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

Estimates of functionals by generalized moduli of continuity generated by the dunkl operators

“…The last equality was established in [11,Remark 7]. Thus, the spaces L p,α (R), p ∈ [1, +∞), belong to B p,α .…”

“…The part of Property T2 concerning the spaces L p,α (R) is contained in [7,9] and CB(R) can be found in [8] (cf. also [11,Corollary 6]). The assertions for the classes L p,α,loc (R) follow from the corresponding assertions for L p,α (R) and the fact that the kernel of the generalized translation operator is finite [11,Corollary 5].…”

“…also [11,Corollary 6]). The assertions for the classes L p,α,loc (R) follow from the corresponding assertions for L p,α (R) and the fact that the kernel of the generalized translation operator is finite [11,Corollary 5]. The assertions for the classes W (r) p,α (R) and W (r) p,α,loc (R) are obvious since the operators T y and Λ commute.…”

“…The second inequality for the sum is easily obtained from the positivity of T on even functions [12, Theorem 3.1]. Property T3 is completely established in [11,Corollary 4 and Theorems 3,4].…”

Estimates of functionals by deviations of Steklov type averages generated by Dunkl type operators

“…We recall some facts about the harmonic analysis generated by a Dunkl type operator [7]- [9]. The earlier results concerning the Sturm-Liouville operator can be found in [3,10].…”