A weak-type expression of the Orlicz modular
Martin K\v{r}epela, Zden\v{e}k Mihula, Javier Soria
TL;DR
This paper presents a new way to express Orlicz modulars using measure of level sets of difference quotients, complementing existing formulas for fractional seminorms and extending to Orlicz functions.
Contribution
It introduces a weak-type expression for Orlicz modulars that complements and extends the Maz'ya-Shaposhnikova formula to Orlicz functions.
Findings
Provides an equivalent expression of Orlicz modulars in terms of measure of level sets
Complements the Maz'ya-Shaposhnikova formula for fractional seminorms
Extends the formula to the setting of Orlicz functions
Abstract
An equivalent expression of Orlicz modulars in terms of measure of level sets of difference quotients is established. The result in a sense complements the famous Maz'ya-Shaposhnikova formula for the fractional Gagliardo-Slobodeckij seminorm and its recent extension to the setting of Orlicz functions.
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Taxonomy
TopicsApproximation Theory and Sequence Spaces · Analytic and geometric function theory · Mathematical functions and polynomials