Lp estimates for strongly singular integrals on spaces of homogeneous type

A theory for strongly singular integrals on spaces of homogeneous type is developed and used to obtain sharp multiplier theorems for various groups.     

Fourier Expansions of Piecewise Smooth Functions

Although Fourier series or integrals of piecewise smooth functions may be slowly convergent, sometimes it is possible to accelerate their speed of convergence by adding and subtracting suitable combination of known functions.     

Inequalities of Paley type for noncommutative martingales

Summary The purpose of this paper is to study the validity of the Paley inequality on square function, for noncommutative martingales. Let be a regular gage space, and a sequence of von-Neumann algebras such that we prove that for every , where ɛ_n(F) is the conditional expectation of F with respect to the subalgebra : We also consider the case of a martingale arising in the context of harmonic analysis on noncommutative discrete groups, in analogy to the theorem of R.E.A.C. Paley on Fourier-Walsh series. Entrata in Redazione il 26 gennaio 1977. Partially sponsored by C.N.R.     

Some results about smoothing methods of fourier series

Summary In this paper we compare the Fourier polynomials approximation, in C(Tⁿ) or in L^p(Tⁿ), with that one obtained by a class of smoothing methods, which naturally arise in solving ill posed problems. It is also given a sharp evaluation of the above approximation in the space Lip (, C(T)), 0<<1.Work supported by Italian M.P.I.