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A relation between pointwise convergence of functions and convergence of functionals

We show that if f n is a sequence of uniformly L p-bounded functions on a measure space, and if f n → f pointwise a.e., then lim for all 0 < p < ∞. This result is also generalized in Theorem 2 to… Expand

Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities

- E. Lieb
- Mathematics
- 1 September 1983

A maximizing function, f, is shown to exist for the HLS inequality on R': 11 IXI - * fIq < Np f A , Iif IIwith Nbeing the sharp constant and i/p + X/n = 1 + 1/q, 1 <p, q, n/X < x. When p =q' or p = 2… Expand

Two Soluble Models of an Antiferromagnetic Chain

- E. Lieb, T. Schultz, D. Mattis
- Physics
- 1 December 1961

Two genuinely quantum mechanical models for an antiferromagnetic linear chain with nearest neighbor interactions are constructed and solved exactly, in the sense that the ground state, all the… Expand

Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation

- E. Lieb
- Mathematics
- 1977

The equation dealt with in this paper is in three dimensions. It comes from minimizing the functional which, in turn, comes from an approximation to the Hartree-Fock theory of a plasma. It describes… Expand

Harmonic maps with defects

- Ha im Brezis, J. Coron, E. Lieb
- Mathematics
- 1 December 1986

Two problems concerning maps ϕ with point singularities from a domain Ω C ℝ3 toS2 are solved. The first is to determine the minimum energy of ϕ when the location and topological degree of the… Expand

Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum

- E. Lieb
- Physics
- 15 May 1963

We continue the analysis of the one-dimensional gas of Bose particles interacting via a repulsive delta function potential by considering the excitation spectrum. Among other things we show that: (i)… Expand

Best Constants in Young’s Inequality, Its Converse, and Its Generalization to More than Three Functions

- H. J. Brascamp, E. Lieb
- Mathematics
- 1 May 1976

The best possible constant Dmt in the inequality | ∬ dx dyf(x)g(x —y) h(y)| |, 1/p + llq+ 1/t = 2, is determined; the equality is reached if /, g, and h are appropriate Gaussians. The same is shown… Expand

On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation

- H. J. Brascamp, E. Lieb
- Mathematics
- 1 August 1976

We extend the Prekopa-Leindler theorem to other types of convex combinations of two positive functions and we strengthen the Prekopa—Leindler and Brunn-Minkowski theorems by introducing the notion of… Expand

Exact analysis of an interacting bose gas. I. the general solution and the ground state

- E. Lieb, W. Liniger
- Physics
- 15 May 1963

A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the… Expand