Sharp Moser–Trudinger inequalities for the Laplacian without boundary conditions

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 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 Δu=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.