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About the number of critical points of a function

Suppose that $f$ is a totally monotone function on $(0,\infty)$, so that $(-1)^n f^{(n)}\ge0$ for all $n=0,1,\dots$, $f(0+)\in(0,\infty)$, and $f(t)\sim\frac{1}{t^{\frac{3}{2}}}$ as $t\to\infty$. Can ...
Ervand's user avatar
  • 51
4 votes
1 answer
277 views

Eigenvalue of a convolution and a restriction?

Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
227 views

A deceptively simple regularity problem for functions on the plane

By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer: Consider a twice ...
vmist's user avatar
  • 989
3 votes
1 answer
83 views

How irregular can the set of points of non-differentiability for an L1 function's primitive F get, before the FTC fails?

A Fundamental Theorem of Calculus for Lebesgue Integration, J. J. Koliha begins with the passage Lebesgue proved a number of remarkable results on the relation between integration and differentiation....
D.R.'s user avatar
  • 833
2 votes
0 answers
109 views

Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support

This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction? Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
H A Helfgott's user avatar
  • 20.2k
3 votes
0 answers
91 views

About BMO space on smooth open bounded domain

Let $\Omega$ be any open domain in $\Bbb R^d$. Define the $\text{BMO}(\Omega)$ space as $$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\}, $$ ...
Guy Fsone's user avatar
  • 1,101
2 votes
0 answers
79 views

Function that is (essentially) a self-convolution but not a multiple of a self-convolution

Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
H A Helfgott's user avatar
  • 20.2k
3 votes
0 answers
83 views
+50

Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps

Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function $$ m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)}, $$ where $\lambda_{\max}$ denotes the largest eigenvalue....
Ran's user avatar
  • 73
-3 votes
0 answers
69 views

Exercise generalizing (related to) Hölder's inequality

I came across this exercise and feel absolutely stuck: Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
HZA's user avatar
  • 1
0 votes
0 answers
66 views

$L_1$ norm of $f\in L^1(\mathbb{R}^n)$ compactly supported and its change of variable

Let $M\in\mathbb{R}^{n\times n}$ be an invertible matrix, denote its induced linear map on $\mathbb{R}^n$ also by $M$, and let $f\in L^1(\mathbb{R}^n)$ be compactly supported. I am wondering if we can ...
Jens Fischer's user avatar
0 votes
0 answers
69 views

A system of nonlinear Diophantine equations whose positive solutions are not coprime

Consider the following system of Diophantine equations: $$v_1k_1=k_1^3-k_2^3+k_3^3 \\ v_2k_2=k_1^3+k_2^3-k_3^3 \\ v_3k_3=-k_1^3+k_2^3+k_3^3 \tag{1}$$ where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
Amir's user avatar
  • 303
3 votes
0 answers
15 views

On compact embeddings in weighted Riesz potential spaces

I wonder if there is any references for the study of the following type of spaces $$ X_{\delta,\alpha}=\{ u\in L^2_\delta(\mathbb{R}^n):\, u= (-\Delta)^\alpha f \quad\text{for some}\quad f\in L^2_{\...
Ali's user avatar
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