Two-weight estimates for fourier operators and bernstein inequality

Abstract

The norm estimation problem for Fourier operators acting from L-w(p)(T) to L-v(q)(T) where 1 < p <= q < infinity was investigated. These results has been generalized to the two-dimensional case and applied to obtain generalizations of the Bernstein inequality for trigonometric polynomials of one and two variables. Also, the rates of convergence of Cesaro and Abel-Poisson means of functions f is an element of L-w(p)(T) has been estimated in the case p = q and v equivalent to w. The generalized Bernstein inequality applied to estimate the order of best trigonometric approximation of the derivative of functions f is an element of L-w(p)(T) in the space L-v(q)(T).