Continuity of the maximal operator in sobolev spaces

By Hannes Luiro. 2006

112 citations recorded by the source corpus.

We establish the continuity of the Hardy-Littlewood maximal operator on Sobolev spaces W 1,p (R n ), 1 < p < ∞. As an auxiliary tool we prove an explicit formula for the derivative of the maximal function.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/001d3846b6fbee66ff46764df4ddb76f3329467c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

On approximate differentiability of the maximal function (2010, background): “The following result is due to Luiro [20] when p > 1 and is new when p = 1.”
On the variation of the Hardy-Littlewood maximal function (2012, background): “[14] H. Luiro, On the differentiability of directionally differentiable functions and applications, arXiv:1208.3971v2.”
Endpoint Sobolev and BV continuity for maximal operators, II (2017, background): “In the remarkable paper [20], Luiro establishes the continuity of M in the W 1,p(Rd)−setting for p > 1.”
On the endpoint regularity of discrete maximal operators (2013, unknown): “The continuity of M : W 1,p → W 1,p for p > 1 was established by Luiro in [13] for the classical Hardy–Littlewood maximal operator and in [14] for its”
Corrigendum to “On a discrete version of Tanaka’s theorem for maximal functions” (2010, background): “n 1998 by Kinunnen [9] that studies the regularity of maximal operators acting on Sobolev functions. Since then, many authors have contributed to extend the theory, for instance [4], [5], [10], [11], [12], always having in the background the general principle that, for maximal operators, an Lp-bound”
On the regularity of the Hardy-Littlewood maximal operator on subdomains of ℝn (2010, methodology): “The proof starts along the same lines as in [12], but the case of subdomains introduces new fairly subtle difficulties arising from the boundary effects (see Remark 2.”
Derivative bounds for fractional maximal functions (2015, background): “Other interesting works related to this theory are [1, 16, 17, 21].”
On the regularity of maximal operators (2008, methodology): “Proof of Theorem 1: Continuity Here we follow carefully the beautiful proof for the continuity in W (R) of the Hardy-Littlewood maximal operator in [8].”
On the variation of maximal operators of convolution type (2013, background): “The continuity of M : W (R) → W (R) for p > 1 was established by Luiro in [13].”
The variation of the maximal function of a radial function (2017, background): “the sign of h. The conclusion is that the above equality is possible only if (2.12) is valid. Proof of Lemma 2.2. (1) The claim is counterpart for the formula for DM cf, which was ﬁrst time proved in [L]. Suppose that B = B(z,r) ∈ B x and let B h := 6 HANNESLUIRO B(z + hv,r). Then it holds that D vM”