# fourier analysis – Gradient condition implies Lipschitz condition

Let $$K:mathbb R^nsetminus{0}to mathbb C$$ be a smooth function with the estimate $$|nabla K(x)|leq C|x|^{-n-1}$$ for $$xneq 0$$ where $$|.|$$ is the Euclidean distance function and $$nabla$$ gradient. How to prove the following statement? $$|K(x-y)-K(x)|leq C|y|^{delta}/|x|^{n+delta}$$ for $$|x|geq 2|y|$$ and for some choice of $$deltain(0,1)$$ independent of $$x$$ and $$y.$$ Using mean value theorem I can show that $$|K(x-y)-K(x)|leq C|y|/|zeta|^{n+1}$$ where $$zeta$$ is in the line segment joining $$x-y$$ and $$x.$$ I’ll be done if $$|zeta|geq |x|$$ whenever $$|x|geq 2|y|$$.