A new look at subrepresentation formulas

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].