哈代空間

在複分析中，哈代空間（或哈代類） $H^{p}$ 是單位圓盤或上半平面上的某類全純函數。高德菲·哈羅德·哈代首先在1915年考慮這類問題。在實分析中，實哈代空間是複哈代空間的成員在實數軸上的邊界值。對於 $1<p<\infty$ ，實哈代空間基本上等於 $L^{p}$ 空間。當 $p\leq 1$ 時， $L^{p}$ 空間較難操作，而哈代空間的性質就比較容易掌握。

在較高維的情況，我們可考慮管狀域（複數情形）及 $\mathbb {R} ^{n}$ 上的函數，從而得到相應的定義。

哈代空間在數學分析、控制論及散射理論中有所應用。

單位圓盤的哈代空間

對 $0<p<\infty$ ，哈代空間 $H^{p}$ 定義為開單位圓盤上滿足下述性質的全純函數 $f$

\sup _{0<r<1}\left({\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f(re^{i\theta })\right|^{p}\;d\theta \right)^{\frac {1}{p}}<\infty .

左側的數定義為範數 $\|f\|_{H^{p}}$ 。

若 $0<p<q<\infty$ ，可證明 $H^{q}\subset H^{p}$ 。

上半平面的哈代空間

藉凱萊變換，可將單位圓盤的定義翻譯到上半平面的情形。此時哈代空間等於上半平面上滿足下述性質的全純函數 $F$

\sup _{y>0}\left(\int _{\mathbb {R} }|F(x+iy)|^{p}dx\right)^{\frac {1}{p}}<\infty

左側的數定義為範數 $\|F\|_{H^{p}}$ 。

參考文獻

Cima, Joseph A.; Ross, William T., The Backward Shift on the Hardy Space, American Mathematical Society, 2000, ISBN 0-8218-2083-4
Colwell, Peter, Blaschke Products - Bounded Analytic Functions, Ann Arbor: University of Michigan Press, 1985, ISBN 0-472-10065-3
Duren, P., Theory of $H^{p}$ -Spaces, New York: Academic Press, 1970
G.B. Folland, Hardy spaces, Hazewinkel, Michiel (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
Hardy, G. H., On the mean value of the modulus of an analytic function, Proceedings of the London mathematical society series 2, 1915, 14: 269–277
Hoffman, Kenneth, Banach spaces of analytic functions, New York: Dover Publications, 1988, ISBN 0-486-65785-X
Riesz, F., Über die Randwerte einer analytischen Funktion, Math. Z., 1923, 18: 87–95, doi:10.1007/BF01192397
S.V. Shvedenko, Hardy classes, Hazewinkel, Michiel (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4

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