Mean Growth, $H_p $ Spaces and Subharmonic Functions in the Upper Half-Plane

Abstract

A function f analytic in the upper half-plane is said to be of class $H_p (0 < p < \infty )$ if the $L_p $ integrals of $f_y (x) = f(x + iy)$ along lines parallel to the real axis are uniformly bounded. In this paper we give alternate proofs of two results of B. F. Logan [SIAM J. Math. Anal., 10 (1979), pp. 733–740; 741–751] on $H_p $ spaces plus a subharmonic version of his second result. Our method of proof extends his $H_p $ results from $1 \leq p < \infty $ to $0 < p < \infty $. We show that if $f \in H_p (0 < p < \infty )$, then \[ \mathop {\lim }\limits_{y \to \infty } \left\| {f_y } \right\|_p = 0 \] and \[ | {f(x + iy)} | \leq A_p y^{{{ - 1} / p}} \left\{ {\int_{ - \infty }^\infty {| {f(t)} |^p } dt} \right\}^{{1 / p}} \] with $A_p = (4\pi )^{{{ - 1} /p}} $. The constant $A_p $ is best possible and necessary and sufficient conditions are given for equality. The method of proof in each case consists in proving it for $p = 2$ by means of the one-sided Paley–Wiener theorem and then extending it to $0 < p < \infty $ by using the Blaschke factorization for $H_p $ functions. The subharmonic analogue for the second result shows that for a continuous nonnegative subharmonic function $g(z)$ defined in the upper half-plane, the condition \[ \int_{ - \infty }^\infty {g(x + iy)} dx \leq M < \infty \] for $y > 0$ implies \[ g(x + iy) \leq M(\pi y)^{ - 1} . \] The constant $\pi ^{ - 1} $ is best possible.