Sobolev spaces, Lebesgue points and maximal functions

On the variation of the Hardy-Littlewood maximal function

“…For other related results, see also e.g. [2,4,5,6,12]. For results considering other concepts than the weak differentiability, see [7,14].…”

Maximal potentials, maximal singular integrals, and the spherical maximal function

“…In our main result (Theorem 1.1) we will prove that the operator A * Ω : L p →Ẇ 1,p is bounded for all n/(n − 1) < p < n. We do not know what happens when 1 < p ≤ n/(n − 1), but we will show that the positive answer to the questions about boundedness for 1 < p ≤ n/(n − 1) would imply a positive answer to a question about boundedness of the spherical maximal function in the Sobolev space, [5], [6], see Proposition 1.2. Now we can state our main result.…”

“…Using this result it is easy to prove, see [5], [6], that actually the spherical maximal operator is bounded in the Sobolev spaces S :…”

“…In particular S : W 1,p →Ẇ 1,p is bounded for n/(n − 1) < p < n. It was posted as an open problem in [5], [6], whether S : W 1,p →Ẇ 1,p is bounded for 1 < p ≤ n/(n − 1). The case of the spherical maximal function has also been discussed in [8].…”

A Marcinkiewicz integral type characterization of the Sobolev space

“…Indeed, according to a celebrated result of Stein [29], and Bourgain [4], S : L p (R n ) → L p (R n ) is bounded for p > n/(n − 1). It was conjectured in [18] that in the range 1 < p ≤ n/(n − 1) the spherical maximal operator is a bounded operator from W 1,p to the homogeneous Sobolev spaceẆ 1,p ; see [16,17] for results supporting this conjecture. The next result which is a direct consequence of Lemma 2.1 provides another support for this conjecture as it allows to represent S f as a Hardy-Littlewood type maximal function.…”