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10 votes
0 answers
656 views

“Taylor series” is to “Volterra series” as “Laurent series” is to _________?

Preamble My question is similar to an earlier MathOverflow question: “Taylor series” is to “Volterra series” as “Padé approximant” is to _________? which I just answered (hopefully my first ever ...
1 vote
1 answer
252 views

GNS Representation — A theorem from Thirring’s book

After the GNS representation for $C^{*}$-algebras is presented in Thirring's book Quantum mathematical physics, the author states the following theorem. The Spectral Theorem: For any given Hermitian (...
2 votes
2 answers
316 views

Properties of the topology of sequential convergence $\tau_\text{seq}$

Let $(X,\tau)$ be a Hausdorff space. Denote by $\tau_\text{seq}$ the topology on $X$ whose closed sets are the sequentially $\tau$-closed subsets of $X$. I have read that $\tau_\text{seq}$ has the ...
4 votes
2 answers
392 views

Lebesgue differentiation theorem at boundary points for Sobolev traces

$\newcommand{\R}{\mathbb R}$ Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$. Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then $$ u(x)...
2 votes
1 answer
336 views

Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?

Let $\Omega$ be a metric space, $C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and $\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
3 votes
1 answer
162 views

Approximation in Bochner spaces

Is there any result like the Bramble-Hilbert lemma for Bochner spaces? More specifically: let $H$ be a (e.g.) Hilbert space, $I\subset \mathbb R$ a bounded interval, and $L \in \mathcal L(H^k(I;H), Y)...
6 votes
1 answer
135 views

Small shifts of weakly converging sequences in $L^1$

$\newcommand\R{\mathbb R}$Let $(f_n)$ be a sequence in $L^1(\R)$ converging weakly to some $f\in L^1(\R)$. Let $(a_n)$ be sequence in $\R$ converging to $0$. For each natural $n$, let $g_n$ be the $...
1 vote
0 answers
128 views

Surjectivity of perturbed linear operators

Consider two Banach spaces $X$ and $Y$ and two linear bounded operators $A,B:Y\rightarrow Y$. Suppose the following: (1) Y is reflexive (or even uniformly convex); (2) $X\cap Y$ is dense in $X$ and $Y$...
5 votes
0 answers
103 views

What conditions guarantee differentiability of higher-order functions?

Sorry if my language is not precise. Let $f_i:\mathbb{R}^m\to \mathbb{R}^n$ be some differentiable functions in the sense of auto-diff. How can you construct new functions $g(f_1,f_2,f_3,...)$ such ...
4 votes
2 answers
191 views

Reference request: "Tangent relation" in metric spaces

Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x_0 \in X$. Let us say that $f$ and $g$ are tangent at $x_0$ if the following condition is satisfied: For every $\epsilon > 0$ there ...
6 votes
0 answers
110 views

Heat Flows and spatial singularities

While working on an abstract problem, I came up with the following question: Let $\Omega_1 := \mathrm B(-1, 1)$ and $\Omega_2 := \mathrm B(1, 1)$, where $\mathrm B(x, r) \subseteq \mathbb R^2$ denotes ...
1 vote
1 answer
262 views

Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?

Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p := L_p(X, \mu, E)$ be the Bochner space of all $\mu$-integrable ...
4 votes
2 answers
374 views

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

I'm reading a proof of below theorem from this paper. Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
3 votes
1 answer
181 views

General form of bounded linear functionals on Banach spaces

It is a famous result due to Riesz that every bounded linear functional $f$ on a Hilbert space $\mathcal{H}$ is of the form $f(x)=\langle x,z \rangle$ for a unique $z\in H$. On p.188 of Introductory ...
2 votes
1 answer
241 views

Strategy of the proof of the "minimal entropy condition" for scalar conservation laws

Combining Theorem 2.3 and Corollary 2.5 of this paper gives that, for a strictly convex conservation law $$u_t + f(u)_x = 0,$$ satisfying the entropy condition $$\eta(u)_t + q(u)_x \le 0$$ in the ...
1 vote
1 answer
294 views

Spectral perturbation theory of discrete spectra in presence of continuous spectrum

This is a 2 part question: 1). I am looking for a (hopefully accessible to beginning grad student who knows matrix perturbation theory) reference for doing concrete calculations of perturbed discrete ...
1 vote
0 answers
74 views

"N-waves" (source-type solutions) for Hamilton-Jacobi equation $v_t + (v_x)^2 = 0$

Let us consider the Burgers equation $$u_t + (u^2)_x = 0$$ In Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ...
5 votes
0 answers
135 views

Criteria for tightness of Gaussian measures on Banach spaces

In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert ...
15 votes
5 answers
2k views

Between Tietze's and Dugundji's extension theorems

The celebrated Tietze extension theorem asserts that any continuous real-valued function defined on a closed subset of a normal space, can be extended to a continuous function on the whole space. Seen ...
3 votes
0 answers
282 views

Left ideals of $\ell^{\infty}(A)$ containing all weakly null sequences in a Banach algebra $A$

Let $A$ be a Banach algebra. $\ell^{\infty}(A)$, the space of all bounded sequences in $A$, is a Banach algebra with pointwise operations. Let $w_0(A)$ be the subspace of all weakly null sequences in $...
1 vote
0 answers
52 views

Sherman-Davis type inequalities for non-negative operator in a Hilbert space with trivial kernel

Recently I read Rupert L. Frank's paper "Eigenvalue Bounds for the Fractional Laplacian: A Review". For a domain $\Omega\subset\mathbf R^n$, there are two different definitions of ...
4 votes
0 answers
134 views

Weighted logarithmic Sobolev inequality

$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that $$ \Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu $$ where the entropy $$ \Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
1 vote
0 answers
100 views

N-wave solution of conservation law $u_t + (u - u^2)_x = 0$

How can we compute the "N-wave" source-solution of the conservation law $$u_t + (u - u^2)_x = 0, $$ that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...
3 votes
1 answer
466 views

Equivalence between two fractional Sobolev spaces

For $s \in (0,1)$, we consider the spectral fractional Laplacian \begin{align} (-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k \end{align} where \begin{align*} \begin{cases} ...
7 votes
2 answers
2k views

Heat kernel estimates and Gaussian estimates for semigroups, good reference?

Hi, it seems like a big field and I'm having trouble getting some solid/classic references to get me started. If $U \subset \mathbb{R}^d$ is a bounded domain with, say, $C^2$-boundary $\partial U$ ...
19 votes
1 answer
3k views

Infinite convex combinations in a Banach space

Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds: For any sequence $(x_k)_{k\ge0}$ in $C$, and for any sequence of non-negative real numbers $(\...
1 vote
1 answer
404 views

Is Schwartz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$? [closed]

Can someone kindly confirm that the Schwartz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all ...
3 votes
0 answers
160 views

Ergodic diffeomorphisms of the circle

From the paper Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889. the following result is known: Let $(E,\Sigma, \mu)$ be a ...
4 votes
0 answers
68 views

Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$

For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that $$ |f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
0 votes
1 answer
734 views

Infinite composition of continuous functions

Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ be a sequence of functions and define $F_n:= f_n\circ \dots\circ f_1$. Then $F_n$ is continuous. However, the pointwise limit need not be (consider Mateusz'...
2 votes
0 answers
104 views

Schwarz space on the upper half-plane

Let us think of the Schwartz space $\mathcal{S}(\mathbb{R}^2_+)$ on the upper half-plane $\mathbb{R}^2_+=\mathbb{R}\times(0,+\infty)$ defined as $$ \mathcal{S}(\mathbb{R}^2_+)=\left\{f\in C^\infty(\...
5 votes
2 answers
321 views

If the Hausforff dimension of the graph of a function $u$ is $N$ and $\tilde u = u$ a.e. then $\dim_H \mathrm{graph} \, \tilde u = N$ too

Let $\Omega$ be an open (non empty) set and $u:\Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a function such that the Hausdorff dimension of its graph is $N$. Let $\tilde u = u$ a.e. Is it true ...
3 votes
0 answers
115 views

Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?

The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
4 votes
2 answers
610 views

Unit ball of the sum space

Let $V$ be a vector space and $\|\cdot \|_1$ and $\|\cdot\|_2$ two norms on $V$. Let $\|\cdot\|_+$ be given by $$ \|v\|_+ := \inf_{v = v_1 + v_2} \|v_1\|_1 + \|v_2\|_2 $$ It is well-known that $\|\...
1 vote
1 answer
305 views

What's the name of this semi-group theorem?

I encountered this theorem, that for a bounded linear transform $L$ and a real parameter $t$ and initial data $u_0$, we have $$\frac{d}{dt} \exp(Lt)[u_0] = L \exp(Lt)[u_0].$$ What is the name of this ...
1 vote
0 answers
47 views

Scaling limit of transport equation with double-well potential

Let us consider the transport PDE $$ u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon) $$ where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the PDE ...
2 votes
0 answers
64 views

Scaling limit of ODE with double-well potential

Let us consider the ODE $$ \frac{d}{dt}x_\epsilon(t) = -\frac{1}{\epsilon} W'(x_\epsilon(t)) $$ where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the ODE reads $$...
1 vote
0 answers
49 views

Banach algebras satisfying $pq=qp=q \Rightarrow \|q\|\leq\|p\|$ for all idempotents $p$ and $q$

This question could be way below the level of MO, so apologies in advance. I posted the same question in MS about 10 days ago without a definitive answer so far. Let $A$ be a Banach algebra with the ...
9 votes
3 answers
3k views

The classical Krein-Rutman theorem

The classical Krein-Rutman theorem states that any positive compact linear endomorphism $T:X \to X$ on a Banach space $X$ with positive spectral radius $r(T)$ has an eigenvalue $r(T)$ with a positive ...
13 votes
2 answers
2k views

When can we divide continuous functions?

Let $X$ be a compact Hausdorff topological space such that for every continuous $f,g:X\to\mathbb{R}$ with $0\le f\le g$ there is a continuous $h:X\to\mathbb{R}$ such that $f=gh$. What can be said ...
12 votes
1 answer
1k views

Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$

Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure ...
1 vote
0 answers
148 views

BMO estimates of singular integral operators on torus

I have the following elliptic problem: $$ \Delta u = \operatorname{div}\operatorname{div}S, $$ where $S=(S_{i,j})\colon \mathbb{T}^n\to \mathbb{R}^{n\times n} $ is bounded and $\mathbb{T}^n$ is the $n$...
0 votes
1 answer
157 views

Oleinik inequality (one-sided Lipschitz condition) implies $BV_{\mathrm{loc}}$ for solution of conservation law

Consider the scalar conservation law $$u_t+f(u)_x=0, \hspace{0.4 cm} \text{in $\hspace{0.2 cm}$ $\mathbb{R} \times (0,\infty)$}$$ where $f \in C^{2}(\mathbb{R})$ is a strictly convex function ($f''>...
9 votes
1 answer
669 views

Reference for "Every compact quasinilpotent operator is the limit of nilpotent ones"

It was mentioned on Page 916 Problem 7 of Halmos's "Ten Problems in Hilbert space" that there is a proof for "Every compact quasinilpotent operator is the limit of nilpotent ones" ...
3 votes
1 answer
558 views

Conditions for chain rule for Gateaux derivatives

Let $X,Y,Z$ be locally convex topological vector spaces over $\mathbb R$ (not necessarily Banach), $D_X \subseteq X$, $D_Y \subseteq Y$ and let $f \colon D_X \to D_Y$, $g \colon D_Y \to Z$. Let us ...
2 votes
1 answer
351 views

References on duality of fractional order Sobolev spaces

I would like to ask you for any good references regarding fractional order Sobolev spaces. I know Hitchhiker's guide to the fractional Sobolev spaces is a very popular one, and I found it to be quite ...
9 votes
0 answers
210 views

Why and how is a representation "continuously decomposable"?

What I am asking may apply to a much more general setting and I am interested in the underlying level of generality of these statements, mostly with canonical references. However I state the question ...
3 votes
1 answer
577 views

A constant ratio of integrals? Part II

This question is a follow up on my latest MO post which was addressed kindly by Iosif Pinelis. What is new here is that I need to correct the assumption by including a missing hypothesis. The context ...
4 votes
1 answer
379 views

A constant ratio of integrals? Part I

Let $u(x)$ be a harmonic polynomial in the unit ball $B_1(0)\subset\mathbb{R}^n$ with $u(0)=0$. For $0<r\leq1$, consider the average of its Dirichlet integral $$A(r):=\frac1{\vert B_r(0)\vert}\int_{...
5 votes
1 answer
630 views

Uniqueness of Kantorovich potentials?

$\newcommand{\R}{\mathbb R}$Take $\Omega\subset \R^d$ bounded, convex, and smooth. Consider the optimal transport problem with cost $c(x,y)=\lvert x-y\rvert^2$, leading to the quadratic Wassersein ...

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