In this talk, we present recent work with Cong Hoang and Kabe Moen, where we extend the classical Sobolev-type subrepresentation formula

$|f(x)| \le c_n\,I_1(|\nabla f|)(x)$,

in different ways. $I_1$ is the classical fractional integral operator or Riesz Potential of order one. First, we will show how to generalize $I_1$ to a family of $A_1$-potential type operators. Next, we show how the right-hand side can be further refined by using fractional derivatives instead of the gradient, incorporating the Bourgain-Brezis-Mironescu factor. We will also discuss the applications of these results to rough singular integral operators.
   We will highlight the connections to  isoperimetric inequalities and extrapolation theory.