Related papers: A Lieb-Thirring inequality for singular values

For a positive semidefinite matrix $H= \begin{bmatrix} A&X\\ X^{*}&B \end{bmatrix} $, we consider the norm inequality $ ||H||\leq ||A+B|| $. We show that this inequality holds under certain conditions. Some related topics are also…

This paper extends the work of El Karoui [Ann. Probab. 35 (2007) 663--714] which finds the Tracy--Widom limit for the largest eigenvalue of a nonsingular $p$-dimensional complex Wishart matrix $W_{\mathbb{C}}(\Omega_p,n)$ to the case of…

Let $p>1$ and $1/p+1/q=1$. Consider H\"older's inequality $$ \|ab^*\|_1\le \|a\|_p\|b\|_q $$ for the $p$-norms of some trace ($a,b$ are matrices, compact operators, elements of a finite $C^*$-algebra or a semi-finite von Neumann algebra).…

For $0\leq \alpha<1$ and prime number $p$ let $L(\alpha,p)$ be the sum of the first $[\alpha p]$ values of Legendre symbol modulo $p$. We study positivity of $L(\alpha,p)$ and prove that for $|\alpha-\frac13|<2\cdot 10^{-6}$ and for…

Motivated by Horn's log-majorization (singular value) inequality $s(AB)\underset{log}{\prec} s(A)*s(B)$ and the related weak-majorization inequality $s(AB)\underset{w}{\prec} s(A)*s(B)$ for square complex matrices, we consider their…

For odd primes $p$ we consider the factors \[ A(p)=\frac{p-\chi_4(p)}{p+\chi_4(p)}, \qquad \chi_4(p)= \begin{cases} 1,&p\equiv 1\pmod 4, \\ -1,&p\equiv 3\pmod 4, \end{cases} \] and study products of $A(p)$ restricted to unions of residue…

To determine the sharp constants for the one dimensional Lieb--Thirring inequalities with exponent gamma in (1/2,3/2) is still an open problem. According to a conjecture by Lieb and Thirring the sharp constant for these exponents should be…

The purpose of this paper is to establish new log-majorization results concerning eigenvalues and singular values which generalize some previous work related to a conjecture and an open question which were presented by R. Lemos and G.…

We prove Lieb-Thirring-type bounds for fractional Schr\"odinger operators and Dirac operators with complex-valued potentials. The main new ingredient is a resolvent bound in Schatten spaces for the unperturbed operator, in the spirit of…

For a prime $p$, we show that uniqueness of factorization into irreducible $\Sigma_{p^2}$-invariant representations of $\mathbb{Z}/p \wr \mathbb{Z}/p$ holds if and only if $p=2$. We also show nonuniqueness of factorization for…

We study distinct $(0,1)$ matrices $A$ and $B$, called \textit{Gram mates}, such that $AA^T=BB^T$ and $A^TA=B^TB$. We characterize Gram mates where one can be obtained from the other by changing signs of some positive singular values. We…

We study a multilinear singular integral obtained by taking averages of simplex Hilbert transforms. This multilinear form is also closely related to Calder\'on commutators and the twisted paraproduct. We prove $L^p$ bounds in dimensions two…

We consider inverse boundary value problems for the Schrodinger equations in two dimensions. Within less regular classes of potentials, we establish a conditional stability estimate of logarithmic order. Moreover we prove the uniqueness…

Let $\mathcal{W}$ be a closed dilation and translation invariant subspace of the space of $\mathbb{R}^\ell$-valued Schwartz distributions in $d$ variables. We show that if the space $\mathcal{W}$ does not contain distributions of the type…

Let $p_n$ denote the $n$-th prime number, $\{q_n\}$ be a sequence of positive numbers and $x\in\mathbb{R}$. In this note we prove that the inequality $$q_n p_{n+1}^{x}-q_{n+1}p_{n}^{x}<p_{n}^{x}p_{n+1}^{x-1}, $$ holds for infinitely many…

For the maximal operator $ M $ on $ \mathbb R ^{d}$, and $ 1< p , \rho < \infty $, there is a finite constant $ D = D _{p, \rho }$ so that this holds. For all weights $ w, \sigma $ on $ \mathbb R ^{d}$, the operator $ M (\sigma \cdot )$ is…

We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights $w$ having finitely many zeros and singularities (i.e., points where $w$ becomes infinite) on an interval and not too ``rapidly…

Various parametrizations of sigma(gamma p) are tried out on the small-x fixed target and HERA data. A two-Pomeron type parametrization is found to give the best reproduction of the data. The data indicate that the value of lambda(eff) for…

We present an overview over recent results concerning semi-classical spectral estimates for magnetic Schroedinger operators. We discuss how the constants in magnetic and non-magnetic eigenvalue bounds are related and we prove, in an…

We characterize two-weight inequalities for certain maximal truncations of the Hilbert transform in terms of testing conditions on simpler functions. For 1<p<2 and two positive Borel measures u, v on R, we assume that u is doubling, and we…