  In this paper K closedness is proved in the case of the couple of real Hardy
spaces in the corresponding couple of Lebesgue spaces. This means roughly that
any measurable decomposition of an analytic function gives rise to an
"analytic" decomposition with summands of roughly the same size. The proof uses
Bourgain's method, the atomic decomposition for Hardy spaces and the
subharmonic property of the gradient of a system of conjugate harmonic
functions.
