An Invariant Volume-Mean-Value Property
If ƒ is harmonic and integrable over the open unit disc U then so is ƒ ∘ ψ for every Moebius transformation ψ of U, and therefore 1 π ∫ U (ƒ ∘ ψ) d A = ƒ(ψ(0) for every ψ. Conversely, does this mean-value property imply that ƒ is harmonic? A more general question, with the unit ball B n of C (for ar...
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| Veröffentlicht in: | Journal of functional analysis 1993-02, Vol.111 (2), p.380-397 |
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| container_start_page | 380 |
| container_title | Journal of functional analysis |
| container_volume | 111 |
| creator | Ahern, P. Flores, M. Rudin, W. |
| description | If ƒ is harmonic and integrable over the open unit disc
U then so is ƒ ∘ ψ for every Moebius transformation ψ of
U, and therefore
1
π
∫
U (ƒ ∘ ψ) d A = ƒ(ψ(0)
for every ψ. Conversely, does this mean-value property imply that ƒ is harmonic? A more general question, with the unit ball
B
n
of
C
(for arbitrary
n≥ 1) in place of the disc, is investigated in the present paper. The answer is found to be affirmative if
n ≤ 11, negative if
n ≤ 12. |
| doi_str_mv | 10.1006/jfan.1993.1018 |
| format | Article |
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U then so is ƒ ∘ ψ for every Moebius transformation ψ of
U, and therefore
1
π
∫
U (ƒ ∘ ψ) d A = ƒ(ψ(0)
for every ψ. Conversely, does this mean-value property imply that ƒ is harmonic? A more general question, with the unit ball
B
n
of
C
(for arbitrary
n≥ 1) in place of the disc, is investigated in the present paper. The answer is found to be affirmative if
n ≤ 11, negative if
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U then so is ƒ ∘ ψ for every Moebius transformation ψ of
U, and therefore
1
π
∫
U (ƒ ∘ ψ) d A = ƒ(ψ(0)
for every ψ. Conversely, does this mean-value property imply that ƒ is harmonic? A more general question, with the unit ball
B
n
of
C
(for arbitrary
n≥ 1) in place of the disc, is investigated in the present paper. The answer is found to be affirmative if
n ≤ 11, negative if
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U then so is ƒ ∘ ψ for every Moebius transformation ψ of
U, and therefore
1
π
∫
U (ƒ ∘ ψ) d A = ƒ(ψ(0)
for every ψ. Conversely, does this mean-value property imply that ƒ is harmonic? A more general question, with the unit ball
B
n
of
C
(for arbitrary
n≥ 1) in place of the disc, is investigated in the present paper. The answer is found to be affirmative if
n ≤ 11, negative if
n ≤ 12.</abstract><cop>Orlando, FL</cop><cop>San Diego, CA</cop><cop>Brugge</cop><pub>Elsevier Inc</pub><doi>10.1006/jfan.1993.1018</doi><tpages>18</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0022-1236 |
| ispartof | Journal of functional analysis, 1993-02, Vol.111 (2), p.380-397 |
| issn | 0022-1236 1096-0783 |
| language | eng |
| recordid | cdi_crossref_primary_10_1006_jfan_1993_1018 |
| source | Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; ScienceDirect Journals (5 years ago - present) |
| subjects | Exact sciences and technology Mathematical analysis Mathematics Potential theory Sciences and techniques of general use |
| title | An Invariant Volume-Mean-Value Property |
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