SOME ESTIMATES OF HARMOMC MAJORAI\TS

The problem of the harmonic majorization of a given subharmonic function in an unbounded domain in Äd involves the Dirichlet problem with the given boundary values and hence tle use of harmonic measure. In the plane case there is the following connection with the theory of Hardy spaces. Let F be a univalent analytic function from the unit disc onto a domain D in the z'plane and let O=P<-. Then F€Hp+lzle has a harmonic majorant in D (see e.g. [4, p. 28]). We note that the implicationeholds without F being univalent. This classical result has been associated with the theory of Brownian exit times by Burkholder, [2], [3]. Let r denote the exit time from D (an open connected set in Rd) of a Brownian motion, starting atzero time from a point x in D. Burkholder proved (for 0=p< that the expectation E*rPtz is finite if and only if lxle has a least harmonic majorant u(x) in D. In fact (see [2, p. 191 (3.3)]), kr,alxle ko,au(x) < dplzExxplz*lxle Kr,au(x), with constants kr,a and Ke,a depending only on p and. d. Thus, one can ask the question whether tle order of magnitudeof u(x) andhence tb4t of E*{tz canbe greater than that of lxle. One purpose of this paper is to consider an estimate of the type u(x) Clxle log lxl, l*l = 1r = 1, which can be shown to be sharp in suitable domains; see Section 3. The majorization of some other subharmonic functions is also treated; see Section 4. A preliminary version of the results was given in t8l