We prove that if is harmonic and there exists a polynomial such that f + W is nonnegative, then f is a polynomial.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba57-3-5, author = {Piotr Nayar}, title = {On Polynomially Bounded Harmonic Functions on the $Z^d$ Lattice}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, volume = {57}, year = {2009}, pages = {231-242}, zbl = {1191.31007}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba57-3-5} }
Piotr Nayar. On Polynomially Bounded Harmonic Functions on the $Z^d$ Lattice. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 57 (2009) pp. 231-242. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba57-3-5/