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1 vote
0 answers
78 views

Trace theorem for $L^2([0,1]; H^k(S^2))$

Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer. Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
2 votes
0 answers
75 views

Regularity of solutions to an elliptic boundary value problem

Let $M = [1,\infty)\times S^2$. For an integer $k \geq 2$ and number $\tau<0$, define the space $L^2_{\tau}([1,\infty);H^k(S^2))$ to be all $H^k(S^2)$-valued functions $u$ on $[1,\infty)$ with $\...
0 votes
0 answers
22 views

Approximation of Lipschitz functions by convex combinaison of Lipschitz functions depending on projections

Let $K\subset \mathbb R^2$ be compact. For any $c>0$, denote by ${\rm Lip}_c(K)$ the collection of Lipschitz functions $f:K\to\mathbb R$ whose Lipschitz constant is less than or equal to $c$. Set $...
4 votes
0 answers
158 views

Measurability of $L^{p}(L^{q})$ integrable functions

Let $ F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ be a function with the property that $ \int_{\mathbb{R}^n} \big[ \int_0^\infty |F(x,r) |^q \, dr \big]^{p/q} \, dx < \infty $ In addition we ...
7 votes
2 answers
2k views

Method of characteristics for higher order PDEs in more than two variables

I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
6 votes
1 answer
528 views

A functional equation

I am working on some physics problem and got stuck with the following equation: Let $a$ be a very small positive number. Is there a bounded function $F$, $0 \leq F \leq 1$, such that for all $x \in \...
0 votes
0 answers
53 views

Vectors of complex exponentials span $\mathbf{C}^N$

Let $Q = [0,1]\times [0,1]$. Let $1\leq k < \infty$ and $\{(x_l,\xi_l)\}_{l=1}^{k}\subseteq Q$ be such that $(x_i,\xi_i)\ne (x_j, \xi_j)$ for $i\ne j$. Additionally, for $1\leq l \leq k$, let $n_l\...
11 votes
1 answer
676 views

Entropy arguments used by Jean Bourgain

My question comes from understanding a probabilistic inequality in Bourgain's paper on Erdős simiarilty problem: Construction of sets of positive measure not containing an affine image of a given ...
1 vote
1 answer
121 views

An asymptotic integral with complex phase

Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds $$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
2 votes
1 answer
197 views

Prékopa-Leindler style inequality?

Does anyone know a simple proof of the following Prékopa-Leindler style inequality: If we have $f_1,f_2,g_1,g_2$ strictly positive functions on $\mathbb{R}$ such that, for any $x_1,x_2 \in \mathbb{R}$,...
0 votes
1 answer
53 views

Rate of convergence of the minimum point over a product space

Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that $f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$. $f(\theta, \epsilon) > 0$ for ...
0 votes
1 answer
77 views

Decay rate of minimum point over a product space

Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that $f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$. $f(\theta, \epsilon) > 0$ for ...
-1 votes
1 answer
139 views

$L^1$ convergence

Setting For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
0 votes
0 answers
57 views

Projection measure and an integral formula for Lipschitz functions

Let $n\geq m\geq 0$ be integers and put $k=n-m$. Let $A\subset\mathbb{R}^n$ be Borel measurable, we define the projection measure of $A$ as $$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
0 votes
1 answer
102 views

On weighted Fourier transforms

Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that $$ \left|\int_0^1 e^{i \xi x} e^{-|\xi|^{-1}x}f(x)\,dx \right| \leq c_1 e^{-c_2|\xi|} \quad \forall\, |\xi|>1.$$ ...
2 votes
1 answer
286 views

Are these conditions regarding products of consecutive terms in a sequence of positive numbers equivalent?

Assume $w_n$ is a bounded (weight) sequence of positive numbers. We want to consider products of consecutive terms in this sequence. For $i,j\in \mathbb{N}$, define $M_i^j = w_i w_{i+1}\cdots w_{i+j-1}...
1 vote
0 answers
128 views

Sum of upper semi continuous and lower semi continuous functions

Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. ...
0 votes
1 answer
127 views

Continuous extensions of tangent vector fields

Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
0 votes
1 answer
80 views

Continuous modification of tangent vector fields

Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
2 votes
0 answers
138 views

Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?

The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm $$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$ The space $L^2([a,b]\times S^2)$ ...
2 votes
1 answer
321 views

A strange functional inequality

Let $f,g \in C([-2,2],\mathbb R_+^*)$ even and concave real functions. Is it true that $$ \int_0^1 f\big(\cos(x^{-1})+\sin(x^{-1})\big) \cdot g\big(\cos(x^{-1})-\sin(x^{-1})\big) \mathrm{d}x\\ \leq f(...
0 votes
0 answers
44 views

Are there probability densities $\rho, f_n$ such that $\lim_n \frac{[\rho * f_n]_\alpha}{\|\rho * f_n\|_\infty} = \infty$?

We fix $\alpha \in (0, 1)$. Let $[f]_\alpha$ be the best $\alpha$-Hölder constant of $f: \mathbb R^d \to \mathbb R^k \otimes \mathbb R^m$, i.e., $[f]_\alpha := \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x-...
4 votes
2 answers
360 views

Functions with asymmetrically decreasing Fourier transform?

$\def\ii{{\rm i}}\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\bbNo{\mathbb N_0}\def\Fou{\mathscr F}$Specifically, I would like to have a compactly supported continuous function $f=u+\ii\,v:\bbR\to\bbC$ ...
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 (...
1 vote
1 answer
57 views

Lower bound the best $\alpha$-Hölder constant of a convolution

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 (...
5 votes
1 answer
246 views

An asymmetric quadrilinear estimate

Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$ where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+...
0 votes
0 answers
112 views

Characterization for the multipliers of Schwartz space

Is the following true? A function $m:\mathbb R^n\to\mathbb C$ is a Schwartz multiplier (i.e. $[f\mapsto mf]:S(\mathbb R^n)\to S(\mathbb R^n)$ is bounded linear) iff the following: For every $\alpha$ ...
9 votes
1 answer
428 views

The cardinality of projections of subsets of the Hilbert cube by inner products

I have three related questions. Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
2 votes
1 answer
183 views

Example of a conditionally convergent series $\sum_{n=1}^\infty b_n$ such that $n^2(b_n-b_{n+1})$ is bounded

Let $(b_n)_{n \in \mathbb{N}}$ be a real sequence such that $(nb_n)$ is bounded. I know that if the series $\sum_{n=1}^\infty b_n$ is conditionally convergent, then $(n^2b_n)_n$ is not bounded. But, ...
0 votes
1 answer
117 views

How to understand the unique continuation result

Let $E$ be the closure of $C_c^{\infty}\left(\mathbb{R}^N\right)$ ($N \geqq 3)$ under the norm $$ \|u\|_E=\left(\int_{\mathbb{R}^N}|\nabla u|^2\right)^{1 / 2}. $$ Suppose $K(x) \in C^1\left(\mathbf{R}^...
2 votes
1 answer
112 views

On compactly supported functions with prescribed sparse coordinates

Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. ...
4 votes
1 answer
188 views

Bound in terms of harmonic oscillator

I wonder if the following is true: Let $\alpha >0$ be a positive real number, do we have $$\Vert H^{\alpha} \psi''\Vert \le \Vert H^{\alpha+1} \psi\Vert,$$ where $H = -\frac{d^2}{dx^2} + x^2$ is ...
1 vote
0 answers
43 views

If a weighted Laplacian's eigenfunction is zero in an open set, when is it identically zero?

Let $m, s \in ([0, 1]^d \rightarrow \mathbb{R}_{\geq 0}$). Define a weighted Laplacian $\Delta_{m, s}f$ evaluated at $x \in [0, 1]^d$ to be: $m(x) \cdot \text{div} ( s(x) \nabla f(x))$. What ...
5 votes
1 answer
351 views

Does the Poincaré inequality hold on annular domains?

Does the following Poincaré inequality hold $$\int_{B_{r_2}\setminus B_{r_1}} |f-\bar{f}|^2 dx \leq C(r_2-r_1)^2 \int_{B_{r_2}\setminus B_{r_1}} |\nabla f|^2 dx,$$ where $B_r$ denotes a ball of radius ...
0 votes
1 answer
121 views

A simple bilinear estimate

Let $2\leq p,q <\infty$ and fix $0<\alpha<1$ such that $\frac{1}{p}+\frac{1}{q}\leq 2-\alpha$. Suppose that $f\in L^{p}([0,1])$ and $g\in L^{q}([0,1])$. What is the optimal value of $t=t(\...
2 votes
1 answer
320 views

Fourier series but different waveform

Given a nondegenerate smooth simple closed convex curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$, and $f$ have zero mean. Let $f_n: [0,2\pi]\to \mathbb C \...
3 votes
2 answers
392 views

Monotonicity of matrix conjugation

Let $A$ and $B$ be positive-definite matrices such that $A \le B.$ By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$ I am now curious under ...
0 votes
0 answers
80 views

Verifying the Cauchy behavior of a sequence

Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
2 votes
1 answer
128 views

On the existence of a complicated fractal-like set of finite perimeter

Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
2 votes
2 answers
235 views

$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?

Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has $$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
1 vote
1 answer
133 views

A question about the maximal function

Let $n>4$, $f\in C^{\infty}(\mathbb{R}^{n})$ and 0 denote the origin of $\mathbb{R}^{n}$. We define a weighted maximal function by $$Mf(x)=\sup_{0<r<1}r^{4-n}\int_{B_{r}(x)}|f|$$ which is ...
0 votes
0 answers
28 views

Metric entropy of mixed norm spaces with exponent-free bounds

Suppose $\mathcal{F}\subset L^p([0,1]^d)$ is a subset with the following property: The $L^q$-covering number of $\mathcal{F}$ is independent of $q$, for all $1\le q\le\infty$. An example of $\mathcal{...
3 votes
0 answers
84 views

About the naturality of Krasnoselskii genus on Variational Methods

I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
0 votes
3 answers
278 views

A generalisation of Tchebychev inequality

Let $f,g \in C(\mathbb R)$ with $\exists M \in \mathbb R^*, \forall (x,y) \in \mathbb R^2, M\times (f(x)-f(y))(g(x)-g(y)) \geq 0$. Is it true that exists $ u$ any real function, and $a,b$ monotone ...
2 votes
1 answer
272 views

Decompose a function into a bounded part and a Lipschitz part

Let $f: \mathbb R^d \to \mathbb R^d$ be a measurable function such that $$ \sup_{x,y \in \mathbb R^d} \frac{|f(x) - f(y)|}{\max \{1, |x-y| \}} < \infty. $$ Are there functions $g,h: \mathbb R^d \...
2 votes
2 answers
197 views

$L^p$ domination of mixed partial derivatives by the unmixed ones?

Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has $$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
2 votes
1 answer
154 views

Are these two norms on localized versions of $L^p_q$ equivalent?

$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$. Let $E$ be the space of all real-valued ...
4 votes
1 answer
279 views

Schroedinger operator in 2 dimensions with singular potential

Consider the Schroedinger operator $$H = -\Delta + \frac{c}{\vert x \vert^2}$$ in two dimensions with $c >0$ This operator has a self-adjoint realization, since it is a positive symmetric operator ...
0 votes
1 answer
139 views

Singular integral bounded by Dirichlet form?

We define for some fixed $L$ $$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$ in particular $x_1,x_2 \in \mathbb R^2.$ Let $f \in C_c^{\infty}(\Omega)$, then I am ...
2 votes
1 answer
140 views

Invertibility of message passing with invertible parametrization

Consider the message passing framework defined by, $$f(\boldsymbol{x}_i)= \boldsymbol{x}_i + \sum_{j \neq i} (\boldsymbol{x}_i -\boldsymbol{x}_j) g(\|\boldsymbol{x}_i -\boldsymbol{x}_j\|^2),$$ for $i=...

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