Related papers: Trace arithmetic--$\kappa_p$ inequality

Let kappa be a regular uncountable cardinal and lambda >=kappa^+ . The principle of stationary reflection for P_kappa lambda has been successful in settling problems of infinite combinatorics in the case kappa=omega_1. For a greater kappa…

For $1<p<\infty$ we prove an $L^p$-version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $ W^{1,\,p}_0(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$. More precisely, let…

A duality of $\kappa$-normed topological vector spaces is defined and investigated. For such spaces the analog of the Mackey-Arens theorem is proved. There are investigated cases, when $\kappa$-normability of a topological vector space…

Let $0<s<1$ and $p>1$ be such that $ps<N$. Assume that $\Omega$ is a bounded domain containing the origin. Staring from the ground state inequality by R. Frank and R. Seiringer we obtain: 1- The following improved Hardy inequality for $p\ge…

We prove a stochastic Gronwall lemma of the following type: if $Z$ is an adapted nonnegative continuous process which satisfies a linear integral inequality with an added continuous local martingale $M$ and a process $H$ on the right hand…

In this paper we use infinitary Turing machines with tapes of length $\kappa$ and which run for time $\kappa$ as presented, e.g., by Koepke \& Seyfferth, to generalise the notion of type two computability to $2^{\kappa}$, where $\kappa$ is…

We present some operator inequalities for positive linear maps that generalize and improve the derived results in some recent years. For instant, if $A$ and $B$ are positive operators and $m,m^{'},M,M^{'}$ are positive real numbers…

For a class of non-symmetric non-local L\'evy-type operators $\mathcal{L}^{\kappa}$, which include those of the form $$ \mathcal{L}^{\kappa}f(x):= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left<z,\nabla f(x)\right>)\kappa(x,z)J(z)\,…

If kappa is strongly compact, lambda > kappa is regular, then (2^{< lambda})^+ --> (lambda+eta)^2_theta holds for eta,theta<kappa.

In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for $1<p,q<\infty$, $0<r<\infty$ with $p+q\geq r$, $\delta\in[0,1]\cap\left[\frac{r-q}{r},\frac{p}{r}\right]$ with $\frac{\delta…

We introduce a generalization of relative entropy derived from the Wigner-Yanase-Dyson entropy and give a simple, self-contained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for the map…

Consider percolation on the triangular lattice. Let $\kappa(p)$ be the free energy at the zero field. We show that $$|\kappa'''(p)| \leq |p-p_c|^{-1/3+o(1)} \mbox{ if } p \neq p_c.$$ Furthermore, we show that there exists a sequence…

In this paper we study the joint convexity/concavity of the trace functions \[ \Psi_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R}, \] where $A$ and $B$ are positive definite matrices and $K$ is any…

The paper studies properties of twisted sums of a Banach space $X$ with $c_0(\kappa)$. We first prove a representation theorem for such twisted sums from which we will obtain, among others, the following: (a) twisted sums of $c_0(I)$ and…

We prove that for a positive integer $c$ and any given $\varepsilon$, $0<\varepsilon<1$, the number $N(c)$ of equations $c=a+b$, $a<b$, with positive coprime integers $a$ and $b$, which satisfy the inequality $$c <…

We show that for all $A, B \subseteq \{0,1,2\}^{d}$ we have $$ |A+B|\geq (|A||B|)^{\log(5)/(2\log(3))}. $$ We also show that for all finite $A,B \subset \mathbb{Z}^{d}$, and any $V \subseteq\{0,1\}^{d}$ the inequality $$ |A+B+V|\geq…

In this paper, we prove the convexity of trace functionals $$(A,B,C)\mapsto \text{Tr}|B^{p}AC^{q}|^{s},$$ for parameters $(p,q,s)$ that are best possible, where $B$ and $C$ are any $n$-by-$n$ positive definite matrices, and $A$ is any…

In this paper, we prove that the inequalities $\alpha [1/3 Q(a,b)+2/3 A(a,b)]+(1-\alpha)Q^{1/3}(a,b)A^{2/3}(a,b)<M(a,b) <\beta [1/3 Q(a,b)+2/3 A(a,b)]+(1-\beta)Q^{1/3}(a,b)A^{2/3}(a,b)$ and $\lambda [1/6 C(a,b)+5/6…

Recently, Yanyan Li and Xukai Yan showed the following interesting Hardy inequalities with anisotropic weights: Let $n\geq 2$, $p \geq 1$, $p\alpha > 1-n$, $p(\alpha + \beta)> -n$, then there exists $C > 0$ such that…

In this paper we give a streamlined proof of an inequality recently obtained by the author: For every $\alpha \in (0,1)$ there exists a constant $C=C(\alpha,d)>0$ such that \begin{align*} \|u\|_{L^{d/(d-\alpha),1}(\mathbb{R}^d)} \leq C \|…