On fourier multipliers in weighted Triebel–Lizorkin spaces

The paper is devoted to two-weighted estimates for multipliers of Fourier transforms. Apparently this problem has not been studied before. Easy to verify conditions for the pairs of weights ensuring two-weight estimates for several classes of multipliers in Triebel–Lizorkin spaces are derived. 'Ikvo-weighted boundedness criteria for the Riemann–Liouville and Weyl transforms on the line are established. The authors' recent results on trace inequalities for one-sided potentials in the "diagonal" case are applied. The method of proof consists of the representation of the operators under consideration in the form of compositions of certain "elementary" transformations. Various examples of pairs of weights governing two-weighted estimates for Fourier multipliers are presented.