SHARP BOUNDS FOR FRACTIONAL TYPE OPERATORS WITH Lα,s-HORMANDER CONDITIONS

We provide the sharp bound for a fractional type operator given by a kernel satisfying the L-alpha,L-s-Hormander condition and certain fractional size condition, 0 < alpha < n and 1 < s <= infinity. In order to prove this result we use a new appropriate sparse domination. Examples of these operators include the fractional rough operators. For the case s = infinity we recover the sharp bound of the fractional integral, I-alpha, proved by Lacey et al. [J. Functional Anal. 259 (2010), no. 5, 1073-1097].