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

100%

Bonilla A.

Colloquium Mathematicum

2000

tom 83

nr 2

155-160

We prove that, if μ>0, then there exists a linear manifold M of harmonic functions in $ℝ^N$ which is dense in the space of all harmonic functions in $ℝ^N$ and lim_{{‖x‖→∞} {x ∈ S}} ‖x‖^{μ}D^{α}v(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 prove that there exists a linear manifold M of harmonic functions in the unit ball 𝔹 of $ℝ^N$, which is dense in the space of all harmonic functions and each function in M has zero nontangential limit at every point of the boundary of 𝔹.

Derivative and antiderivative operators and the size of complex domains

100%

Bernal-González L.

Annales Polonici Mathematici

1994

tom 59

nr 3

267-274

We prove some conditions on a complex sequence for the existence of universal functions with respect to sequences of certain derivative and antiderivative operators related to it. These operators are defined on the space of holomorphic functions in a complex domain. Conditions for the equicontinuity of those sequences are also studied. The conditions depend upon the size of the domain.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Colloquium Mathematicum

1 Bernal-González L.

1 Bonilla A.

1 2000

1 1994

Wyniki wyszukiwania

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

Derivative and antiderivative operators and the size of complex domains