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We prove that, if μ>0, then there exists a linear manifold M of harmonic functions in which is dense in the space of all harmonic functions in 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...
A compact set satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that
if dist(z,K) ≤ 1. (LS)
Here 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).
We introduce and study a class of random walks defined on the integer lattice -a discrete space and time counterpart of the symmetric α-stable process in . When 0 < α <2 any coordinate axis in , 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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