Abstract
Trudinger and Moser, interested in certain nonlinear problems in differential geometry, showed that if |⊇u| q is integrable on a bounded domain in R n 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.
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