Sobolev inequalities on product Sierpinski spaces

TL;DR

This paper establishes Sobolev inequalities involving mutually singular measures on product Sierpinski spaces, providing conditions for their validity and calculating sharp exponents for specific measures, advancing analysis on fractals.

Contribution

It introduces Sobolev inequalities with singular measures on fractals, offering necessary and sufficient conditions and explicit exponent calculations for product Sierpinski spaces.

Findings

Established Sobolev inequalities involving singular measures.

Derived necessary and sufficient conditions for inequalities.

Computed sharp exponents for the product Kusuoka measure.

Abstract

On fractals, different measures (mutually singular in general) are involved to measure volumes of sets and energies of functions. Singularity of measures brings difficulties in (especially non-linear) analysis on fractals. In this paper, we prove a type of Sobolev inequalities, which involve different and possibly mutually singular measures, on product Sierpinski spaces. Sufficient and necessary conditions for the validity of these Sobolev inequalities are given. Furthermore, we compute the sharp exponents which appears in the sufficient and necessary conditions for the product Kusuoka measure, i.e. the reference energy measure on Sierpinski spaces.