Discrete analogues in harmonic analysis II: Fractional integration

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L^p boundedness of discrete singular Radon transforms (2005, background): “Wainger [25] in connection with sharp boundedness properties of certain discrete fractional integral operators (see also [23] and [16]).”
Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications (2016, background): “We recall that Stein and Wainger showed in [27, Proposition (a)] that the operator acting on functions g : Z → R , is bounded from r into l , whenever 1 /l ≤ 1 /r − 1 + λ and 1 r l ∞ .”
Harmonic analysis associated with a discrete Laplacian (2014, unknown): “Wainger [19, 20]; see also [13], where a brief history and a nice exposition of recent progress on discrete analogues can be found.”
Corrigendum to “On a discrete version of Tanaka’s theorem for maximal functions” (2010, methodology): “a technique from number theory pioneered in the context of discrete analogues by Bourgain [2], [3], and further developed in a number of interesting cases (see for example [7], [8], [13], [14], [16], [17]). In this paper we introduce the study of the regularity theory of discrete maximal operators i”