Weak type inequalities in Hardy space (Q1114041)

The major result of the paper deals with an extension of the celebrated imbedding theorem of Carleson and its generalization by P. Duren and consists of establishing the equivalence of the condition \[ (*)\quad \mu \{z=x+iy\in {\mathbb{R}}^ 2_+:\quad | x-x_ 0| <\delta y_ 0,\quad | y-y_ 0| <\delta y_ 0\}\leq Cy^{\alpha}_ 0 \] for all \(z_ 0=x_ 0+iy_ 0\in {\mathbb{R}}^ 2_+\) for a measure \(\mu\) and the weak type inequality \[ \mu \{z\in {\mathbb{R}}^ 2_+:\quad | f(z)| (Im z)^{(1-\alpha)/p}>t\}\leq C(t^{-1}\| f\|_ p)^ p \] for all \(f\in H^ p\), \(0<p<\infty\) for the Hardy class \(H^ p\). Here, \(\alpha >0\), \(\alpha\neq 1\). Note that for \(1<\alpha <\infty\) (*) is equivalent to the Carleson condition of order \(\alpha\) introduced by Duren, but is weaker than that for \(\alpha <1\).