3
votes
0answers
119 views

Reference for Hodge decomposition

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...
0
votes
1answer
295 views

For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that $$ \sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty, $$ where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
0
votes
1answer
113 views

Is the span of those vectors dense in $\ell_2$?

For all $x \in \mathbb{R}^n$ and $\alpha \in \mathbb{Z}_{\geq 0}^n$ let $x^\alpha=x_1^{\alpha_1} \cdots x_n^{\alpha_n}$. Let $$\ell^2=\{z=(z_\alpha)_{\alpha \in \mathbb{Z}_{\geq 0}^n}:\, z_{\alpha} ...
0
votes
0answers
120 views

When is there a polynomial transformation? [closed]

First part: given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}=\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(f(x_1,x_2,\dots,x_n)))|$$ where $P_i$ is polynomial ( that ...
1
vote
0answers
69 views

Structure of the zero set of analytic maps (Lojasiewicz’s Structure Theorem for Varieties)

The Lojasiewicz’s Structure Theorem for Varieties states that for $\Phi:\mathbb{R}^n\rightarrow \mathbb{R}$ real analytic with $\Phi(0)=0$ the set $\Phi^{-1}(0)$ is locally a union of subvarieties of ...
9
votes
2answers
564 views

Functions that Calculate their $L_p$ Norm

are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ ...
4
votes
1answer
58 views

Reference: Hardy space regularity of the Jacobian determinant

I'm looking for a reference, expository in nature, for the proof of the following theorem of Coifman, Lions, Meyer and Semmes. Theorem: For all $u\in W^{1,n}(\mathbb{R}^n;\mathbb{R}^n)$, ...
2
votes
2answers
96 views

formula for repeated finite differences

I am looking for a proof of a well-known fact, whose proof must be very easy, though I've been struggling to find it. Let $\Delta$ be the map from real-valued functions of a real variable, given by ...
3
votes
0answers
83 views

Generalized family of Holder inequalities

Is the "only if" direction of the following fact known? For fixed sequences $(a)_i = a_1, \dots, a_r$, $(b)_i = b_1, \dots, b_r$ and $(c)_i = c_1, \dots, c_r$, the inequality $\prod_{i = 1}^r ...
1
vote
2answers
109 views

reference needed for sobolev type estimates

I'm reading a paper and the authors applied the following sobolev type estimates $$ ||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2} $$ for $\alpha>\frac{1}{4}$, where $v$ ...
0
votes
2answers
119 views

Analyticity of Logarithmic Integrals

Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...
3
votes
1answer
303 views

Does there exist a function such that $\int_{\mathbb{R}_+^{\star} } t^nf(t)dt=0$? [closed]

Let $f\in C([a,b],\mathbb{R})$ such that $\displaystyle\int_{a}^{b} t^nf(t)dt=0$ for all integer n. We know that $f\equiv 0$. It's call Hausdorff theorem. This theorem is wrong on $\mathbb{R^+}$, a ...
2
votes
1answer
63 views

radius of the ball where the inverse of Lipschitz maps exists

I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $δ_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke: Teorem. Let $f : U ⊂ \mathbb{R}^n ...
16
votes
2answers
606 views

Felix Klein on infinitesimals

This is a reference request prompted by some intriguing comments made by Felix Klein. In 1908, Felix Klein formulated a criterion of what it would take for a theory of infinitesimals to be ...
3
votes
1answer
278 views

An elementary inequality: reference request

Consider the problem of minimizing $\sum_{i=1}^{n}{x_{i}}$ under the constraints $\sum_{i=1}^{n}{x_{i}^{2}}=1$ and $x_{i} \geq 0$. Obviously the solution is given by the vector $(1,0,\ldots,0)$. Now ...
0
votes
0answers
79 views

Detailed Taxonomy of Multivariate Real Functions: $\mathbb{R}^n\rightarrow\mathbb{R}$

I want to classify the following functions: $$ f:(x,y)\in\mathbb{R}\times\mathbb{R}\rightarrow\sqrt{(x+1)^2+y^2}+\sqrt{(x-1)^2+y^2}-2$$ $$ ...
6
votes
1answer
223 views

Alternative proof of Lojasiewicz inequality

is there a "brute force proof" of the Lojasiewicz inequality? By "brute force" i mean without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e. standard ...
1
vote
1answer
150 views

Lebesgue's integrability condition in several variables

The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable $f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
1
vote
2answers
223 views

Banach algebra of BV functions

I would like to find a reference for the proof that functions of bounded variation make a Banach algebra. Same question for $BV\cap L^\infty$.
6
votes
1answer
230 views

“Values” of divergent integrals

Are there existing theories of integration in which $I_0 = \int_0^{\infty} dx$ and $I_1 = \int_0^{\infty} x \ dx$ are well-defined infinite elements in a non-archimedean extension of the reals? I can ...
1
vote
2answers
248 views

Is there a reference for compact imbedding theory of Hölder space?

This question is posted and unanswered from math.stackexchange. Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to ...
4
votes
1answer
220 views

Norms for complex measures

I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...
2
votes
1answer
82 views

Young transform reference

The Young transform of nonnegative function $f(x)$, $x \in \mathbb R^n_+$ is defined to be $$ (\mathscr Yf)(y) = \inf \left[ \left. \frac{x_1 y_1 + \ldots + x_n y_n}{f(x)} \; \right|\; x \colon ...
0
votes
3answers
274 views

$L_p$ space embedding (reference request)

There is a result in the wikipedia article about $L_p$ space embedding: a. Let $0 ≤ p < q ≤ ∞$. $L_q(S, μ)$ is contained in $L_p(S, μ)$ iff $S$ does not contain sets of arbitrarily large ...
4
votes
2answers
550 views

Tails of sums of Weibull random variables

Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d random variables distributed according to Weibull distribution with shape $0 < \epsilon < 1$ (it means that $\mathbf{Pr}[X_i \geq t] = ...
4
votes
1answer
211 views

Morse lemma with least amount of regularity.

I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across ...
3
votes
1answer
148 views

Subharmonic envelope

I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me ...
5
votes
1answer
494 views

Extending an assignment property from Q to R (or C)

Property of any odd number of nonnegative integers: Given $x_1 \leq \ldots \leq x_{2n + 1}$ with each $x_i \in \mathbb{Z}_{\geq 0}$, suppose that for any $x_i$ we remove, the remaining numbers can be ...
4
votes
2answers
394 views

Can we calculate the inner product of a semicontinous function with the Dirac delta function?

Dear all, It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...
3
votes
3answers
455 views

When is the infimum of an arbitrary family of measurable functions also measurable?

Let $(X,\Sigma,\mu)$ be a measure space and consider a family of $\mu$-measurable functions $f_i:X \to \mathbb{R}$ for $i$ lying in some index set $I$. Define $$f(x) = \inf_{i \in I} f_i(x)$$ I think ...
3
votes
1answer
351 views

Ask for theory about the weighted L^2(R^d) space.

Dear MOs, I am now considering the following norm: $$ ||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:. $$ where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative ...
0
votes
1answer
316 views

Asymptotic equivalence for functions with zeros

I am considering the relative asymptotic behavior of a pair of real functions on the positive real axis, say $f$ and $g$. There is no $x_0$ such that $f$ and $g$ are non-zero for all $x>x_0$. ...
7
votes
2answers
1k views

Divergent series expansion in Apéry's proof of the irrationality of $\zeta(2)$ and $\zeta(3)$

UPDATE. I am now making this a CW in the hope someone can improve the content of this question and/or correct the text. This is a concise version of this math.SE question of mine. I've got an answer ...
4
votes
2answers
304 views

Articles with examples of Darboux functions without fixed points

A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a < b \in I$ and for every $\lambda$ between $f(a)$ and $f(b)$ there exists $c ...
1
vote
0answers
348 views

Uniform convergence of convex functions - references

Inspired by the following question on stackexchange: http://math.stackexchange.com/questions/126142/uniform-convergence-of-convex-sequence-of-functions, I thought of asking whether anyone knows of ...
5
votes
1answer
488 views

Cosets of groups of functions

Let's consider an interval $I\subseteq\mathbb R$, and let $\mathcal F(I)$ be the set of bijective functions $f:I\to I$ so that the graph of $f$ is a analytic curve in $I\times I$. The set $\mathcal ...
0
votes
2answers
512 views

Is there dual space of the distributions $\mathcal{D}'(R)$?

Dear MOs, Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak ...
3
votes
4answers
806 views

analysis over non-Archimedean ordered fields

Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...
5
votes
2answers
647 views

A dual theory to the theory of currents?

The k-currents are defined as dual space to the spaces of all smooth k-forms. (These monsters are used to work with the minimal k-surfaces.) Assume I want to look at the generalized k-forms; they can ...
7
votes
1answer
487 views

Proof of the “Neo-classical Inequality”

I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1, n$: $\frac{1}{p^2}\sum_{j=0}^n\frac{a^{j/p}b^{(n-j)/p}}{(j/p)!((n-j)/p)!}\leq ...
5
votes
1answer
473 views

Quantitative bounds for multivariate central limit theorem

Hi, For the univariate central limit theorem, the Berry-Esseen theorem gives a quantitative bound on the rate of convergence of distributions to the Normal distribution under Kolmogorov distance: ...
5
votes
2answers
589 views

Is the inclusion of Lebesgue spaces compact?

[Disclaimer: this may be a very trivial question; it certainly looks like it ought to have been studied and understood. I started thinking about it this morning when writing some notes for ...
1
vote
1answer
899 views

Power series with non-integer exponents

Motivation: For the sake of concreteness, I'll state a very particular context, but my question is a little more general. I'm trying to find a function $\gamma\colon [0,\delta) \to [0,\delta')$ that ...
3
votes
1answer
273 views

approximately linear functions — more

Suppose $f,g$ are continuous functions from $\mathbb R$ to $\mathbb R$, with the property that $$f(x)+f(y)=g(x+y)$$ for all $x,y$. Taking $x=y=z/2$ implies that $g(x)=2f(x/2)$ so that the above ...
6
votes
1answer
830 views

approximately linear functions

i suppose it's fairly well known that if a (continuous, real-valued) function $f$ on the real line satisfies $f(x-y)=f(x)-f(y)+const$ then it is necessarily linear. are there any general ...
6
votes
1answer
637 views

Inverse function theorem for DC-functions

I would like to have an inverse (or/and) implicite function theorem for DC-functions. It seems that I have right definitions, but I fail to prove it... Definitions: Let $h:\mathbb R^n\to\mathbb R$ ...
3
votes
2answers
561 views

Analyzing the solution to a second-order, non-linear ODE

Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...
5
votes
1answer
292 views

The set of non-smooth points of a convex function is (m - 1)-rectifiable

I am looking for a reference to the following result. Let $f:\mathbb R^m\to\mathbb R$ be a convex function. Then $f$ is differentiable at all points of outside of a countable union of ...