On discrete fractional integral operators and mean values of Weyl sums

On Discrete Fractional Integral Operators and Mean Values of Weyl Sums

In this paper we prove new ` → ` bounds for a discrete fractional integral operator by applying techniques motivated by the circle method of Hardy and Littlewood to the Fourier multiplier of the operator. From a different perspective, we describe explicit interactions between the Fourier multiplier and mean values of Weyl sums. These mean values express the average behaviour of the number rs,k(...

Weights for Commutators of the One-sided Discrete Square Function, the Weyl Fractional Integral and Other One-sided Operators

Abstract. The purpose of this paper is to prove strong type inequalities with one-sided weights for commutators (with symbol b ∈ BMO) of several one-sided operators, such as the one-sided discrete square function, the one-sided fractional operators, or one-sided maximal operators given by the convolution with a smooth function. We also prove that b ∈ BMO is a necessary condition for the bounded...

A Note on Discrete Fractional Integral Operators on the Heisenberg Group

We consider the discrete analogue of a fractional integral operator on the Heisenberg group, for which we are able to prove nearly sharp results by means of a simple argument of a combinatorial nature.

φ-factorable operators and Weyl-Heisenberg frames on LCA groups

Multilinear integral operators and mean oscillation

Let b ∈ BMO(Rn) and T be the Calderón–Zygmund singular integral operator. The commutator [b,T ] generated by b and T is defined by [b,T ] f (x) = b(x)T f (x)−T (b f )(x). By a classical result of Coifman et al [6], we know that the commutator is bounded on Lp(Rn) for 1 < p < ∞. Chanillo [1] proves a similar result when T is replaced by the fractional integral operators. In [9], the boundedness ...