ON SINGULAR MOSER-TRUDINGER INEQUALITY FOR EMBEDDING INTO EXPONENTIAL AND MULTIPLE EXPONENTIAL SPACES

Abstract

Let Omega subset of R-n, n >= 2, be a bounded domain containing the origin and let alpha < n - 1. We prove the Moser-Trudinger inequality with a singular weight 1/vertical bar x vertical bar(beta), beta epsilon (0,n), for the embedding of the space W0L(n) log L-alpha into the Orlicz space corresponding to a Young function that behaves like exp (n/t(n-1-alpha)) for large t.

We also give the result for the Orlicz-Sobolev spaces embedded into multiple exponential spaces. The ConcentrationCompactness Alternative for the singular Moser-Trudinger inequality is established to.

Keywords

• Orlicz spaces
• Orlicz-Sobolev spaces
• embedding theorems
• sharp constants
• Concentration-Compactness Principle