In the following $(\cdot,\cdot)$ denotes the usual Euclidean inner product.
I am struggling to understand a claim in this paper https://projecteuclid.org/download/pdfview_1/euclid.aoap/1216677126#page=5&zoom=100,0,0 . Although the paper is in the area of stochastics my misunderstanding seems to be a simple multi-variable calculus confusion.
The setup is this (Assumption $2.1$), $V:\mathbb{R}^d \to \mathbb{R}^d$ is locally lipshitz with polynomial growth. $V$ is continuously differentiable and with $DV$ its Jacobian satisfies (for constants $K_V,R_0>0$) $$ (h,DV(x)h)\leq -K_V ~~\forall ||h||=1,x\geq R_0.$$
The authors then claim (Lemma $2.2$), for all $x\in \mathbb{R}^d$ there exists $K>0$ such that $$ (x-y,V(x)-V(y))\leq K||x-y||^2. $$
$\textbf{My question :}$ So I understand the proof of this claim except the statement (top of page 1384) that : "by continuity of $DV$, there exists $K>0$ such that
$$ (h,DV(x)h)\leq K~~ \forall ||h||=1, x\in \mathbb{R}^d"$$
Is this really obvious? Sorry for my stupidity. Thanks in advance.
Yes, that's obvious. You have an upper bound for $(h,DV(x)h)$ for every $x$ with $||x|| \ge R_0$, by assumption.
So $DV$, which is assumed to be continous, has to be estimated only on a compact set ($||x|| \le R_0$), where, by continuity, it will have bounded norm: $$||DV||_{B_{R_0}}\le M$$ say. Now since $h$ is taken from the unit sphere the claim follows.