# Tagged Questions

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### 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 ...
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### 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 $f(x)=e^{-|x|}$, ...
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### 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 ...
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### 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 ...
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### 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 $BV\cap L^\infty$.
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### “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 ...
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### 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 ...
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### 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 ...
The Young transform of nonnegative function $f(x)$, $x \in \mathbb R^n_+$ is defined to be $$(\mathscr Yf)(y) = \inf \left[ \left. \frac{x_1 y_1 + \ldots + x_n y_n}{f(x)} \; \right|\; x \colon ... 3answers 274 views ### L_p space embedding (reference request) There is a result in the wikipedia article about L_p space embedding: a. Let 0 ≤ p < q ≤ ∞. L_q(S, μ) is contained in L_p(S, μ) iff S does not contain sets of arbitrarily large ... 2answers 550 views ### Tails of sums of Weibull random variables Suppose that X_1, X_2, \ldots, X_n are i.i.d random variables distributed according to Weibull distribution with shape 0 < \epsilon < 1 (it means that \mathbf{Pr}[X_i \geq t] = ... 1answer 211 views ### Morse lemma with least amount of regularity. I recently came across with C^2 Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across ... 1answer 148 views ### Subharmonic envelope I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me ... 1answer 494 views ### Extending an assignment property from Q to R (or C) Property of any odd number of nonnegative integers: Given x_1 \leq \ldots \leq x_{2n + 1} with each x_i \in \mathbb{Z}_{\geq 0}, suppose that for any x_i we remove, the remaining numbers can be ... 2answers 394 views ### Can we calculate the inner product of a semicontinous function with the Dirac delta function? Dear all, It is clear that if f:R\mapsto R is a continuous function, than < f, \delta_x >=f(x). Now, if f is only semicontinous, can we say that < f, \delta_x >=f(x)? I think this ... 3answers 455 views ### When is the infimum of an arbitrary family of measurable functions also measurable? Let (X,\Sigma,\mu) be a measure space and consider a family of \mu-measurable functions f_i:X \to \mathbb{R} for i lying in some index set I. Define$$f(x) = \inf_{i \in I} f_i(x)$$I think ... 1answer 351 views ### Ask for theory about the weighted L^2(R^d) space. Dear MOs, I am now considering the following norm:$$ ||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 ... 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. ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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: ... 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 ... 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 ... 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)$$for all x,y. Taking x=y=z/2 implies that g(x)=2f(x/2) so that the above ... 1answer 830 views ### approximately linear functions i suppose it's fairly well known that if a (continuous, real-valued) function f on the real line satisfies f(x-y)=f(x)-f(y)+const then it is necessarily linear. are there any general ... 1answer 637 views ### Inverse function theorem for DC-functions I would like to have an inverse (or/and) implicite function theorem for DC-functions. It seems that I have right definitions, but I fail to prove it... Definitions: Let h:\mathbb R^n\to\mathbb R ... 2answers 561 views ### Analyzing the solution to a second-order, non-linear ODE Let \psi : [0,\infty] \to \mathbb R be a strictly positive, continuously differentiable function, and consider the non-linear ODE$$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...
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 ...