A general notion of lifting properties for families of sesquilinear forms is formulated. These lifting properties, which appear as particular cases in many classical interpolation problems, are studied for the Toeplitz kernels in Z, and applied for refining and extending the Nehari theorem and the Paley lacunary inequality.

We consider two standard group representations: one acting on functions by translations and dilations, the other by translations and modulations, and we study local Toeplitz operators based on them. Local Toeplitz operators are the averages of projection-valued functions $g\mapsto P_{g,\varphi }$, where for a fixed function ϕ, $P_{g,\varphi }$ denotes the one-dimensional orthogonal projection on the function $U_g\varphi$, U is a group representation and g is an element of the group. They are defined as integrals ${ʃ}_W{P}_{g,\varphi }dg$, where W is an open, relatively...