Imbeddings and bounds for functions with -Laplacians

Spaces of locally integrable functions on R n that vanish at ∞ and whose gradient and Laplacian are in L p ( R n ; R n ) , L q ( R n ) respectively are defined. A representation theorem for such functions is described and properties of the fundamental solution of the modified Laplacian operator are used to prove L r and supremum norm inequalities when n = 3 . Imbedding results for these spaces into L r ( R 3 ) and C 0 ( R 3 ) when 1 ⩽ p < 3 are described. The case p = q = 2 yields a reproducing kernel Hilbert space of functions on R 3 . Some different estimates for solutions of the finite mass and energy solutions of Poissonʼs equation on R 3 are found using these results.