On the norms of generalized translation operators generated by Dunkl-type operators

The paper establishes the integral representation and improves the norm estimate for the generalized translation operators generated by Dunkl-type operators
$$ \Lambda f(x)=f'(x)+\frac{A'(x)}{A(x)}\,\frac{f(x)-f(-x)}2 $$
in the spaces $L_p(\mathbb R)$ with weight $A$. Under some natural conditions on the function $A$, it is proved that these norms do not exceed two.