Abstract. In this paper we study the space of functions in the unit ball in C n annihilated by the differential operators ∆ α,β , α, β ∈ C, given by ∆ α,β = (1 − |z| 2 )

We study positive measures on 𝔹n satisfying that . for any ƒ ∈ where is the Hardy-Sobolev space in the unit ball. We obtain several computable sufficient conditions as well as some necessary conditions and establish their sharpness. We study the same problem for Besov-Sobolev spaces and give some applications to multipliers.

Abstract. We study trace inequalities of the typein the "upper triangle case" 1 ≤ q < p for integral operators T k with positive kernels, where dσ and dµ are positive Borel measures on R n . Our main tool is a generalization of Th. Wolff's inequality which gives two-sided estimates of the energyassociated with the kernel k and measures dµ, dσ. We initially work with a dyadic integral operator with kernelwhere D = {Q} is the family of all dyadic cubes in R n , and K : D → R + . The corresponding continuous versions of Wolff's inequality and trace inequalities are derived from their dyadic counterparts.

carme cascante, joaquõ Ân m. ortega and igor e. verbitsky 1 1 À zz n À a ; for 0 < a < n; z P B n ; z P S n :In general, (1.6) is not equivalent to (1.5) even for p q (see [2,3,6,7], and [5] for q < p). In § 4 we prove that if 0 < n À ap < 1 and 1 < q < p, then problems (1.5) and (1.6) are indeed equivalent. For n À ap > 1 some related results are obtained. Trace inequalities of p; q-type with 1 < q < pLet m be a positive Borel measure on R n , 1 < q < p < 1, and 0 < a < n. In this section we give a new characterization of the class of measures m such that the trace inequality kI a à f k L q dm < C k f k L p d x 2:1 carme cascante, joaquõ Ân m. ortega and igor e. verbitsky

Abstract. We obtain characterizations of positive Borel measures µ on B n so that some weighted Hardy-Sobolev are imbedded in L p (dµ), where w is an