ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES

By P. Hajłasz, Jani Onninen. 2004

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View Paper: https://www.semanticscholar.org/paper/5005b5a8f8486af6503d6c652166c6299db320fa

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Derivative bounds for fractional maximal functions (2015, background): “ltilinear, local and fractional contexts in [7, 12, 13]. Due to the lack of reﬂexivity of L1, results for p = 1 are subtler. Examples of such results were motivated by the following question posed in [10]: QuestionA.(Hajla sz and Onninen [10]) Is the operator f → |∇Mf| bounded from W1,1(Rd) to L1(Rd”
On the regularity of the Hardy-Littlewood maximal operator on subdomains of ℝn (2010, background): “1) is an easy corollary of the result of [5].”
On the variation of the Hardy-Littlewood maximal function (2012, background): “approachable than the case p>1. Because Mf /∈ L1 whenever f is non-trivial, Kinnunen’s result fails for p= 1. Still, one can ask whether the maximal function of f∈ W1,1 belongs locally to W1,1. In [8], the authors posed the following question. Question 1.1 (Hajla sz and Onninen). Is the operator ”
Corrigendum to “On a discrete version of Tanaka’s theorem for maximal functions” (2010, methodology): “ion of f. For the centered maximal operator M we prove that, given a function f : Z→ Rsuch that f ∈ ℓ1(Z), Var(Mf) ≤ Ckfkℓ1(Z). This provides a positive solution to a question of Hajl asz and Onninen [6] in the discrete one-dimensional case. 1. Introduction It is natural to expect that an averaging ”
On the variation of maximal operators of convolution type (2013, background): “variation, discrete operators. *corresponding author email: carneiro@impa.br. 1 2 E. CARNEIRO AND B. F. SVAITER the operator f→ ∇Mfis bounded from W1,1(Rd) to L1(Rd), posed by Hajla sz and Onninen in [7], remains open in its general case (see also [6]). Partial progress was achieved in the discrete ”