SOME CONJECTURES ABOUT INTEGRAL MEANS OF ∂f AND ∂f

Here we denote by Ẇ (C,C) denotes the “homogeneous” Sobolev space of complex valued locally integrable functions in the plane whose distributional first derivatives are in L on the plane. Integrals without specified variables are understood to be with respect to Lebesgue measure. Since ∂(f) = ∂f, Conjecture 1 is true if and only if (1.1) always holds when L(∂f, ∂f) in the integral is replaced by L(∂f, ∂f). The function L, with a plus 1 added to the right hand side, was introduced by Burkholder [Bu4], [Bu5, p.20]. In his setting, the variables z and w are taken from an arbitrary Hilbert space. It