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3 votes
1 answer
353 views

Sequential separability on $C_p(X)$

Definition. Let $E$ be a topological space. Suppose that $E$ contains a sequence $\{x_n\}$ such that for every $x\in E$, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ with $x=\lim x_{n_k}$. ...
4 votes
1 answer
274 views

Eigenvalue of a convolution and a restriction?

Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
0 votes
1 answer
115 views

Bounding $(\int_{S^1}\left|\partial_r u(r\omega)\right|^2 d\omega)^{1/2}$ with $(\iint \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy)^{1/2} $?

The following inequality is trivially true $$\left(\int_{S^1}\left|\frac{\partial u}{\partial r}(r\omega)\right|^2 d\omega\right)^{1/2} \le \left(\int_{S^1}\left|\nabla u(r\omega)\right|^2 d\omega\...
3 votes
0 answers
91 views

About BMO space on smooth open bounded domain

Let $\Omega$ be any open domain in $\Bbb R^d$. Define the $\text{BMO}(\Omega)$ space as $$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\}, $$ ...
2 votes
0 answers
225 views

A deceptively simple regularity problem for functions on the plane

By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer: Consider a twice ...
2 votes
0 answers
77 views

Function that is (essentially) a self-convolution but not a multiple of a self-convolution

Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
4 votes
2 answers
871 views

Decay of eigenfunctions for Laplacian

Consider the discrete second derivative with Dirichlet boundary conditions on $\mathbb C^n$. Its eigendecomposition is fully known: see wikipedia It seems like the largest eigenvalue $\lambda_1$ is ...
0 votes
1 answer
124 views

Holomorphic functions of certain blow up at origin

Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
1 vote
0 answers
146 views

integral over the unit sphere of $\Bbb C^n$

Please, is there a way to calculate this integral $$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$ where $ z $ is a fixed point in the complex unit ball ...
3 votes
2 answers
352 views

General version of Weyl's lemma

The classical Weyl's lemma say, suppose $u \in L^1_{loc}(\Omega­)$ satisfies $$\int_{\Omega}u \Delta \phi dx=0\ \ \forall \phi\in C_c^{\infty}(\Omega),$$ then $u$ is harmonic in $\Omega.$ What I want ...
0 votes
0 answers
53 views

Spectral theory of compact operator for quasi-Banach spaces

Let $X$ be a Banach space and let $Y\subset X$ be a quasi-Banach space (with compact inclusion). Suppose $T:X\to X$ is a compact operator such that $1$ is not its eigenvalue and $T|_{Y}:Y\to Y$ is ...
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 \...
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:...
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?
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 ...
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 ...
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\...
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 ...
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(\...
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
1 answer
86 views

Lattice of functions and their minimal separating set upto topological equivalence

There is a very wide series of questions I have been thinking about and I am wondering if there is any literature on this type of structures. Let's start with the set of all functions $F: \mathbb{R} \...
2 votes
0 answers
90 views

Representation of Dirac-delta distribution in subspace of functions

Suppose I have a subspace $V\subset L^2(\Omega)$ where $\Omega\subset \mathbb{R}^d$ is a bounded and closed set. $V$ is defined by \begin{align} V=\text{span}(\{\varphi_i(x): i=1,2,\dots,n\}) \end{...
0 votes
0 answers
57 views

Double-periodic functions with (possible) poles

Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
1 vote
1 answer
157 views

Is finding the CDF from the Laplace transform well-posed?

In my study of Dynamic Light Scattering, I came across the following inverse problem. Let $F(s):[0,T]\rightarrow[0,T]$ be the Laplace transform of a probability distribution $f(t)$ on the real line ...
1 vote
0 answers
45 views

Existence of optimal entropic weights for empirical modeling

Let $\mathcal{X} = [0,1]^n$ be the input space and $\mathcal{Y} = \{1, ..., n_c\}$ be a discrete output space. Let $D = \{(x_i, y_i)\}_{i=1}^N \subset \mathcal{X} \times \mathcal{Y}$ be a training ...
0 votes
0 answers
68 views

Family of separable Hilbert spaces over locally compact form a continuous field of Hilbert space?

Let $\{H_{x}\}_{x\in G^{0}}$ be a family of separable Hilbert spaces and $G^{0}$ be a locally compact second countable topological space. Let $\mathbb{B}_{x}$ be the orthonormal basis of $H_{x}$. If ...
3 votes
1 answer
220 views

What we know about the function in Fefferman's Theorem

In Fefferman's many papers on Whitney's theorem he, amongst other things, constructs the existence of a smooth function $F$ which extends a function $f$ on a (say) finite set $E\subseteq \mathbb{R}^n$ ...
2 votes
0 answers
189 views

Smoothing property of the heat kernel on the one-dimensional torus

Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation} G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
2 votes
2 answers
154 views

Closure of $C([0,1]^2)$ via weak*-topology [closed]

Let $C([0,1]^2)$ denote the set of continuous functions on $[0,1]^2$. Let $L^1([0,1]^2)$ be the set of all Lebesgue integrable functions on $[0,1]^2$. The dual space of $C([0,1]^2)$, denoted by $C^*([...
7 votes
0 answers
619 views

Lavrentiev Phenomenon

Does there exist a (onedimensional) integral functional of calculus of variations $$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt
 $$ such that not only $$ \inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
0 votes
1 answer
231 views

Questions on the compactness of $L_1([0,1]^2)$'s unit sphere

Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
0 votes
1 answer
101 views

Limit sequence of regular function in $L_1$‘s unit sphere

Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
3 votes
1 answer
490 views

Space derivative of flow of ODE with monotone source

Consider the ODE $$ \begin{cases} \partial_t\Phi(t,x) = f(t,\Phi(t,x)), &\ t>0, \ x \in \mathbb R \\ \Phi(0,x) = x, & x \in \mathbb R \end{cases} $$ where $f$ is function which is a non-...
0 votes
0 answers
121 views

Is there a good or commonly accepted short notation for the set of differentiable, but not necessarily continuously differentiable maps?

Every once in a while I find myself in need of some short notation for the set of differentiable, but not continuously differentiable maps, say, $X \to Y$. Always having to specify "...
-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 $\mathcal{X}$ ...
1 vote
1 answer
330 views

Does $\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$ converge?

I am trying to study the converge of the series $$\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$$ But $e^{\sin{n}}$ is not monotone, and the Abel's test rule fails here. Can someone help me? ...
1 vote
1 answer
127 views

approximating differentiable functions with double trigonometric polynomials

Let $Q = [0,1]^2$. For sake of notation, let $$ f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi). $$ Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if $$ \|...
1 vote
1 answer
130 views

Existence of solutions to a series of integral equations

I am trying to solve the following integral equation analytically: $$ \sum_{n \geq 1} \left( \int_0^te^{-n^2(t-s)} f_n(s) \, ds \right) = g(t), \quad t \in [0, T], $$ where $(f_n(t))_n$ is the unknown ...
0 votes
0 answers
100 views

Construct a bi-Lipschitz mapping that maps a cube to a ball which has the same center with the cube

A mapping $f: \mathbb{R}^n\to \mathbb{R}^n$ is said to be $K$-bi-Lipschitz, $K>1$, if \begin{equation*} \dfrac{1}{K}\leqslant \dfrac{|f(x)-f(y)|}{|x-y|}\leqslant K, \end{equation*} for any $x,y\in \...
2 votes
1 answer
474 views

Polynomial $f(x)$ has positive coefficients and only real roots. How many polynomials formed from terms of $f(x)$ also have only real roots?

Let $$f(x)=a_n \ x^n+a_{n-1} \ x^{n-1}+\cdots+a_1 \ x+a_0$$ be a $n$-th degree polynomial with positive coefficients such that all of its roots are real. Choose any number terms from this expression ($...
10 votes
1 answer
518 views

Inverse function theorem for $W^{2,n}\cap W^{1,\infty}$ functions

Let $n\ge 2$, $f:B_1\subset \mathbb R^n\rightarrow \mathbb R^n$, $f\in W^{2,n}\cap W^{1,\infty}(B_1)$, $\text{det}(Df)>c>0$, where $B_1$ is the unit ball. Can we show that $f$ is a homeomorphism ...
3 votes
1 answer
187 views

Is this property preserved under weak$^*$ convergence?

Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
7 votes
0 answers
249 views

Proving this function is convex

Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
2 votes
0 answers
120 views

On mollifiers acting between $L^2$ and Sobolev spaces

(I'm reposting here this question from MSE as it didn't receive any answer for two weeks.) Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by $$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{...
2 votes
1 answer
128 views

Density of smooth functions in weighted Sobolev space

Let $\rho(x)=e^{-\phi(x)}$, where $\phi$ is an even polynomial with positive leading coefficient. I am interested in a proof of the fact that the space of smooth compactly supported functions $\...
4 votes
0 answers
140 views

Condition under a function is uniquely identifiable by the supremum values

Let $f(x),g(x)$ be two real-valued functions on $\mathbb{R}$ and $h(x,y)$ be a real-valued function on the plane. We can assume continuity (maybe piecewise differentiability also) of these functions. ...
0 votes
0 answers
84 views

Question on approximation of norms

Suppose that $E\in Int[L_{p},L_{q}]$ for some $1<p<q<\infty$ and $E$ is $w$-concave with $1<w<\infty$. It is well-known that for each $r\geq w$, we have $E=L_{r}\odot F_{r}$ for some ...
0 votes
0 answers
36 views

Derivate involving Bessel function of second type

Let. $$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$ Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
2 votes
0 answers
57 views

Mappings that preserve local or global minimum

In the most general form, I'm interested in any non-trivial results of the following question. Consider metric space $X$ and $Y$, denote all real valued functions on $X$ and $Y$ as $\mathbb{R}^{X}$ ...
3 votes
1 answer
176 views

Question about Lebesgue Bochner spaces

Let $T>0$ and $\Omega\subset\mathbb{R}^N$ be a bounded domain. Also $p\in (1,\infty)$ is any number. I know that $u\in L^{p}((0,T);L^p(\Omega))$ and $\nabla u\in L^{p}((0,T);L^p(\Omega))^N$. How ...

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