Recently, q-Bernstein polynomials have been intensively investigated by a number of authors. Their results show that for q = 1, q-Bernstein polynomials possess of many interesting properties. In this paper, the convergence rate for iterates of both q-Bernstein polynomials and their Boolean sum are estimated. Moreover, the saturation of {B n (·, q n )} when n → ∞ and convergence rate of B n ( f , q; x) when f ∈ C n−1 [0, 1], q → ∞ are also presented.

We prove sharp weak type weighted estimates for a class of sparse operators that includes majorants of standard singular integrals, fractional integral operators, and square functions. These bounds are known to be sharp in many cases, and our main new result is the optimal bound [w,σ ] Mathematics subject classification (2010): 42B20, 42B25.

In this paper we prove several weighted estimates for iterated product commutators generated by BMO -functions and the bilinear fractional integral operators on Morrey spaces. As a corollary we obtain new weighted estimates for Adams type inequality.