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Let (ϕt)t≥ 0 be a semigroup of holomorphic functions in the unit disk $\mathbb {D}$ and K a compact subset of $\mathbb {D}$ . We investigate the conditions under which the backward orbit of K under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of K are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that K forms with the unit disk.

By potential theoretic methods involving the Cartan fine topology a recent result by two of the authors is extended as follows: The Riesz charge of the lower envelope of a family of 3 or more δ-subharmonic functions (no longer supposed continuous) in the plane equals the infimum of the charges of the lower envelopes of all pairs of functions from the family. As a key to this it is shown in two different ways that the (fine) harmonic measures of any 3 pairwise disjoint finely open planar sets have Borel supports with empty intersection. One proof of this uses the Jordan curve theorem and the fact that the set of inaccessible points of the fine boundary of a fine domain is Borel and has zero harmonic measure; the other involves Carleman-Tsuji type estimates together with a fine topology version of a recent result of P. Jones and T. Wolff on harmonic measure and Hausdorff dimension.

Though there have been extensive works on the existence of maximizers for sharp Trudinger-Moser inequalities under homogeneous and inhomogeneous constraints, and sharp Adams inequalities under homogeneous constraints, much less is known for that of the maximizers for Adams inequalities under inhomogeneous constraints. Furthermore, whether exists equivalence between subcritical and critical Adams inequalities in $W^{m,2}(\mathbb {R}^{2m})$ also remains open. In this paper, we shall give partial answers to these problems. We first establish the equivalence of subcritical Adams inequalities and critical Adams inequalities under inhomogeneous constraints through exploiting the scaling invariance of Adams inequalities in $W^{m,2}(\mathbb {R}^{2m})$ (see Theorem 1.1). Then we consider the existence and non-existence of extremals for sharp Adams inequalities under inhomogeneous constraints in Theorem 1.2, 1.3 and 1.4. Our methods are based on Fourier rearrangement inequality and careful analysis for vanishing phenomenon of radially maximizing sequence for Adams inequalities in $W^{m,2}(\mathbb {R}^{2m})$ .

In this paper, the main aim is to discuss the existence of the extreme points and support points of families of harmonic Bloch mappings and little harmonic Bloch mappings. First, in terms of the Bloch unit-valued set, we prove a necessary condition for a harmonic Bloch mapping (resp. a little harmonic Bloch mapping) to be an extreme point of the unit ball of the normalized harmonic Bloch spaces (resp. the normalized little harmonic Bloch spaces) in the unit disk 𝔻. Then we show that a harmonic Bloch mapping f is a support point of the unit ball of the normalized harmonic Bloch spaces in 𝔻 if and only if the Bloch unit-valued set of f is not empty. We also give a characterization for the support points of the unit ball of the harmonic Bloch spaces in 𝔻.

If Exc is the set of all excessive measures associated with a submarkovian resolvent on a Lusin measurable space and B is a balayage on Exc then we show that for any m∈Exc there exists a basic set A (determined up to a m-polar set) such that Bξ=(BA)*ξ for any ξ∈ Exc, ξ ≪ m. The m-quasi-Lindelöf property (for the fine topology) holds iff for any B there exists the smallest basic set A as above. We characterize the case when any B is representable i.e. there exists a basic set such that B=(BA)* on Exc.

Hunt’s hypothesis (H) and the related Getoor’s conjecture is one of the most important problems in the basic theory of Markov processes. In this paper, we investigate the invariance of Hunt’s hypothesis (H) for Markov processes under two classes of transformations, which are change of measure and subordination. Our first theorem shows that for two standard processes (Xt) and (Yt), if (Xt) satisfies (H) and (Yt) is locally absolutely continuous with respect to (Xt), then (Yt) satisfies (H). Our second theorem shows that a standard process (Xt) satisfies (H) if and only if $(X_{\tau _{t}})$ satisfies (H) for some (and hence any) subordinator (τt) which is independent of (Xt) and has a positive drift coefficient. Applications of the two theorems are given.