Geodesic
On the norms of generalized translation operators generated by Dunkl-type operators
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 26, Tome 392 (2011), pp. 5-31

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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. 
@article{ZNSL_2011_392_a0,
     author = {O. L. Vinogradov},
     title = {On the norms of generalized translation operators generated by {Dunkl-type} operators},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--31},
     publisher = {mathdoc},
     volume = {392},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_392_a0/}
}
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O. L. Vinogradov. On the norms of generalized translation operators generated by Dunkl-type operators. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 26, Tome 392 (2011), pp. 5-31. http://geodesic.mathdoc.fr/item/ZNSL_2011_392_a0/