Related papers: Some inequalities for norms in $\mathbb{ R}^n$ and…
$ \newcommand{\schs}{\scriptstyle{\mathsf{S}}_1} $For all $n \ge 1$, we give an explicit construction of $m \times m$ matrices $A_1,\ldots,A_n$ with $m = 2^{\lfloor n/2 \rfloor}$ such that for any $d$ and $d \times d$ matrices…
If a pair of functions nearly extremizes Young's convolution inequality for R^d, with all three exponents finite and strictly greater than 1, then each function is close in norm to a Gaussian. The proof relies on the Riesz-Sobolev…
We extend the Moser-Trudinger inequality to any Euclidean domain satisfying Poincar\'e's inequality. We find out that the same equivalence does not hold in general for conformal metrics on the unit ball, showing counterexamples. We also…
The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous…
In this note, we investigate the regularity of extremal solution $u^*$ for semilinear elliptic equation $-\triangle u+c(x)\cdot\nabla u=\lambda f(u)$ on a bounded smooth domain of $\mathbb{R}^n$ with Dirichlet boundary condition. Here $f$…
We prove that for any two quasi-Banach spaces $X$ and $Y$ and any $\alpha>0$ there exists a constant $\gamma_\alpha>0$ such that $$ \sup_{1\le k\le n}k^{\alpha}e_k(T)\le \gamma_\alpha \sup_{1\le k\le n} k^\alpha c_k(T) $$ holds for all…
We prove that, if the closed unit ball of a normed space $X$ has sufficiently many extreme points, then every mapping $\Phi$ from $X$ into itself with the following property is affine: For any pair of points in $X$, there exists a (not…
We present the best constant and the existence of extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in $\mathbb{R}^N$. We also…
We prove a uniform bound on the topological Tur\'an number of an arbitrary two-dimensional simplicial complex $S$: any $n$-vertex two-dimensional complex with at least $C_S n^{3-1/5}$ facets contains a homeomorphic copy of $S$, where $C_S >…
In this paper we prove the existence of extremal functions for the Adams-Moser-Trudinger inequality on the Sobolev space $H^{m}(\Omega)$, where $\Omega$ is any bounded, smooth, open subset of $\mathbb{R}^{2m}$, $m\ge 1$. Moreover, we extend…
We consider numerically the roughness of a planar crack front within the long-range elastic string model, with a tunable disorder correlation length $\xi$. The problem is shown to have two important length scales, $\xi$ and the Larkin…
It is shown that a simple closed curve in $\mathbb C^n$ that is a uniform limit of rectifiable simple closed curves each of which has nontrivial polynomial hull has itself nontrivial polynomial hull. In case the limit curve is rectifiable,…
We present five-loop results for the renormalization of various models with a cubic interaction (in ${d = 6 - 2 \varepsilon}$ dimensions). For the scalar model and its ${O(n)}$-symmetric extension we provide renormalization constants,…
We establish a connection between the uniform infinite planar triangulation and some critical time-reversed branching process. This allows to find a scaling limit for the principal boundary component of a ball of radius R for large R (i.e.…
We consider the set of extremal points of the generalized unit ball induced by gradient total variation seminorms for vector-valued functions on bounded Euclidean domains. These are central to the understanding of sparse solutions and…
We give sharp conditions for the limiting Korn-Maxwell-Sobolev inequalities \begin{align*} \lVert P\rVert_{{\dot{W}}{^{k-1,\frac{n}{n-1}}}(\mathbb{R}^n)}\le…
We consider Hotelling's T^2 statistic for an arbitrary d-dimensional sample. If the sampling is not too deterministic or inhomogeneous, then under zero means hypothesis, T^2 tends to \chi^2_d in distribution. We show that a test for the…
Norm-resolvent convergence with order-sharp error estimate is established for Neumann Laplacians on thin domains in $\mathbb{R}^d,$ $d\ge2,$ converging to metric graphs in the limit of vanishing thickness parameter in the resonant case. The…
We introduce a family of conformal invariants associated to a smooth metric measure space which generalize the relationship between the Yamabe constant and the best constant for the Sobolev inequality to the best constants for…
Several applications of Abel's partial summation formula to the convergence of series of positive vectors are presented. For example, when the norm of the ambient ordered Banach space is associated to a strong order unit, it is shown that…