  We derive a sharp Moser-Trudinger inequality for the borderline Sobolev
imbedding of W^{2,n/2}(B_n) into the exponential class, where B_n is the unit
ball of R^n. The corresponding sharp results for the spaces W_0^{d,n/d}(\Omega)
are well known, for general domains \Omega, 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. Our proof is based on general
abstract results recently obtained by the authors, and 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 \Delta 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.
We will also consider the situation where the exponential class is endowed with
more general Borel measures, and obtain corresponding sharp Moser-Trudinger
inequalities of trace type.
