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3 votes
2 answers
247 views

Morse approximation with bounded number of critical points

Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-...
3 votes
0 answers
158 views

Gowers' dichotomy for quotients

Gowers' dichotomy establishes that every infinite dimensional Banach space contains a closed infinite dimensional subspace that has an unconditional basis or it is hereditarily indecomposable. A ...
1 vote
0 answers
84 views

Does sets of positive capacity rule out constant functions?

Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by \begin{align*} \text{Cap}_{p}(K, U) := \inf \left\{ \int_U |\...
1 vote
1 answer
142 views

Operator norm of some type of discrete Fourier matrix

Let $N$ be a natural number and let $w$ be a complex number. We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows, $$ a_{k,l}=\begin{cases}1 & l=k+1\\ w &...
3 votes
1 answer
128 views

Comparing two different principles of premeasure-to-measure extension

It is well-known that a premeasure $\mu_0$ (possibly taking infinite values) on a ring of subsets $\Omega_0$ of a set $X$ can be extended to a complete measure space $(X, \Omega_C, \mu_C)$ ($C$ for ...
2 votes
0 answers
194 views

Functions such that the *integral* of the Fourier transform is non-negative?

Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that $$\int_{-\infty}^x \widehat{f}(t) dt \...
0 votes
1 answer
170 views

Is the evolution family self-adjoint?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \newcommand{\qtext}[1]{\quad\text{#1}} \newcommand{\qtextq}[1]{\quad\text{#1}\quad} $ I am reading Roland Schnaubelt's survey ...
1 vote
1 answer
59 views

Embedding of domain of fractional power of Laplacian into Sobolev space for cylindrical domains

On a bounded domain $\Omega \subset \mathbb R^d, d\geq 2$ with smooth boundary, it is well known that for the Dirichlet Laplacian $\Delta_D$, $D((-\Delta_D)^\frac12) = H^1_0(\Omega)$. I'm interested ...
2 votes
0 answers
52 views

On distributions and kernels

Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
3 votes
0 answers
155 views

Colimits in commutative Banach algebras?

Let $K$ be a complete non-Archimedean field. It is known that the category $\mathrm{Ban}_K$ of $K$-Banach spaces with bounded linear maps does not have infinite colimits. The usual argument for $\...
3 votes
1 answer
158 views

Upper and lower bounds for a Rademacher-type expectation

Suppose that $\varepsilon_i$ are independent Rademacher random variables (that is, $ \mathbb{P}(\varepsilon_i=-1) = \mathbb{P}(\varepsilon_i=1) =1/2 $. Fix an $a\in\mathbb{R}^n$ and define the random ...
5 votes
1 answer
206 views

Compactness in trace class operators space

Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$. Are there easy ...
20 votes
2 answers
2k views

Sobolev and Poincaré inequalities on compact Riemannian manifolds

Let $M$ be an $n$-dimensional compact Riemannian manifold without boundary and $B(r)$ a geodesic ball of radius $r$. Then for $u\in W^{1,p}(B(r))$, the Poincare and Sobolev–Poincaré inequalities are ...
0 votes
0 answers
55 views

Compactness and Leray-Schauder degree

What's the relationship between compactness of solutions in partial differential equations (PDEs) and the Leray-Schauder degree?
0 votes
1 answer
188 views

Does the second Bourgain–Delbaen space belong to C_p?

The second Bourgain–Delbaen space $Y$ is a separable $\mathcal{L}_\infty$ space such that $Y$ contains no copy of $c_0$ or $\ell_1$. The dual space is isomorphic to $\ell_1$. An operator $T:X\to Y$ ...
3 votes
0 answers
130 views

A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)

As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
0 votes
1 answer
65 views

Strict positive definite function gradient tuple

I have a (Gaussian) random function (aka "stochastic process" or "random field") $(f(t))_{t\in \mathbb{R}^d}$. I now want to consider the vector valued random function $g=(f, \...
25 votes
2 answers
2k views

Functional approach vs jet approach to Lagrangian field theory

Context: I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts ...
0 votes
0 answers
113 views

Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets?

Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$. We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if $$\forall_{u \in A(\...
3 votes
1 answer
189 views

Ribe's Theorem: finitely representability between two uniformly homeomorphic Banach spaces

An infinite-dimensional Banach space $X$ is said to be crudely finitely representable (with constant $\lambda$) in an infinite-dimensional Banach space $Y$ if there is a constant $\lambda>1$ such ...
2 votes
0 answers
70 views

Is the hypothesis "uniformly equivalent" needed?

I am reading S. Shimorin's paper titled Complete Nevanlinna-Pick property of Dirichlet-type spaces. My question concerns Lemma 2.3. which is as follows: Assume $\mathscr{H}$ is a Hilbert space of ...
5 votes
1 answer
630 views

Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution

Examples of infinite dimensional involutions Edit 2/25/23, as suggested by YCOR below: (Start) The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
1 vote
0 answers
111 views

References on the partial trace

For the Hilbert space $H^N:=L((\mathbb R^{3})^N,\mathbb C)$, consider the projection operator $D: H^N\to H^N$ as follows : $$D(\Phi):=\left(\int_{(\mathbb R^{3})^N}\overline{\Psi(x_1,\ldots, x_N)}\Phi(...
0 votes
1 answer
119 views

Nonstationary phase method for oscillatory integral

I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth. The stationary phase method says that if $t_0\in [a,b]$ is such that ...
3 votes
0 answers
95 views

Commutator of $A\otimes I$ and $I \otimes B$ vanishes?

Consider two Hilbert spaces $H_1$ and $H_2$, and $A$, $B$ unbounded operators on $H_1$, $H_2$ respectively. $(A \otimes I)$ is classically defined as the closure of the operator defined on the set of ...
6 votes
0 answers
213 views

Hölder's inequality for trace-class maps of $p$-liquid spaces and a related conjecture of Grothendieck

In Condensed Math and Complex Geometry Proposition 8.8, Clausen-Scholze describe trace-class maps between projective objects in the $p$-liquid category as sums of rank 1 operators against ${<}p$-...
10 votes
1 answer
2k views

Chain rule for distributional derivative

Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$). Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...
2 votes
0 answers
43 views

Distributions and time-kernels

Let $U\subset\mathbb{R}^{d}$ be an open subset and set $M:=I\times U$, where $I=(a,b)\subset\mathbb{R}$ is some open subset. Lets consider a linear operator $B:C^{\infty}_{c}(M)\to C^{\infty}(M)$ that ...
-1 votes
0 answers
53 views

convergence of convolution in Bochner space

I want to prove a well-known fact in $L^p(R^n)$ namely that, the convolution of an element in $L^p$ with an element of $L^1$ is in $L^p$ let: if $u∈L^p (R;X) , f∈L^1 (R)$ and $X$ is Separable and ...
-1 votes
0 answers
94 views

Why define Schwartz by supremum rather than limit?

The Schwartz space is defined as the set of all indefinitely differentiable functions such that the supremum over the free variable of any (order) derivative times any (order) power is finite. However,...
9 votes
5 answers
2k views

Convexity of distance-to-boundary function

Let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded convex domain. Denote $d_{\Omega}:\Omega\rightarrow\mathbb{R}$ the distance-to-boundary function, that is, $$ d_{\Omega}\left(x\right):=\inf\left\...
11 votes
7 answers
1k views

What are some interesting ways of making new metrics out of old metrics?

If $d(x,y)$ and $e(x,y)$ are metrics then $d(x,y)+e(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$ are metrics. If $d_i(x,y)$ for $i=1,\dots,n$ are metrics then so is $\sqrt{\sum_{i=1}^n{d_i^2(x,y)}}$ Are ...
1 vote
1 answer
90 views

Sobolev inequality with weight in the case $1<n\leq p$

Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in ...
21 votes
7 answers
2k views

Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
2 votes
1 answer
316 views

Periodic solution for linear parabolic equation - existence, regularity

I am interested in proving the existence and regularity of solution to the following problem: $$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-\Delta y(t,x)+c(t,x)y(t,x)=f(t,x), & (t,x)\in (0,T)...
3 votes
1 answer
307 views

Approximate square root of Dirac delta function on $\mathrm{SL}_2(\mathbb{R})$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\AdS{AdS}$I hope to find a sequence of complex-valued functions $\{f_i(g)\}$ on the group element $g$ of a locally compact group $\SL(2,\mathbb{R})$ so ...
3 votes
1 answer
198 views

Can gradient zero implies that a function is constant with Hörmander vector fields

Let $X=(X_1,\cdots,X_m)$ be a system of Hörmander vector fields defined on $\mathbb{R}^n$. The Sobolev space $W_{X}^{1,p}(\Omega)$ is defined by $$W_{X}^{1,p}(\Omega):=\{u\in L^p(\Omega)|X_iu\in L^p(\...
2 votes
1 answer
2k views

Monge–Ampère operator

I'm studying the article of Bedford–Taylor "Fine topology, Šilov boundary…" but I don't understand the proof of the following proposition. Let $u$, $v$ be plurisubharmonic functions defined ...
1 vote
1 answer
117 views

Lower bound for a commutator trace

I have this Hilbert space of square-integrable complex-valued functions on a square, $\mathbb{L}^2([0,1]^2)$. And let $M_x$, $M_y$, and $M_{x+y} = M_x+M_y$ be the operators of multiplication by the ...
3 votes
1 answer
1k views

Borel-Cantelli lemma for general measure spaces (those with infinite measure)

The Borel-Cantelli lemma is often stated for a probability space or spaces with finite measure. But it seems to me that it still holds if the space $X$ is of infinite measure. I seem to be able to ...
16 votes
1 answer
970 views

Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis

While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
0 votes
1 answer
169 views

Existence of a "universal" measure-preserving transformation on the unit interval

Let $I = [0,1]$ be the unit interval equipped with the Lebesgue measure $\lambda$. Let $\mathcal{M}$ be the set of all Lebesgue measure-preserving transformations $T: I \to I$. We say a transformation ...
3 votes
2 answers
154 views

On nowhere differentiability of functions that just barely fail to be Lipschitz

By Rademacher’s theorem, Lipschitz functions are differentiable almost everywhere. I am wondering how badly this pointwise differentiability fails for functions that “just barely” fail to be Lipschitz....
4 votes
1 answer
227 views

Problem in Probability Theory and Functional Analysis

Let's consider the vector space V of bounded scalar functions, which includes the constant function 1. We assume that any uniform limit of a bounded monotonic sequence of functions from V also ...
7 votes
2 answers
394 views

Tangent space to infinite dimensional manifolds

In finite dimensional geometry, there is a single invariant of a vector space - its dimension. This characterizes finite dimensional manifolds as being glued from Euclidean balls. This situation is ...
1 vote
1 answer
329 views

Hölder continuity of Radon transform of smooth function

Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by $$ R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
0 votes
0 answers
146 views

On the pointwise limit of a sequence of analytic functions

I have been confused with this problem for weeks now. Suppose I have Banach spaces $E$ and $F$ and a sequence of functions $f_{n}: U \subset E \to F$, where $U$ is open and nonempty. Let $x \in U$ be ...
0 votes
1 answer
431 views

Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$

$\def\zbar{\smash{\overline z}\vphantom z}$I would like to construct an approximation of the product \begin{equation} f(z) = \frac{1}{\zbar-a} \frac{1}{z-b}, \end{equation} where $a, b \in \mathbb{C}$,...
1 vote
2 answers
117 views

If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $g\geq f$ on $[0,\delta]$ and $g(0)=f(0)$ exist?

The question is the following: Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...
0 votes
0 answers
55 views

reference request: conditions for pointwise and operator-norm convergence of kernel projections

At a very high level, I’m interested in the following question. Suppose $X$ is a (separable) Hilbert space, and $T_n : X \rightarrow X$ is a sequence of finite rank self-adjoint maps that converges (...