Abstract

Trudinger and Moser, interested in certain nonlinear problems in differential geometry, showed that if |∇u|^q is integrable on a bounded domain in Rⁿ with q ≥ n ≥ 2, then u is exponentially integrable there. Symmetrization reduces the problem to a one-dimensional inequality, which Jodeit extended to q > 1. Carleson and Chang proved that this inequality has extremals when q ≥ 2 is an integer. Hence, so does the Moser-Trudinger inequality (with q = n).

This paper extends the result of Carleson and Chang to all real numbers q > 1. An application and some related results involving noninteger q are also discussed.

Authors