This paper by Linda Preiss Rothschild and E. M. Stein explores hypoelliptic differential operators and nilpotent groups. The authors develop a class of singular integral operators modeled on convolution operators on certain nilpotent Lie groups to construct parametrices and obtain sharp regularity results in terms of Sobolev and Lipschitz spaces. The main objectives include finding suitable nilpotent groups for analyzing operators, studying related questions on these groups, and applying this analysis to original partial differential operators to derive regularity results. The paper covers topics such as the Campbell-Hausdorff formula, proofs of lemmas, properties of the map $\Theta$, $L^p$ inequalities for operators of type $\lambda$, operators of type $\lambda$ and vector fields, parametrices, and spaces $S^0_p$, $L^p_\alpha$, and $\Lambda_\alpha$. The authors also discuss hypoelliptic operators, including sums of squares of vector fields and operators of Hörmander type, and provide estimates for $\Box_b$. The paper is supported by grants from the National Science Foundation and is published in Acta Mathematica.This paper by Linda Preiss Rothschild and E. M. Stein explores hypoelliptic differential operators and nilpotent groups. The authors develop a class of singular integral operators modeled on convolution operators on certain nilpotent Lie groups to construct parametrices and obtain sharp regularity results in terms of Sobolev and Lipschitz spaces. The main objectives include finding suitable nilpotent groups for analyzing operators, studying related questions on these groups, and applying this analysis to original partial differential operators to derive regularity results. The paper covers topics such as the Campbell-Hausdorff formula, proofs of lemmas, properties of the map $\Theta$, $L^p$ inequalities for operators of type $\lambda$, operators of type $\lambda$ and vector fields, parametrices, and spaces $S^0_p$, $L^p_\alpha$, and $\Lambda_\alpha$. The authors also discuss hypoelliptic operators, including sums of squares of vector fields and operators of Hörmander type, and provide estimates for $\Box_b$. The paper is supported by grants from the National Science Foundation and is published in Acta Mathematica.