Bounds for $W^{2,\infty}$ functions

Question

For a $W^{2,\infty}$ function $h$, how do you prove the following inequality?

$$|h(x) +h(y) - 2h(z)| \le \|h\|_{W^{2,\infty}} \left(|x-z|^4 + |y-z|^4 + |x+y-2z|^2\right)^{\frac 12}$$

Answer 1

Depending on the exact definition of the Sobolev norm $\|\cdot\|_{W^{2,\infty}}$, the statement might involve a slightly different constant. In the following, we will assume $$ \|h\|_{W^{2,\infty}} = \|h\|_\infty + \|h'\|_\infty + \|h''\|_\infty. $$

Using repeatedly the foundamental theorem of calculus we have $$ \begin{align} |h(x) + h(y) - 2 h(z)| &= |h(x) - h(z) + h(z) - h(y)|\\ &=\left| \int_z^x h'(\xi) \, \mathrm{d} \xi + \int_z^y h'(\xi) \, \mathrm{d} \xi \right|\\ &=\left| \int_z^x (h'(\xi) - h'(z)) \, \mathrm{d} \xi + \int_z^y (h'(\xi)-h'(z)) \, \mathrm{d} \xi + (x+y-2z)h'(z) \right|\\ &=\left| \int_z^x \int_z^\xi h''(t) \, \mathrm{d} \xi \mathrm{d}t + \int_z^y \int_z^\xi h''(t) \, \mathrm{d} \xi \mathrm{d}t + (x+y-2z)h'(z) \right|. \end{align} $$

Using the triangular inequality and the definition of $\|\cdot\|_{W^{2,\infty}}$ we deduce $$ \begin{align} |h(x) + h(y) - 2 h(z)| &\leq \|h''\|_\infty \frac{|x-z|^2}{2} + \|h''\|_\infty \frac{|y-z|^2}{2} + \|h'\|_\infty |x+y-2z|\\ &\leq \|h\|_{W^{2,\infty}} \left( \frac{|x-z|^2}{2} + \frac{|y-z|^2}{2} + |x+y-2z| \right). \end{align} $$

Finally, recalling the inequality $a + b + c \leq \sqrt 2 (a^2+b^2+c^2)^{1/2}$ for $a,b,c >0$, we deduce $$ |h(x) + h(y) - 2 h(z)| \leq \|h\|_{W^{2,\infty}} \left( |x-z|^4 + |y-z|^4 + 2|x+y-2z|^2 \right)^{1/2} $$ which is the desired inequality up to a constant.

P.S.: The desired constant can be achieved in this setting by considering the following improvements on the above argument:

1) a more careful treatment of the estimate $\max( \|h'\|_\infty, \|h''\|_\infty) \leq \|h\|_{W^{2,\infty}}$ noting that this step can be improved by writing $$ |h(x) + h(y) - 2 h(z)| \leq \|h\|_{W^{2,\infty}} \left( \epsilon \frac{|x-z|^2}{2} + \epsilon \frac{|y-z|^2}{2} + (1-\epsilon) |x+y-2z| \right) $$ with $0 < \epsilon < 1$;

2) a finer inequality such as $a+b+c \leq (\alpha a^2 + \alpha b^2 + \beta c^2)^{1/2}$ in the last passage which holds true for $\alpha \geq 2$ and $\beta \geq \alpha / (\alpha -1)$.