Conjugate functions on spaces of parabolic Bloch type

Abstract

Let H be the upper half-space of the (n+1)-dimensional Euclidean space. Let 0 < α ≤ 1 and m(α) = min {1, 1/(2α)}. For σ > −m(α), the α-parabolic Bloch type space ${\cal B}$_α(σ) on H is the set of all solutions u of the equation (∂/∂t + (−Δ_x)^α)u = 0 with finite Bloch norm || u ||_{${\cal B}$α}(σ) of a weight t^σ. It is known that ${\cal B}$_1/2(0) coincides with the classical harmonic Bloch space on H. We extend the notion of harmonic conjugate functions to functions in the α-parabolic Bloch type space ${\cal B}$_α(σ). We study properties of α-parabolic conjugate functions and give an application to the estimates of tangential derivative norms on ${\cal B}$_α(σ). Inversion theorems for α-parabolic conjugate functions are also given.