A note on the density of the parabolic area integral (Q5935766)

A solution to the heat equation is called a caloric function. Its boundary behavior can be studied via the heat area integral. Following the lines of \textit{J. Brossard}'s paper [Invent. Math. 93, No. 2, 297--308 (1988; Zbl 0655.31004)] about the Laplace equation the author considers the density \(D_T^r(\theta)\) of the heat area integral at level \(r\). For \(1< p < \infty\) it is shown that the \(L^p\)-norm of \(D^\ast(\theta):= \sup_{r,T} D_T^r(\theta)\) can be estimated from below and above by appropriate multiples of the \(L^p\)-norm of the heat area integral. The probabilistic methods of Gundy and Brossard are used together with the Barlow-Yor inequalities to prove the desired estimates.