Slow Growth for Universal Harmonic Functions
<p/> <p>Given any continuous increasing function <inline-formula> <graphic file="1029-242X-2010-253690-i1.gif"/></inline-formula> such that <inline-formula> <graphic file="1029-242X-2010-253690-i2.gif"/></inline-formula>, we show th...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2010-01-01
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| Series: | Journal of Inequalities and Applications |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/2010/253690 |
| Summary: | <p/> <p>Given any continuous increasing function <inline-formula> <graphic file="1029-242X-2010-253690-i1.gif"/></inline-formula> such that <inline-formula> <graphic file="1029-242X-2010-253690-i2.gif"/></inline-formula>, we show that there are harmonic functions <inline-formula> <graphic file="1029-242X-2010-253690-i3.gif"/></inline-formula> on <inline-formula> <graphic file="1029-242X-2010-253690-i4.gif"/></inline-formula> satisfying the inequality <inline-formula> <graphic file="1029-242X-2010-253690-i5.gif"/></inline-formula> for every <inline-formula> <graphic file="1029-242X-2010-253690-i6.gif"/></inline-formula>, which are universal with respect to translations. This answers positively a problem of D. H. Armitage (2005). The proof combines techniques of Dynamical Systems and Operator Theory, and it does not need any result from Harmonic Analysis.</p> |
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| ISSN: | 1025-5834 1029-242X |