A new look at subrepresentation formulas

TL;DR

This paper broadens subrepresentation formulas by incorporating general A_1-potential operators, fractional derivatives, and rough singular integrals, enhancing the understanding of pointwise inequalities in harmonic analysis.

Contribution

It introduces new extensions of subrepresentation formulas involving A_1-potentials, fractional derivatives, and rough singular integrals, advancing previous results.

Findings

Established local and global pointwise inequalities for A_1-potential operators.

Improved subrepresentation formulas using fractional derivatives.

Extended results to rough singular integral operators.

Abstract

We extend the subrepresentation formula $$ |f(x)|\le c_n\,I_1(|\nabla f|)(x) $$ in several ways. First, we consider more general $A_1$-potential operators on the right-hand side and prove local and global pointwise inequalities for these operators. Second, we show that we can improve the right-hand side using fractional derivatives. Finally, we extend our results to rough singular integral operators, similar to the main result in [HMP1].