For which $ l \in\mathbb{N} $ and $ \alpha\in[0,1] $ does u belong to the hoelder space $ C^{l,\alpha}(\overline {B_1(0)}) $?

Question

$$ u:\overline {B_1(0)}\subset\mathbb{R^2}\to\mathbb{R} $$ defined by : $$ u(x_1,x_2)=x_1x_2(1-\sqrt{x_1^2+x_2^2}) $$ Can I at first consider $$ x_1x_2 $$ and look if $$ x_1x_2\in C^{l,\alpha}(\overline {B_1(0)})\ ? $$ , because the Hoelder space is a vector space .