L estimates for angular maximal functions associated with Stieltjes and Laplace transforms

Maximal angular operator sends a function defined in a sector of the complex plane to a function of modulus obtained by maximizing over all admissible values of the argument for the given modulus. The compositions of the so obtained maximal angular operator with the Poisson, Stieltjes and Laplace transform (in the sectors of their respective ranges) are shown to be bounded (nonlinear) operators from Lp to Lq for the naturally expected values of p and q. 1. Suppose g(z) is a complex-valued function defined in the sector C(θ1,θ2) = {z ∈ C | θ1 < arg z < θ2}. The angular maximal function for g with respect to C(θ1,θ2) is a function R+ → [0,∞] defined as follows: M12 g (ρ) = ess sup θ∈(θ1,θ2) |g(ρe)|. (1) In this work, we will only deal with continuous (in fact, harmonic) functions g(z), so the question whether g(ρe) is measurable and in what sense will never arise. Also, in this situation ess sup in (1) can be replaced by sup. For convenience we introduce shorter notation for the angular maximal functions corresponding to the three sectors that will be predominantly used: the plane cut along the real axis C = C(0,2π), the upper half-plane H = C(0,π), and the right half-plane C+ = C(−π/2,π/2): Mg(ρ) = sup 0<θ<2π |g(ρe)|, Mg (ρ) = sup 0<θ<π |g(ρe)|, Mg (ρ) = sup |θ|<π/2 |g(ρe)|. 2. The objects of our study are the angular maximal functions associated with the Poisson transform, the Stieltjes transform, and the Laplace transform.