A weak-type inequality of subharmonic functions

We prove the weak-type inequality lambda mu(u + \upsilon\ greater than or equal to lambda) less than or equal to (alpha + 2) integral(partial derivative D) u d mu, lambda > 0, between a non-negative subharmonic function u and an H-valued smooth function upsilon, defined on an open set containing the closure of a bounded domain D in a Euclidean space R-n, satisfying \upsilon(0)\ less than or equal to u(0), \del upsilon\ less than or equal to \del u\ and \Delta upsilon\ less than or equal to alpha Delta u, where alpha less than or equal to 0 is a constant. Here mu is the harmonic measure on partial derivative with respect to 0. This inequality extends Burkholder's inequality in which alpha = 1 and H = R-nu, a Euclidean space.