Narrow Results

Some properties of distributions f satisfying x · ∇ f ∈ L^p (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 , x · ∇ f ∈ L^p (ℝⁿ) and |x|^n/p|f(x)| vanishes at infinity, then f belongs to L^p (ℝⁿ). One of the Sobolev type inequalities is shown to be equivalent to the Hardy inequality in L² (ℝⁿ).