Weighted total variation

Question

Suppose $ P = \{x_1, x_2, \dots, x_n \} $ is a partition of $ [0, 1] $, i.e., $0 = x_1 < x_2 < \cdots < x_n = 1 $. Suppose $ g $ and $ w \geq 0 $ are smooth functions on $ [0, 1] $. I'd like to know whether $$ \sum_{i=1}^{n-1} |g(x_{i+1})-g(x_i)|w(i/n) \leq c\int_0^1 |g'(x)|w(x)dx. $$ for some universal positive constant $ c $. It seems related to total variation, but I'm not sure how to proceed. Any help would be very much appreciated!

Answer 1

Let $g$ and $w$ be "bump functions" with disjoint supports. For instance, $w$ can be supported in $[0,1/2]$ with its peak at $x=1/4$ and $g$ can be supported in $[1/2,1]$ with its peak at $x=3/4$. Clearly $$\int_0^1 |g'(x)| w(x) \, dx = 0.$$

If you let $n=4$ and use the partition $\{0,\frac 34,\frac 78, 1\}$ you'll find that $$\sum_{i=1}^{3} |g(x_{i+1})-g(x_i)|w(i/4) \ge |g(x_2) - g(x_1)|w(1/4) = g(3/4)w(1/4) > 0.$$