Beyond Local Maximal Operators

Abstract

Abstract. We obtain (essentially sharp) boundedness results for certain generalized local maximal operators between fractional weighted Sobolev spaces and their modifications. Concrete boundedness results between well known fractional Sobolev spaces are derived as consequences of our main result. We also apply our boundedness results by studying both generalized neighbourhood capacities and the Lebesgue differentiation of fractional weighted Sobolev functions.

A note on the weak* and pointwise convergence of BV functions

“…we then deduce from [12,Lemma 6.4] (based essentially on the facts that A i is an open set and that the Hardy-Littlewood maximal operator is bounded from…”

Local potential operator and uniform resolvent estimate for generalized Schrödinger operator in Orlicz spaces

“…This enables us to derive the classical good‐λ inequality by letting $$\delta \rightarrow \infty$$. Note that local maximal functions of forms similar to Equation (1.6) arise also in other settings, where the underlying spaces are endowed with some specific geometric structure (see [29–31, 37], and references therein).…”

On density of smooth functions in weighted fractional Sobolev spaces

“…Let us also mention other articles on similar topics. In [4] Luiro and Vähäkangas considered slightly different fractional Sobolev spaces, that are equipped with the seminorm…”

“…, where the kernel K does not have to be radial. The authors find some condition which is sufficient for the space [4], (3.8) and Lemma 3.4). We obtain a similar result, Theorem 15, with more general sets Ω, but less general kernels K.…”