227
2
votes
1
answer
169
views
How to choose minimisers in a continuous way
Let $\langle X, X' \rangle$ be a dual pair equipped with the weak and weak* topologies.
Let $C$ be a weak* compact subset of $X'$ with nonempty interior. For each $x \in X$, let $M(x)$ be the set of ...
- 869
2
votes
0
answers
279
views
Relationship between $p$-capacity and Riesz $s$-capacity of a set
What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set?
Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
- 839
2
votes
0
answers
199
views
Convergence of the difference quotient of a BV function
Consider a BV function $u \in BV(\mathbb{R}^N; \mathbb{R}^N)$.
What can be said about the difference quotient
$$
\frac{u(x+\epsilon y)-u(x)}{\epsilon}
$$
regarding its convergence as $\epsilon \to 0$...
- 839
2
votes
1
answer
404
views
Feynman-Kac formula for lattice heat equation with non-diagonal potential
Suppose that $X$ is the continuous-time simple symmetric random walk on the lattice $\mathbb Z^d$ (i.e., a simple symmetric random walk with i.i.d. exponential jump times), and let
$$u(t,x):=\mathbf E\...
- 891
1
vote
2
answers
220
views
A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$
Let $A\subset \ell^2$ consist of all $x\in \ell^2$ with $|x|_2=1$ which does not belong to any $\ell^p$ for all $0<p<2$.
Note that $A$ is non-empty with a Baire category argument.
I ...
- 356
1
vote
0
answers
164
views
How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?
(May be this is very basic question for MO)
(For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
- 1,051
1
vote
0
answers
304
views
Harmonic coordinates on asymptotically flat manifold
I am studying the existence of harmonic coordinates at infinity on an asymptotically flat manifold. My Reference papers are, The Mass of Asymptotically Flat Manifold, by Bartnik [B] and The Yamabe ...
- 914
1
vote
0
answers
99
views
Minimize $\langle(1-\kappa)^{-1}f,f\rangle$ for a parameter-dependent integral operator $\kappa$
I've got a contractive self-adjoint linear integral operator $\kappa$ of the form $$(\kappa g)(x):=g(x)+\int\lambda({\rm d}y)k(x,y)(g(y)-g(x))\;\;\;\text{for }g\in L^2(\mu),$$ where $k$ depends on the ...
- 167
1
vote
1
answer
113
views
Is $I-S$ in my attempt of Fredholm alternative injective?
Let $E$ be a Banach space. Let $\mathcal K(E)$ be the space of all compact (bounded linear) operators from $E$ to $E$. For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. Let ...
- 657
1
vote
1
answer
433
views
Why complex conjugate in definition of the Fourier transform?
Let $G$ be a locally compact abelian group and $f:G \to \mathbb{C}$ a function. Its Fourier transform (when it exists) is defined to be
$$\widehat{f}(\chi) = \int_G f(g) \bar{\chi}(g) \mathrm{d} g,$$
...
- 21.3k
1
vote
2
answers
424
views
Regular Lagrangian flow for the problem $\frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x))$
Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x)), &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
where $\chi$ denotes the ...
- 839
1
vote
1
answer
242
views
Can (how) one distinguish germs of continuous functions by a countable set of params?
Continuous functions can be distinguished by their values at say rational points of [0 1].
Germs of analytic functions can be distinguished by derivatives at a point.
So in both cases we see ...
- 24.9k
1
vote
0
answers
133
views
Does the Gaussian Poincare inequality hold for infinite dimensional measure metric spaces?
This is a question subsequent to the one:
Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?
There, I received a very helpful answer that the Gaussian poincare inequality for any ...
- 3,477
1
vote
1
answer
205
views
Is the canonical map $\mathfrak L(X,E)\:\hat\otimes_\pi\:\mathfrak L(Y,F)\to\mathfrak L(X\:\hat\otimes_\pi\:Y,E\:\hat\otimes_\pi\:F)$ injective?
If $A,B$ are $\mathbb R$-Banach spaces, let $A\:\hat\otimes_\pi\:B$ denote the completion of the algebraic tensor product of $A$ and $B$ with respect to the projective norm. Let $X,Y,E,F$ be $\mathbb ...
- 167
1
vote
1
answer
128
views
Infinite direct sum decomposition of the heat semigroup on $\mathbb{R}^n$
Consider the heat semigroup $Q_t$ on $L^2(\mathbb{R}^n)$ generated by the Laplace operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x^2_i}$. Does there exist a direct sum decomposition
$$\oplus_{...
- 407
1
vote
1
answer
301
views
Vague convergence VS Laplace transform convergence?
If we assume that $\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}\mu(dx), \forall s\geq0$, it is possible to show that $\mu_n\to\mu$ vaguely. Where $\mu_n$ is a measure. Please check here for ...
1
vote
0
answers
177
views
A consequence of De Giorgi oscillation lemma
The following lemma is true (see DeGiorgi oscillation lemma)
Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$
where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\...
- 839
1
vote
2
answers
604
views
Generalized limits
Cross-posted from Math SE.
The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question:
...
- 2,203
1
vote
0
answers
877
views
Changing the order of integration of double integral: references and theorems
The Fubini's theorem states that if we have $ \int_0^{\infty} \int_0^{\infty} |f(t,x)| dt dx$ well defined (i.e. function is absolutely integrable) then we can interchange order of integration:
$$ \...
- 1,199
1
vote
0
answers
100
views
Conditions on a measure to satisfy certain relation on moments.
Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>-1$ $t^s\in L^1(\mathrm d\mu(t))$.
I'd like to impose some conditions on $\mu$ so the function
$$f:p\to \frac{\int_0^\infty t^...
- 173
1
vote
1
answer
387
views
$L^p$ compactness for a sequence of functions from compactness of product with cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
1
vote
1
answer
112
views
A bilinear estimate with a simple one-dimensional oscillatory integral kernel
Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$.
I am trying to show that
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
- 852
1
vote
0
answers
100
views
Weak estimate for difference quotient of BV function
In an answer to the question Weak Lebesgue spaces and an estimate for BV functions it was remarked that if $u\in BV(\mathbb R^N)$ then there exists a Lebesgue negligible set $F \subset \mathbb R^N$ ...
- 839
1
vote
0
answers
92
views
Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case
By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user123457
1
vote
1
answer
353
views
A question about PDE argument involving monotone convergence theorem and Sobolev space
I'm reading this paper. In it there is the following argument (see page 240).
Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. $b(\...
- 266
1
vote
3
answers
435
views
Operator norm of shift operator for finite measure spaces
Let $\nu$ be a finite Borel measure on $\mathbb{R}^n$ and define the shift operator $T_a$ on $L^p_{\nu}(\mathbb{R}^n)$ by $f\to f(x+a)$ for some fixed $a\in \mathbb{R}^n-\{0\}$. Suppose moreover that ...
- 5,405
1
vote
0
answers
104
views
Amenability of $\textrm{w}_0(L^1(G))$
Let $G$ be an infinite compact group and $A=L^1(G)$. It is known that $c_0(A)$ is amenable [Runde2020, p.80] while $\ell^{\infty}(A)$ is not [Daws2009] .
Let $\textrm{w}_0(A)$ denote the subspace of $\...
- 2,605
1
vote
1
answer
192
views
Characterization of a subset of $[0,1]$
Let $T\subseteq[0,1]$ be a subset containing $1$. Now we know that $T$ satisfies the following property:
For every $t\in [0,1)$, if there exists a decreasing sequence $\{t_n\}_{n\ge 1}\subset T$ such ...
- 1,835
1
vote
1
answer
511
views
Convergence in $C_c$ but not in $C$
Let $C_c(\mathbb{R})$ be the set of compactly-supported continuous functions on $\mathbb{R}$. We can view this with a number of different topologies but I have my eye on two in particular. Let $X$ ...
- 5,405
1
vote
1
answer
131
views
Optimal constant comparing $f(1/2)$ and $\|f\|_2$ when $f$ is $t$-Hölder?
Suppose that $f \colon [0, 1] \to \mathbb{R}$ is $k$ times continuously differentiable and Holder in the sense that for some
$t = k + \beta$, where $\beta \in (0, 1]$ and $k$ is a nonnegative integer ...
- 410
1
vote
2
answers
275
views
Again, proving that specific preorder on the set of measurable functions is symmetric
This question is followup to the previous similar question. There I was trying to find good sufficient condition for abstract preorder to be symmetric, but now, as I have found good formalization of ...
- 105
1
vote
1
answer
121
views
Complemented subspace constructed from finite pieces
Suppose $Y=\overline{\cup E_n}$ is a closed subspace of a Banach space, where each $E_n$ is a $n$-dimensional subspace, $K$-complemented in $X$, and for any $n$, $E_n\subseteq E_{n+1}$. Can one ...
- 247
1
vote
3
answers
359
views
For a tempered distribution $F$ on $\mathbb{R}^2$, what exactly does it mean by $\lvert F(x,y) \rvert \leq \lvert x-y \rvert^{-n}$?
Let $F$ be a tempered distribution on $\mathbb{R}^2$ and $n \in \mathbb{N}$ be a fixed natural number. I wonder what exactly it means by
$\lvert F(x,y) \rvert \leq \lvert x-y \rvert^{-n}$ where $x,y \...
- 3,477
1
vote
2
answers
90
views
Is the difference between $\alpha$-Hölder constants of $f*\rho$ and $g*\rho$ controlled by $\|f-g\|_\infty$?
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
- 835
1
vote
1
answer
217
views
Open set of geodesics implies the set of starting points is open
Let $X$ be a complete and separable metric space, let $G(X) \subset C([0,1],X)$ be the space of continuous curves from $[0,1]$ to $X$ with constant speed, i.e.
$$ d(f(t),f(s)) = |t-s| d(f(0), f(1)). $$...
- 41
1
vote
0
answers
73
views
Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
1
vote
1
answer
317
views
The continuous convergence given the a.e. convergence
Suppose that $f_n: \mathbb{R} \times [0,\infty) \to \mathbb{R}$ is a uniformly bounded sequence (i.e., there exists $C>0$: $|f_n| < C$ for every $n$) such that
$$ f_n \in C^2_x \times C^1_t, $$
...
- 41