Imbeddings and bounds for functions with Lq-Laplacians

Spaces of locally integrable functions on Rn that vanish at ∞ and whose gradient and Laplacian are in Lp(Rn;Rn), Lq(Rn) 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 Lr and supremum norm inequalities when n=3. Imbedding results for these spaces into Lr(R3) and C0(R3) when 1⩽p<3 are described. The case p=q=2 yields a reproducing kernel Hilbert space of functions on R3. Some different estimates for solutions of the finite mass and energy solutions of Poissonʼs equation on R3 are found using these results.