Abstract
In this paper we investigate the endpoint regularity of the discrete m-sublinear fractional maximal operator associated with ell^{1}-balls, both in the centered and uncentered versions. We show that these operators map ell^{1}(mathbb{Z}^{d})timescdotstimes ell^{1}(mathbb{Z}^{d}) into operatorname{BV}(mathbb{Z}^{d}) boundedly and continuously. Here operatorname{BV}(mathbb{Z}^{d}) represents the set of functions of bounded variation defined on mathbb{Z}^{d}.
Highlights
1 Introduction 1.1 Background The regularity theory of maximal operators has been the subject of many recent articles in harmonic analysis
The first work was contributed by Kinnunen [1] who investigated the Sobolev regularity of the centered Hardy–Littlewood maximal function M and proved that M is bounded on the first order Sobolev spaces W 1,p(Rd) for all 1 < p ≤ ∞
The above result was extended to a local version in [3], to a fractional version in [4], to a multisublinear version in [5, 6] and to a one-sided version in [7]
Summary
1.1 Background The regularity theory of maximal operators has been the subject of many recent articles in harmonic analysis. The first work was contributed by Kinnunen [1] who investigated the Sobolev regularity of the centered Hardy–Littlewood maximal function M and proved that M is bounded on the first order Sobolev spaces W 1,p(Rd) for all 1 < p ≤ ∞. It was noticed that the W 1,p-bound for the uncentered maximal operator M holds by a simple modification of Kinnunen’s arguments or [2, Theorem 1]. In the remarkable work [20], Carneiro et al proved that the operator f → (Mf ) is continuous from W 1,1(R) to L1(R).
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