# Fefferman’s original proof of Open image in new window

• Akihito Uchiyama
Part of the Springer Monographs in Mathematics book series (SMM)

## Abstract

$$\left| {\smallint _{{\text{R}}^n } \vec Rf\left( {x.} \right) \cdot \vec g\left( x \right)dx} \right| \leqslant C\left( n \right)\left\| {\vec Rf} \right\|_{L^1 } \left\| {\vec g} \right\|_{{\text{BMO}}}$$
(19.1)
where
$$f \in L^2 \left( {{\text{R}}^n ,{\text{R}}} \right),\vec g \in L^2 \left( {{\text{R}}^n ,{\text{R}}^{n + 1} } \right)$$
(19.2)
In Sections 19‐20, we let
$$\begin{gathered} \Delta = D_t^2 + D_{x_1 }^2 + \cdots + D_{x_n }^2 ,\nabla = \nabla _{t,x} = \left( {D_t ,D_{x_1 } \cdots ,D_{x_n } } \right) \hfill \\ \left| {\nabla _u } \right| = \left\{ {\sum\limits_{j = 0}^n {\left( {D_{x_j } u} \right)^2 } } \right\}^{1/2} , \hfill \\ \left| {\nabla ^2 u} \right| = \left\{ {\sum\limits_{j = 0}^n {\sum\limits_{i = 0}^n {\left( {D_{x_j } D_{x_i } u} \right)^2 } } } \right\}^{1/2} ,{\text{where }}D_{x_0 } = D_t . \hfill \\ \end{gathered}$$

## Keywords

Euclidean Space Harmonic Function Number Theory Algebraic Geometry Hardy Space
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.