The Lieb–Thirring Inequality for Interacting Systems in Strong-Coupling Limit

We consider an analogue of the Lieb–Thirring inequality for quantum systems with homogeneous repulsive interaction potentials, but without the antisymmetry assumption on the wave functions. We show that in the strong-coupling limit, the Lieb–Thirring constant converges to the optimal constant of the one-body Gagliardo–Nirenberg interpolation inequality without interaction.

On the Existence of Extremals for Moser-Type Inequalities in Gauss Space

The existence of an extremal in an exponential Sobolev-type inequality, with optimal constant, in Gauss space is established. A key step in the proof is an augmented version of the relevant inequality, which, by contrast, fails for a parallel classical inequality by Moser in the Euclidean space.

Equivalent Conditions of a Hilbert-Type Multiple Integral Inequality Holding

Let ∑i=1n1/pi=1pi>1, in this paper, by using the method of weight functions and technique of real analysis; it is proved that the equivalent parameter condition for the validity of multiple integral Hilbert-type inequality ∫R+nKx1,⋯,xn∏i=1nfixi dx1⋯dxn≤M∏i=1nfipi,αi with homogeneous kernel Kx1,⋯,xn of order λ is ∑i=1nαi/pi=λ+n−1, and the calculation formula of its optimal constant factor is obtained. The basic theory and method of constructing a Hilbert-type multiple integral inequality with the homogeneous kernel and optimal constant factor are solved.

On a Property of Harmonic Measure on Simply Connected Domains

Let $D\subset \mathbb{C}$ be a domain with $0\in D$ . For $R>0$ , let $\widehat{\unicode[STIX]{x1D714}}_{D}(R)$ denote the harmonic measure of $D\cap \{|z|=R\}$ at $0$ with respect to the domain $D\cap \{|z|<R\}$ and let $\unicode[STIX]{x1D714}_{D}(R)$ denote the harmonic measure of $\unicode[STIX]{x2202}D\cap \{|z|\geqslant R\}$ at $0$ with respect to $D$ . The behavior of the functions $\unicode[STIX]{x1D714}_{D}$ and $\widehat{\unicode[STIX]{x1D714}}_{D}$ near $\infty$ determines (in some sense) how large $D$ is. However, it is not known whether the functions $\unicode[STIX]{x1D714}_{D}$ and $\widehat{\unicode[STIX]{x1D714}}_{D}$ always have the same behavior when $R$ tends to $\infty$ . Obviously, $\unicode[STIX]{x1D714}_{D}(R)\leqslant \widehat{\unicode[STIX]{x1D714}}_{D}(R)$ for every $R>0$ . Thus, the arising question, first posed by Betsakos, is the following: Does there exist a positive constant $C$ such that for all simply connected domains $D$ with $0\in D$ and all $R>0$ , $$\begin{eqnarray}\unicode[STIX]{x1D714}_{D}(R)\geqslant C\widehat{\unicode[STIX]{x1D714}}_{D}(R)?\end{eqnarray}$$ In general, we prove that the answer is negative by means of two different counter-examples. However, under additional assumptions involving the geometry of $D$ , we prove that the answer is positive. We also find the value of the optimal constant for starlike domains.