Abstract :
We derive a sharp Moser–Trudinger inequality for the borderline Sobolev imbedding of W2,n/2(Bn)
into the exponential class, where Bn is the unit ball of Rn. The corresponding sharp results for the spaces
W
d,n/d
0 (Ω) are well known, for general domains Ω, and are due to Moser and Adams. When the zero
boundary condition is removed the only known results are for d = 1 and are due to Chang–Yang, Cianchi
and Leckband. The proof of our result is based on a new integral representation formula for the “canonical”
solution of the Poisson equation on the ball, that is, the unique solution of the equation u = f which is
orthogonal to the harmonic functions on the ball. The main technical difficulty of the paper is to establish
an asymptotically sharp growth estimate for the kernel of such representation, expressed in terms of its
distribution function.
© 2011 Elsevier Inc. All rights reserved.
