Let m ≥ 1 be an integer and N > 2m. Let µ be a positive Radon measure on R N . We study necessary and sufficient conditions on possible distributional solutions of (−∆) m u = µ on R N , that guarantee the validity of the representation formula u(2m a.e. on R N , where l ∈ R and c(2m) is a positive constant depending on m and N . Several consequences are derived. In particular we prove Liouville theorems for systems of higher order elliptic inequalities and weighted form of Hardy-Littlewood-Sobolev systems of integral equations. Mathematics Subject Classification (2000). Primary 35C15; Secondary 31C10, 45G15.

We prove a simple sufficient criteria to obtain some Hardy inequalities on Rie-\ud mannian manifolds related to quasilinear second-order differential operator ∆p u :=\ud div | u|p−2 u . Namely, if ρ is a nonnegative weight such that −∆p ρ ≥ 0, then\ud the Hardy inequality\ud c M\ud |u|p\ud | ρ|p dvg ≤\ud ρp\ud | u|p dvg ,\ud ∞\ud u ∈ C0 (M ).\ud M\ud holds. We show concrete examples specializing the function ρ.\ud Our approach allows to obtain a characterization of p-hyperbolic manifolds as\ud well as other inequalities related to Caccioppoli inequalities, weighted Gagliardo-\ud Nirenberg inequalities, uncertain principle and first order Caffarelli-Kohn-Nirenberg\ud interpolation inequality

The aim of this paper is to study necessary conditions for existence of weak solutions of the inequality L (x, y, D-x, D-y,) u greater than or equal to \u\(q)/\x\(01)\y\(02), xis an element of R^d, y is an element of R^k where L is a quasi-homogeneous differential operator, q>1, theta(1) .theta(2) is an element of R and k, d greater than or equal to 1. As special cases, our results can be applied when L is Tricomi or Grushin-type operators. (C) 2003 Elsevier Science (USA). All rights reserved

In this paper, we prove the following result. Let α be any real number between 0 and 2. Assume that u is a solution of {(-δ)α/2u(x)=0,x∈Rn,lim¯|x|→∞u(x)|x|γ≤0, for some 0≥≥;1 and γα. Then u must be constant throughoutRn. This is a Liouville Theorem for α-harmonic functions under a much weaker condition. For this theorem we have two different proofs by using two different methods: One is a direct approach using potential theory. The other is by Fourier analysis as a corollary of the fact that the only α-harmonic functions are affine