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Is there a dyadic cube decomposition where edge length is comparable to L^2 averages?

Suppose I have some measurable function $f : B_1(0) \to [0,1)$, which is pointwise very small, i.e. $\|f\|_{\infty} << 1$. I'm looking to construct some kind of dyadic cube decomposition or ...
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Semicontinuity of the Lebesgue measure of images of a family of functions

Let $\mu$ be the usual Lebesgue measure on $\mathbb{R}^m$. Suppose $f:\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}^m$ is uniformly continuous, and let $U$ be a measurable subset of $\mathbb{R}^n$....
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Express $\zeta(n)$, in particular the Apéry's constant, in terms of series involving Gregory coefficients or the Schröder's integral

The idea and motivation of this post is to know if it is possible to express integer arguments of the Riemann zeta function $\zeta(n)$, in particular $\zeta(3)$, in terms of series involving the so-...
user142929's user avatar
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Smooth compactly supported function with good scaling with respect to the fractional Laplacian

Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
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479 views

What are the sets on which norm-closedness implies weakly closedness?

Let $X$ be a Banach space. Let $C$ be a convex, and normed-closed subset of $X$. It is well-known that $C$ becomes weakly closed subset of $X$. I want to know is there any well-know class of non ...
Red shoes's user avatar
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159 views

Limiting property of polylogarithm ratio

I try (without success) to figure out what could be the following limit if any... For real $s$ strictly $> 1$ and $x \rightarrow +\infty$ (x real) the limit of the polylogarithm ratio $\frac{Li_{s-...
Gianfranco OLDANI's user avatar
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77 views

Energy-minimizing set of discrete points in a bounded domain

Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain. Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize $$ \sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|} $$ over all ...
Ben Ciotti's user avatar
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54 views

Existence of a solution for the Laplace equation with sub-linear non-linearity

At first, I do apologize if my question is silly. I know that by variational methods it is possible to prove the existence of a solution for $$ \begin{cases} -\Delta u = u^p & \Omega \subset \...
Hheepp's user avatar
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221 views

Positivity of an integral

Let $f(x)=x-\theta(x)$, and for $x\ge 2$ let $J(x)=\int_x^{\infty}\frac{f(y)(1+\log y)}{y^2\log^2 y}dy$. Furthermore, $J(2)>0$. Suppose $J(x)>0$ when $f(x)=0$. Does it follow that $J(x)>0$ ...
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Does there exist $\alpha>0, \beta\in (0,1)$ such that $\dfrac{\sum_{k=1}^n a_k}{n}\le \alpha (a_1\cdots a_n)^{1/n} + \beta \max_i(a_i)$ holds?

Let $a_1\ge a_2\ge \cdots\ge a_n\ge 0$ be given non-negative numbers. My question is the following: Is there any $\beta \in(0,1),\ \alpha>0$, such that $$\dfrac{a_1+\cdots+a_n}{n}\le \alpha (a_1\...
Samrat Mukhopadhyay's user avatar
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63 views

Feller semigroups and fractional operators

Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
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Harnack Inequality for uniformly elliptic PDE via constructing a singularity

I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
David's user avatar
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Looking for example of integral transformations that preserve number of zeros

Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros. I am looking for non-trivial examples of integral transformation \begin{align} g(x)= \int f(t) h(t,x) dt \end{align} such that $f$ and $g$...
Boby's user avatar
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If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$

Let $$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$ that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$ Question: Let $\|\...
Idonknow's user avatar
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Conditions for the embedding of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$

Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$. If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...
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170 views

What functions can one try employing to fit an apparently doubly-periodic real function over $[0,1]$?

I have a cosine-like data curve over $x \in [0,1]$ that I can rather well-fit by a function of the form $a \cos{2 \pi x} +b$. Although good, the fit is still lacking, in that the residuals from the ...
Paul B. Slater's user avatar
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Dense Egoroff theorem

Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given. ...
ABIM's user avatar
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$p$-volume of $n$-dimensional hyper-ellipsoids

I read that the unit hypersphere has maximum volume for dimension five and would like to generalize this result. (If you think that integrating over an $n$-dimensional $p$-hyper-ellipsoid area ($x_1^...
Hauke Reddmann's user avatar
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On certain integrals of exponential functions with respect to Gaussian measures

I have questions about the integral $$F(a,b,c)=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}e^{-bx^4+cx^3-ax^2}dx$$ for $a,b,c>0$. What is the asymptotic behavior of $F(a,b,c)$ for small $a,b,c$? In ...
S.Z.'s user avatar
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60 views

Solution of a functional equation with cosine transform

What are the functions verifying: $$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$ With $\lambda$ a constant ? (Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions ...
Bertrand's user avatar
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Adjoint of differential equation

Motivation: Consider the ODE $$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation $$y'(t)=A^*y(t).$$ I ...
Umberto's user avatar
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58 views

$N-$Green function in $\mathbb R^N$

Let $N \geq 3$. Does there exist solution of the following equation $$-\Delta_N G + G^{N-1} = \delta_0,$$ where $-\Delta_N = - \text{ div}(|\nabla \cdot |^{N-2} \nabla \cdot )$ denotes $N-$Laplace ...
nguyen0610's user avatar
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299 views

When convolution with exponential kernel is bounded

Let $g(t)=e^{-\omega t}$, $\omega>0$. What is, in terms of well-known function spaces, the space $X$, $L_{loc}^2(0,\infty)\subset X$, of all functions $f:\mathbb{R}^+\to \mathbb{R}^+$, satisfying $...
Saj_Eda's user avatar
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Reference for the Hardy maximal function on the torus

I am searching for a reference for the (sharp) Hardy maximal function on the torus $\mathbb{T}^2:=\mathbb{R^2}/\mathbb{Z}^2$, for instance I would need result result of the following type : if $g\in H^...
Ayman Moussa's user avatar
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311 views

Approximations of Polylogarithm and Lerch transcendent?

For the Gamma function $\Gamma(x+1)$, we have beautiful approximations of the function in terms of elementary function, such as the Stirling approximation and its refinements, that give sharp ...
MCH's user avatar
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71 views

Existence of local minimizer

For a $f\in C^3$ function, if there is a sufficiently small $\epsilon$ $$\| \nabla F(x) \| < \epsilon$$ and a sufficiently large $\alpha$ where $$\lambda_{\min}[\nabla^2 F(x)] \ge \alpha$$ Can ...
Nikolayevich's user avatar
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81 views

Differential operator and equivalence

Here is the problem: I have a certain PDE and there is the nonlinear terme $h$, I have as data: $f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$ Now on consider the fnction $$h(...
Gustave's user avatar
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140 views

Lipschitz extensions preserving the convex hull of the range

We assume that $X$ is a metric space and that $A \subseteq X$ is a subset. Let $f : A \rightarrow \mathbb R$ be a Lipschitz-continuous function with Lipschitz constant $L$. By the Kirszbraun theorem, ...
shuhalo's user avatar
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308 views

Invertible operator

We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$ We hope to prove that $T$ is invertible if and only if $L = n\pi $. and for this ...
Gustave's user avatar
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in search of convergent daughter sequences

Let $\{f_n\}\subset L^1(\Omega,\mu)$, where $\mu$ is the Lebesgue measure, and $\Vert f_n\Vert_1\leq M$ and $\Vert Df_n\Vert_{1/2}\leq C$ uniformly in $n$. Question. Is there a subsequence $\{f_{...
T. Amdeberhan's user avatar
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93 views

What is the class of real sequences satisfying these conditions?

I'm interested in finding the class of the real sequences $u_{k}$, $k\in \mathbb{N^*}$ which satify the following conditions: $\displaystyle \sum_{k=1}^{\infty}\frac{1}{u_{k}}$ diverges i.e $\...
zeraoulia rafik's user avatar
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80 views

Comparison of two functions

Given a function $f$ from $R^2$ to $R$ satisfying tha following: $1)$ $f$ is a convex function which vanishes on $(0,1)$ and on $(1,0).$ $2)$ $f$ is a decreasing function on $x$ and on $y$ and $f$...
Khadija Mbarki's user avatar
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42 views

What (analytical or numerical) method can I use to solve scalar optimal problem?

I got the following optimization problem in mind and I am looking for some (analytic or numerical) methods to solve it. Can anyone have any ideas? Here is problem \begin{aligned} & {\text{...
Thomas Edison's user avatar
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59 views

Restriction to Basis of Cadlag function

If $f \in L^2([0,T])$ then it can be written as $$ f(t) \triangleq \sum_{i \in \mathbb{N}} c_i e_i(t), $$ for some sequence $\{c_i\}$ of real numbers and a Schauder basis $\{e_i(t)\}$ of $L^2([0,T])$ ...
ABIM's user avatar
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271 views

Convolution Integral involving an unknown function

I've got the following problem I'm working on which is related to some of my research. I am trying to solve the following equation for the function $f$. $$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 \...
Comic Book Guy's user avatar
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131 views

Measurable sets of probability measures $\{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}$

Let $(X,\mathscr{F})$ be a measurable space, and let $M$ be the set all probability measures $\mu: \mathscr{F} \to [0,1]$. Let us denote with $\mathscr{M}$ the $\sigma$-algebra on $M$ generated by the ...
Nduccio's user avatar
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Some problems about symmetric convolution semigroup on the unit circle

These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes". Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...
yangmengqh's user avatar
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64 views

Approx the jump point of a $BV$ function from both hand side

Let $I=(-1,1)$ be an interval in one dimension. Let $u\in BV(I)$ be defined as $$ u(x)= \begin{cases} 0,&\text{ if }x\in(-1,0)\\ 1,&\text{ if }x\in(0,1) \end{cases} $$ Clearly, we have $u\in ...
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470 views

Derivatives of Mollified functions

I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following: Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
ghjdnkmttrasda's user avatar
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344 views

Beurling density $D(X)$ of $X=\{x_j\in\mathbb R, \ |x_j-x_{i}|>\gamma>0: \ i,j\in\mathbb Z\}$

Beurling density of set $X$ is defined (see, for example "Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces" by Aldroubi and Grochenig) as: $$D(X)=\lim_{r\rightarrow \infty} \inf_{y\in\...
Mark's user avatar
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Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane

We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$. What is ...
Nikita Kalinin's user avatar
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63 views

The union of weighted compact supported continuous function

Let $\Omega\subset \mathbb R^N$ be open. Given a weight function $v\geq 1$ such that $v\in L^1_{\text{loc}}(\Omega)$ and $l.s.c$. Also supposethere exists a Lipschitz continuous sequence $v_n$ such ...
JumpJump's user avatar
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173 views

Is this has anything to do with Riesz representation?

The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49. Here I come up with a question which has similar ...
JumpJump's user avatar
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82 views

Construction of a path of quadratic variation

This question has been posted to Stack Exchange earlier, and no answer is available yet. Consider a path $x: [0,1] \to \mathbb R$. it's $p$-variation on an interval is defined by $$V_{p}(x, [a, b]) =...
kenneth's user avatar
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808 views

Inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$

Consider the inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$. My question is, how can we conclude about the decay properties, support and smoothness of the inverse Fourier transform? I ...
anonymous's user avatar
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152 views

The Lebesgue measure of the low level sets of the two-dimmension Fourier transform of a compactly supported function

Let $f\in {{L}^{1}}\left( {{\mathbb{R}}^{2}} \right)$ be a density function with the support $\operatorname{supp}\left( f \right)\subset \left[ a,b \right]\times \left[ c,d \right]$. Denoted by $\hat{...
Baily's user avatar
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A special approximation of BV functions

Suppose $\Omega:=B(0,1)\subset \mathbb R^2$. Let $u\in BV(\Omega)$ be a radially symmetric, i.e., $u(x)=u(Rx)$ for all $R\in SO(2)$. In addition, suppose $w$ to be an affine function,. i.e., $\...
JumpJump's user avatar
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104 views

Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?

The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question: Given ...
Spencer's user avatar
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145 views

Discrete measures and discrete kernels

This is a cross-post from math.stack. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence $(x_k)_{k\in\...
andy teich's user avatar
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94 views

Do they have the same limit?

Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$ $$ \frac{1}{T}\int_{\mathbb{R}}dx\int_{[-T,T]^2}d\mathbf{v}\int_{[-T,T]^2}...
Uchiha's user avatar
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