INEQUALITIES ASSOCIATED WITH DILATIONS

Abstract. Some properties of distributions f satisfying x · ∇f ∈ Lp(Rn), 1 ≤ p <∞, are studied. The operator x · ∇ is the generator of a semi-group of dilations. We first give Sobolev type inequalities with respect to the operator x ·∇. Using the inequalities, we also show that if f ∈ Lploc(Rn), x · ∇f ∈ Lp(Rn) and |x|n/p|f(x) | vanishes at infinity, then f belongs to Lp(Rn). One of the Sobolev type inequalities is shown to be equivalent to the Hardy inequality in L2(Rn). 1