Conjugate functions on spaces of parabolic Bloch type

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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 + (−Δx)α)u = 0 with finite Bloch norm || u ||${\cal B}$<sub>α</sub>(σ) of a weight tσ. 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.

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