Tagged Questions

0
votes
0answers
26 views

Norm estimation of an area integral

I am solving a certain kind of integral equations using iteration and Volterra series. Now I get a formal solution and in order to prove convergence I need to estimate the $L^1$ an …
3
votes
1answer
163 views

Lipschitz map of the circle onto a triangle

Assume that $f$ is (Euclidean) $L-$biLipchitz mapping of the unit circle onto a triangle $\Delta(A,B,C)$. Can we find a $10000 L$ bi-lipchitz extension of $f$ onto the whole plane. …
3
votes
1answer
273 views

Is a Cauchy principal value invariant under a “change of variables”?

Let $f \in C^{\gamma}_c(\mathbb{R}^n) $. Let $K:\mathbb{R}^n \backslash {\vec{0}} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties: 1) K smoot …
1
vote
1answer
163 views

A question about “nice” functions

Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us c …
3
votes
1answer
116 views

Lipschitz map of the ellipse

Is there a L-Lipschitz homeomorphism of the Elipse $x^2/4+y^2=1$ onto the unit circle $x^2+y^2=1$ such that $L<1$?
0
votes
0answers
55 views

New differintegral formula: how is it related to other differintegral formulas?

Lets define new differintegral formula as $$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ or, equivalently, $$\mathbb{D}^s_xf(x …
0
votes
1answer
134 views

The pth power of a distance function is twice continuously differentiable, for $p>2$?

Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$. Is $\beta^p$, $p>2$ a tw …
1
vote
0answers
46 views

Continuous Differentiability under Expectation

Suppose $f: \Re \rightarrow \Re$ is continuously differentiable a.e.. Let $F(x) = E_D(f(x-D))$, where we assume that the expectation is well defined for each $x \in \Re$ and $D$ is …
1
vote
1answer
195 views

Integral inequality for convex function

Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true: \begin{equation} \frac …
2
votes
1answer
124 views

Lower bounds on derivative around zero set of a positive smooth function.

As part of a different problem, I came across the following simplified question, for which I cannot exhibit a proof nor a counterexample. Note that the assumptions of smoothness an …
4
votes
3answers
360 views

Classic applications of Baire category theorem

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theor …
3
votes
0answers
97 views

Evaluation of an $n$-dimensional integral

I asked the same question on math.se but got no answer there. Since it pertains to my current research, I decided to ask here: Let $n\in 2\mathbb{N}$ be an even number. I want to …
1
vote
1answer
67 views

Does the Border (Boundary) Points of a convex body make a concave function?

Let $\mathbb{S}$ be a closed and bounded convex body in 2-D with some non-empty intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most poin …
2
votes
1answer
138 views

Boundedness of an Oscillating Integral

Let $g(x):\mathbb{R}_{\geq0}\rightarrow\mathbb{R}$ be real analytic s.t. $g(0)\neq 0$ and $g(x)=O(x^{-2})$ as $x\rightarrow\infty$. I think the following integral should be bounded …
0
votes
0answers
42 views

a question on bounds for complex functions

Let $\bf{u}$ be a smooth complex vectorfield defined on the closed unit ball $B_1(0)\subset \mathbb{R}^3$. Let $\phi$ be any smooth complex function defined on $B_1(0)$. My quest …

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