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"Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits

A. Bonilla (2000)

Colloquium Mathematicae

We prove that, if μ>0, then there exists a linear manifold M of harmonic functions in N which is dense in the space of all harmonic functions in N and lim‖x‖→∞ x ∈ S ‖x‖μDαv(x) = 0 for every v ∈ M and multi-index α, where S denotes any hyperplane strip. Moreover, every nonnull function in M is universal. In particular, if μ ≥ N+1, then every function v ∈ M satisfies ∫H vdλ =0 for every (N-1)-dimensional hyperplane H, where λ denotes the (N-1)-dimensional Lebesgue measure. On the other hand, we...

Łojasiewicz-Siciak condition for the pluricomplex Green function

Marta Kosek (2011)

Banach Center Publications

A compact set K N satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that V K ( z ) B ( d i s t ( z , K ) ) β if dist(z,K) ≤ 1. (LS) Here V K denotes the pluricomplex Green function of the set K. We cite theorems where this condition is necessary in the assumptions and list known facts about sets satisfying inequality (LS).

α-stable random walk has massive thorns

Alexander Bendikov, Wojciech Cygan (2015)

Colloquium Mathematicae

We introduce and study a class of random walks defined on the integer lattice d -a discrete space and time counterpart of the symmetric α-stable process in d . When 0 < α <2 any coordinate axis in d , d ≥ 3, is a non-massive set whereas any cone is massive. We provide a necessary and sufficient condition for a thorn to be a massive set.

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