Domination conditions for families of quasinearly subharmonic functions

Domar has given a condition that ensures the existence of the largest subharmonic minorant of a given function. Later Rippon pointed out that a modification of Domar's argument gives in fact a better result. Using our previous, rather general and flexible, modification of Domar's original argument, we extend their results both to the subharmonic and quasinearly subharmonic settings.

then w is locally bounded above in D, and thus w * is subharmonic in D.
As pointed out by Domar, [2, pp. 436-440], and by Rippon, [7, p. 129], the above results are for many particular cases sharp. As Domar also points out, [2, p. 430], the result of his theorem holds in fact for more general functions, that is, for functions which by good reasons might be -and indeed already have been! -called quasinearly subharmonic functions. See below for the definition of this function class. u(y) dm n (y).
The function u ≡ −∞ is considered subharmonic. We say that a function u : Observe that in the standard definition of nearly subharmonic functions one uses the slightly stronger assumption that u ∈ L 1 loc (D), see e.g. [3, p. 14]. However, our above, slightly more general definition seems to be more practical.
We say that a Lebesgue measurable function u : Observe that a function u is 1-quasinearly subharmonic if and only if it is nearly subharmonic (in the above slightly more general sense).
We recall here only that this function class includes, among others, subharmonic functions, and, more generally, quasisubharmonic and nearly subharmonic functions (see e.g. [3, pp. 14, 26]), also functions satisfying certain natural growth conditions, especially certain eigenfunctions, and polyharmonic functions. Also, the class of Harnack functions is included, thus, among others, nonnegative harmonic functions as well as nonnegative solutions of some elliptic equations. In particular, the partial differential equations associated with quasiregular mappings belong to this family of elliptic equations.

AN IMPROVEMENT TO THE RESULTS OF DOMAR AND RIPPON
Our improvement to the results of Domar and Rippon is the following:  Remark. Our theorem is indeed flexible. One corollary is obtained by replacing the above condition (iii) e.g. by (iii') The following integral is convergent: Another corollary is the following: Remark. To get a more concrete corollaries, choose for example p > 0 and φ(t) = t p . The special case p = 1 and K = 1 then gives Domar's and Rippon's results. A more complicated, but still a concrete corollary, is obtained by choosing p > 0, q > 0 and φ(t) = t p (logt) q , say.

SEPARATELY SUBHARMONIC FUNCTIONS
For results on this area, see [1,4,5] and the references therein. Here we state, as an example, only the following partial result: Let Ω be a domain in R m+n , m, n ≥ 2. Let u : Ω → [−∞, +∞) be such that (a) for each y ∈ R n the function Ω(y) x → u(x, y) ∈ [−∞, +∞) is nearly subharmonic, and, for almost every y ∈ R n , subharmonic, (b) for each x ∈ R m the function Ω(x) y → u(x, y) ∈ [−∞, +∞) is upper semicontinuous, and, for almost every x ∈ R m , (nearly) subharmonic, (c) for some p > 0 there is a function v ∈ L p loc (Ω) such that u ≤ v. Then u is upper semicontinuous and thus subharmonic in Ω.