All Questions
949 questions
2
votes
1
answer
339
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
548
views
Can the characteristic function of a Borel set be approached by a sequence of continuous function through a certain convergence in $L^\infty$?
Want to find $f_n$ a sequence of continuous functions, so that for all Borel regular measure $\mu$, we have
$\int f_n d\mu\rightarrow\int \chi_\Delta d\mu$, as $n$ goes to infinity, where $\Delta$ ...
- 31
2
votes
2
answers
223
views
Relating function value to $L^2$ norm in Holder space
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
2
votes
1
answer
378
views
Does the norm on a sequence space have to be monotone?
Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties:
$\rho(\lambda u)=\lambda\rho( u)$, for every $u\in [0,+\infty)^{\mathbb{N}}$ and $\lambda\ge 0$;
$\rho(u+v)\le \...
- 5,529
2
votes
0
answers
184
views
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
1
answer
3k
views
A comparison principle for parabolic equation
(Crossposted from https://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear)
Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for ...
- 266
2
votes
2
answers
867
views
Decomposition of an abelian von Neumann algebra
Hi, I came across the statement below and I couldn't figure out why it is true. I was hoping someone could explain it or give me a good reference. Thank you in advance.
"Let $\pi$ be a non-degenerate ...
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
1
answer
118
views
Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius
This is a follow up from this question.
I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
- 765
2
votes
2
answers
457
views
A Fixed point Theorem that does not need the convexity of set valued map?
I am looking for a fixed point theorem for set valued maps that does not assume the set valued map should be convex valued.
Something like contractiblity or other properties can be replaced with ...
- 383
2
votes
1
answer
1k
views
Existence of a projection operator onto subspace of Hilbert space
Let $V \subset H$ be Hilbert spaces with a continuous, compact and dense imbedding. Let $\{w_j\}_j \subset V$ be a basis of $V$ and of $H$ (so finite linear combinitions are dense) which is not ...
- 35
2
votes
1
answer
328
views
Hausdorff dimension of the graph of a BV function (in 1 dimensional setting)
Let $u: \Omega\subset \mathbb{R} \to \mathbb{R}$ be a function of bounded variation.
Question 1.
How can we prove that the Hausdorff dimension of the essential graph of $u$ equal to $1$?
Question ...
- 839
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
1
answer
72
views
Generalised Lebesgue transform continuous wrt. strict topology?
Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$].
For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, consider the ...
- 463
1
vote
1
answer
113
views
The Fourier projection mappings $\{ P_N \}$ form an equicontinuous family of linear maps on $E'(S^1)$ as well?
Let $S^1=\mathbb{R}/\mathbb{Z}$ and define the Fourier projection operator $P_N$ for each $N \in \mathbb{N}$ as
\begin{equation}
P_N(f)=\sum_{n=-N}^N \langle f, e_n \rangle_{L^2} e_n
\end{equation}
...
- 3,477
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
1
answer
264
views
Is there a version of dominated convergence theorem for local $L^p$ spaces?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
- 835
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
76
views
Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius
I have a smooth function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
- 765
1
vote
0
answers
104
views
Amenability of $\textrm{w}_0(L^1(G))$
Let $G$ be an infinite compact group and $A=L^1(G)$. It is known that $c_0(A)$ is amenable [Runde2020, p.80] while $\ell^{\infty}(A)$ is not [Daws2009] .
Let $\textrm{w}_0(A)$ denote the subspace of $\...
- 2,605
1
vote
1
answer
128
views
Infinite direct sum decomposition of the heat semigroup on $\mathbb{R}^n$
Consider the heat semigroup $Q_t$ on $L^2(\mathbb{R}^n)$ generated by the Laplace operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x^2_i}$. Does there exist a direct sum decomposition
$$\oplus_{...
- 407
1
vote
1
answer
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
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
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
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
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
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
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
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
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
0
answers
922
views
A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it
Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
- 698
1
vote
1
answer
205
views
Is the canonical map $\mathfrak L(X,E)\:\hat\otimes_\pi\:\mathfrak L(Y,F)\to\mathfrak L(X\:\hat\otimes_\pi\:Y,E\:\hat\otimes_\pi\:F)$ injective?
If $A,B$ are $\mathbb R$-Banach spaces, let $A\:\hat\otimes_\pi\:B$ denote the completion of the algebraic tensor product of $A$ and $B$ with respect to the projective norm. Let $X,Y,E,F$ be $\mathbb ...
- 167
1
vote
1
answer
223
views
A linear algebraic q-difference equation [SOLVED]
I would like to solve the following algebraic linear q-difference equation:
\begin{equation}
a\left(x\right)f\left(x\right)=f\left(qx\right)
\end{equation}
The parameter $q$ is real, positive and ...
- 133
1
vote
0
answers
233
views
Fubini: can we interchange integration order on this double integral (with Fourier series product)
Can we interchange the order of integration of following double integral ?
$$I = \int_{0}^{1} \int_{0}^{\infty} F(x,y) \overline{R(x,y)} - R(x,y) \overline{F(x,y)} \; dx \; dy$$
Where $F(x,y)= \...
- 1,199
1
vote
0
answers
304
views
Harmonic coordinates on asymptotically flat manifold
I am studying the existence of harmonic coordinates at infinity on an asymptotically flat manifold. My Reference papers are, The Mass of Asymptotically Flat Manifold, by Bartnik [B] and The Yamabe ...
- 914
1
vote
1
answer
121
views
Complemented subspace constructed from finite pieces
Suppose $Y=\overline{\cup E_n}$ is a closed subspace of a Banach space, where each $E_n$ is a $n$-dimensional subspace, $K$-complemented in $X$, and for any $n$, $E_n\subseteq E_{n+1}$. Can one ...
- 247
1
vote
0
answers
99
views
Minimize $\langle(1-\kappa)^{-1}f,f\rangle$ for a parameter-dependent integral operator $\kappa$
I've got a contractive self-adjoint linear integral operator $\kappa$ of the form $$(\kappa g)(x):=g(x)+\int\lambda({\rm d}y)k(x,y)(g(y)-g(x))\;\;\;\text{for }g\in L^2(\mu),$$ where $k$ depends on the ...
- 167
1
vote
1
answer
231
views
Continuous semigroup homomorphism of composition to additive structure
Let $G$ be the topological semigroup whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with composition as semigroup operation and let $H$ be the topological group whose underlying ...
- 5,405
1
vote
2
answers
604
views
Generalized limits
Cross-posted from Math SE.
The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question:
...
- 2,203
1
vote
1
answer
317
views
The continuous convergence given the a.e. convergence
Suppose that $f_n: \mathbb{R} \times [0,\infty) \to \mathbb{R}$ is a uniformly bounded sequence (i.e., there exists $C>0$: $|f_n| < C$ for every $n$) such that
$$ f_n \in C^2_x \times C^1_t, $$
...
- 41
1
vote
0
answers
177
views
A consequence of De Giorgi oscillation lemma
The following lemma is true (see DeGiorgi oscillation lemma)
Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$
where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\...
- 839
1
vote
0
answers
99
views
Gluing together dense subset of Projective Limit in $Ban_1$
Let $(X_n,\pi_n^{m})$ be a countable projective system in the category Ban$_1$ of Banach spaces and short linear maps (is (continuous) linear constructions). Then (co)-completeness of this category ...
- 5,405
1
vote
1
answer
151
views
If $f(x_1,x_2)=f(x_2,x_1)$, $f(x_1,x_2)=\sum_k \lambda_k f_k(x_1)f_k(x_2)$? [closed]
Consider a symmetric function
$$
f(x_1,x_2):R^n \times R^n \to R
$$
satisying $f(x_1,x_2)=f(x_2,x_1)$. Are there functions $f_k:R^n \to R$ such that
$$
\int_{x\in R^n}f_k(x)f_l(x)dm=\delta_{kl},
$$
...
1
vote
2
answers
424
views
Regular Lagrangian flow for the problem $\frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x))$
Consider the problem
$$(\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}
$$
where $\chi$ denotes the ...
- 839
1
vote
1
answer
505
views
Gaussian measures on non-separable spaces
Let $X$ be a topological affine space which is neither separable nor metrizable. There are plenty of trivial Gaussian measures: each Dirac point-mass $\delta_x$ are the Gaussian measure with zero ...
- 8,512
1
vote
1
answer
178
views
Growth assumption and example of finite (arbitrarily small) time blow up for ODE
Consider the following ODE initial value problem
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \...
- 839
1
vote
0
answers
315
views
"Integration by parts" formula for functionals
We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;V)$ with $u',v' \in L^2(0,T;V')$
then
$$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$
where the $...
- 29
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
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