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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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0answers
20 views

Example of function belongs to Orlicz space but not in Lebesgue spaces

Can you give me please an example of function belongs to Orlicz spaces but not in Lebesgue spaces??
1
vote
0answers
44 views

inverse of sobolev riemannian metric still sobolev?

Given a covariant riemannian metric of certain sobolev class (i.e. with square-integrable weak derivatives up to a large enough integer order) on a (compact, if necessary) finite-dimensional smooth ...
2
votes
0answers
29 views

Continuous Local Martingales under time change under what conditions are they still local martingales?

This question is motivated by reading a section in Continuous Martingales and Brownian Motion by Daniel Revuz, Marc Yor. In Chapter V there is a section on time-change: Definition: A time change $C$...
8
votes
2answers
252 views

Hölder continuity for operators

Let $x,y$ be positive real numbers then $$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$ we obtain $1/...
1
vote
0answers
59 views

Does this chain rule in Sobolev spaces hold?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded set. Let $S \subseteq \mathbb{R}^k$ be an open dense smooth submanifold of $\mathbb{R}^k$. Let $u \in W^{1,p}(\Omega,\mathbb{R}^k) \cap C(\...
1
vote
0answers
113 views

A problem on integrability of derivatives

Let $$f : (0,1) \to \mathbb{R}$$ and $$g(x) = |f(x)|^{r-1} f(x)$$$r \in \mathbb{N}$. It is known that $g\in \mathcal{L}^2(0,1)$ and the $r^{th}$ weak derivative, $ g^{(r)} \in \mathcal{L}^2(0,1)$. I ...
1
vote
1answer
124 views

A simple two variable analytic inequality, inspired by probability

I'm trying to prove the following inequality: $$ bf_1g_1 + (x-b)f_1g_0 + (y-b)f_0g_1 + (1-x-y+b)g_0f_0 \le (|f_1|^p x + |f_0|^p (1-x))^{1/p} (|g_1|^p y + |g_0|^p (1-y))^{1/p} $$ where $0\le xy\le b\le ...
0
votes
0answers
33 views

A variant of the optimal transport

Let $\mu$, $\nu$ and $\gamma$ be three probability measures on $\mathbb R$. Consider the optimisation problem as follows: $$\inf_{(X,Y,Z)}~ \mathbb E\big[|Y-Z|^2\big],$$ where the inf is taken ...
1
vote
0answers
234 views

Is that correct $\mathbb R^2\cong\mathbb R$ as measurable spaces? [closed]

Is that correct $R^2\cong R$ as measurable spaces? If we consider $R$ and $R^2$ with Borel $\sigma$-algebras, is there measurable map from $R$ to $R^2$ with measurable inverse?
2
votes
1answer
185 views

Schwartz space on $\bigcup_{n=1}^CR^n$

I have an application where I need to work with the following idea. Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...
4
votes
2answers
112 views

Can we stay invertible while approximating linear maps in Sobolev spaces?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$. Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...
3
votes
0answers
30 views

Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)

Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail. I ...
1
vote
1answer
89 views

Giving Uniform Bound on Differences of Sums of Converging Polynomials

The title does not quite capture the essence of the difficulty, please allow me to be more explicit here. I thought of this question when I was trying out an open problem by Ovidiu Furdui(See problem ...
3
votes
0answers
50 views

Constant in trace theorem for balls

Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$ The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
1
vote
0answers
55 views

Nonlinear maps in Riesz Thorin theorem

The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear. What I was wondering about is whether this is because otherwise you do ...
7
votes
1answer
174 views

Proof of Green's formula for rectifiable Jordan curves

$\newcommand{\Ga}{\Gamma}$ I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...
11
votes
1answer
365 views

Nonlinear Schrödinger equation with discrete Laplacian

In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
5
votes
2answers
364 views

Existence of Solution, System of Equations

Suppose $P(\lambda, i)$ is the probability that a Poisson random variable with average $\lambda$ is equal to $i$, i.e. $\frac{\lambda^i}{e^{\lambda}i!}$ I think the following system of equations ...
11
votes
2answers
215 views

Semigroup of differentiable functions on real line

Let $D(\mathbb R) $ be the set of all differentiable functions $f: \mathbb R \to \mathbb R$. Then obviously $D(\mathbb R)$ forms a semigroup under usual function composition. Can we characterize (up ...
-1
votes
1answer
123 views

How to choose compactly supported smooth $h$ so $h^2(x)+ h^2(x-1)=1$ for all $x\in [0,1],$ and $\int_{-3/4}^{3/4} |h(x)|^2 dx =3/2$? [closed]

It is known that we may choose smooth $f:\mathbb R \to [0,1]$ such that $f(x)=1$ if $x\geq \frac{3}{4} $ and $f(x)=0$ if $x\leq -\frac{3}{4}+1.$ Define $h(x)= \sin (\frac{\pi}{2} f(x+1))$ if $x\...
2
votes
0answers
37 views

Dependency of the Wasserstein distance on the parameter: a differential perspective

Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$. Consider the $2-$Wasserstein distance below: $$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...
15
votes
1answer
367 views

Can an injective $f: \Bbb{R}^m \to \Bbb{R}^n$ have a closed graph for $m>n$?

Question. Suppose $m>n$ are positive integers. Is there a one-to-one $f: \Bbb{R}^m \to \Bbb{R}^n$ such that the graph $\Gamma_f$ of $f$ is closed in $\Bbb{R}^{m+n}$? Remark 1. The answer to the ...
12
votes
2answers
992 views

The Riemann hypothesis as a problem in analysis

The recent post("Long-standing conjectures in analysis ... often turn out to be false") prompted me to think about a question which I have not given much though before: to what extent the ...
2
votes
1answer
136 views

Reference request for weak solutions of an Elliptic PDE

Edit : I just learned that all weak solutions are $C^\infty$, so this question, by Willie, seems more appropriate than the current one. I want to find weak, non trivial, continuous, solutions of $$\...
4
votes
2answers
116 views

A kind of exponential concavity for polynomials?

Is there $C > 0$ such that the inequality $$ \prod_{n\in\mathbb{N}} p(n)^{a_n} \leq p\left(C\prod_{n\in\mathbb{N}} n^{a_n}\right) $$ holds for all finitely supported sequences $(a_n)$ with $a_n\geq ...
3
votes
0answers
41 views

PDE satisfied by projection of a function onto a subspace

Given an open bounded set $D\subset \mathbb R^N$, let $f\in W^{-1,q}(D)$ and let $u$ be a Sobolev function $u\in W_0^{1,p}(D)$ such that $u$ solves the PDE $$ \begin{cases} -\Delta_p u=f\;\text{in $D$}...
1
vote
1answer
119 views

Applications of the Calderon-Zygmund theory to PDE's

I am planning to build a PDE topics course focussing on the Calderon-Zygmund theory. I know some important applications of the Calderon-Zygmund theory to elliptic PDEs, but I don't know enough to get ...
0
votes
0answers
94 views

Integer reduction of a positive semidefinite matrix

Take $\Sigma $ a real positive semidefinite matrix. Define $P$ to be the smallest projection with the property that for any $\mathbf{a}\in \mathbb{Z}^n$ with $\mathbf{a}^\dagger (I-P)^\dagger \Sigma (...
19
votes
2answers
1k views

Mathematical Evidence Backing $|\mathbb{R}|=\aleph_2$

The "true" size of the real line, $\mathbb{R}$, has been the subject of Hilbert's first problem. Due to the Goedel and Cohen's work on the inner and outer models of $\text{ZFC}$, it turned out to be ...
7
votes
2answers
194 views

Non-separable metric probability space

Let us say a metric probability space $(X,\rho,\mu)$ has property (*) if: the support of $\mu$ is contained in a separable subspace of $X$. Questions: 1. Is there a standard name for this property? ...
0
votes
0answers
66 views

Is there a standard definition for this topology setup?

There are two complete metric spaces $(A,d_A)$ and $(B,d_B)$, also $B \subset A$, so there is a third induced metric space $(B,d_A)$. There is a continuous and onto function $e:A\to B$. For any $b \in ...
1
vote
0answers
72 views

Higher Order Partial Derivatives Test

For a nonconstant analytic function $ℝ→ℝ$, a point is a local minimum iff at that point, the order of the first nonzero derivative is even and that derivative is positive. Is there an analogous test ...
34
votes
5answers
6k views

“Long-standing conjectures in analysis … often turn out to be false”

The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1 His example of a "long-standing conjecture" is the Riemann hypothesis,...
2
votes
0answers
171 views

Absence of fixed points

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$ where $x_0$ is an arbitrary but fixed ...
2
votes
1answer
172 views

Computing minimum / maximum of strange two variable funcion

I want to compute $$\max_{\frac{1}{5 \theta }\leq \alpha \leq \frac{1}{2}} \left(\frac{\alpha\log (\alpha)}{1-\alpha} + \log \left( 1 - \alpha\right) + \frac{1}{1-\alpha} \cdot \left( f\left(\frac{...
1
vote
2answers
63 views

Dirichlet problem for capillary equation over convex domain

Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary. Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function. Let $L$ be a quasilinear elliptic ...
16
votes
1answer
2k views

Differential equation changing sign almost everywhere

Conjecture: Let $f:\mathbb{R}→\mathbb{R}$ be an everywhere differentiable function and assume that $f(x)+f′(x)∈ \{-1,1\}$ almost everywhere and $f'(0)=0$. Then is $f$ necessarily a constant function? ...
3
votes
1answer
229 views

Function square-integrable

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$ where $x_0$ is an ...
2
votes
3answers
241 views

Uniqueness of solution depending on constant?

I am a physicist and I am aware that this forum is for professional mathematical questions, but please be not too hard on my notation. I encountered the following integral equation for functions $f:[...
9
votes
1answer
229 views

Current vs Varifold

I know the basic definitions concerning current and varifold, and they are generalization of submanifolds. What are their respective pros and cons? What are their crucial similarities and differences?
1
vote
1answer
49 views

Generalizations of Pedal Coordinates

I recently "stumbled upon" the article Pedal coordinates, Dark Kepler and other force problems by Petr Blaschke from 2017; further search about Pedal Coordinates didn't bring up any other relevant ...
-2
votes
1answer
131 views

A generalization of Chebyshev's sum inequality

From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference. Inequality: Let $y=f(x,y)$ is ...
0
votes
0answers
103 views

Estimation of the integral $\int_a^b e^{2\pi i f(x)} dx $

Let $f$ be a $C^2$ real-valued function on the interval $[a,b]$. Suppose that $f'(x)$ is monotone on $[a,b]$ and there is $\lambda>0$ such that $$ \min_{x\in [a,b]} |f'(x)|>\lambda $$ It is ...
16
votes
2answers
418 views

Existence of an antiderivative function on an arbitrary subset of $\mathbb{R}$

Let $f:\mathbb{R}\to \mathbb{R}$ be continuous at $x$ for every $x\in I$ where $I\subset \mathbb R$ could be arbitrary. Does there always exist a function $F:\mathbb{R}\to \mathbb{R}$ differentiable ...
0
votes
1answer
100 views

Do functions exist and are they dense? Or does it depend on the basis?

Consider an orthonormal basis $(\varphi_n)_{n \in \mathbb N}$ of $L^2(\mathbb R).$ We consider the functionals $\Phi_n$ given by $$ C^b(\mathbb R) \ni f \mapsto \left\langle \varphi_n, f \varphi_{n+1}...
1
vote
2answers
164 views

Number theory on Banach space $L^2(\mathbb R)$ meets linear independence?

Consider an orthonormal basis $(\varphi_k)$ of $L^2(\mathbb R)$ with Lebesgue measure. I came along a nice number theoretic question in analysis: Write $$f_k(x):=\int_{\left\lvert y \right\rvert \...
-2
votes
0answers
98 views

Oscillating particle motion

Suppose I have a particle with an initial position at $X_0 =0$. Its velocity is given by the following: \begin{align*} \dot{x} = \begin{cases} 1 & \text{if } x = 0 \\ -1 & \text{ otherwise }\...
1
vote
1answer
171 views

An equation in Gamma function has at most (n-1) positive solutions

I have to prove some result. And for that, I need to prove this new problem. To prove, $c_{1}\Gamma(z+b_{1})+c_{2}\Gamma(z+b_{2})+\ldots+c_{n}\Gamma(z+b_{n})=0$ has at most $(n-1)$ real positive ...
0
votes
0answers
60 views

Is $\sum_{m\in \mathbb Z} f(x-m)f(x-n) \in L^2(a,b)$ if $f\in L^2(\mathbb R)$?

Let $f\in L^2(\mathbb R)$ and $0<a<b< \infty,$ put $A= \{x\in \mathbb R: a<|x|<b\}.$ Fix $n\in \mathbb Z.$ Define for all most all $x$ $$ F_n(x)=\sum_{m\in \mathbb Z} f(x-n-m)f(x-m)$$ ...
3
votes
0answers
109 views

Function and distance on bounded set [closed]

Does there exist a bounded set $A \subset \mathbb{R}^ n$ (for some $n$) and a function $f:A\to A$ (not necessarily onto) such that $\forall x,y \in A$ then $\|x-y\| < \|f(x)-f(y)\|$? If such a ...