The well-known general Tauberian theorem of N. Wiener is formulated and proved for distributions in the place of functions and its Ganelius' formulation is corrected. Some changes of assumptions of this theorem are discussed, too.
We investigate the behaviour of Fourier coefficients with respect to the system of ultraspherical polynomials. This leads us to the study of the “boundary” Lorentz space corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients of -functions turn out to behave like the Fourier coefficients of functions in the real Hardy space ReH¹. Namely, we prove that for any the series is the Fourier series of some function φ ∈ ReH¹ with .
We give conditions on pairs of weights which are necessary and sufficient for the operator to be a weak type mapping of one weighted Lorentz space in another one. The kernel is an anisotropic radial decreasing function.
Necessary conditions and sufficient conditions are derived in order that Bessel potential operator is bounded from the weighted Lebesgue spaces into when .
There is a one parameter family of bilinear Hilbert transforms. Recently, some progress has been made to prove Lp estimates for these operators uniformly in the parameter. In the current article we present some of these techniques in a simplified model...