A note on analytic functions in the unit circle
1932 ◽
Vol 28
(3)
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pp. 266-272
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1. Letbe a function regular for |z| < 1. We say that u belongs to the class Lp (p > 0) ifIt has been proved by M. Riesz that, for p > 1, if u(r, θ) belongs to Lp, so does v (r, θ). Littlewood and later Hardy and Littlewood have shown that for 0 < p < 1 the theorem is no longer true: there exists an f(z) such that u(r, θ) belongs to every L1−ε and v(r, θ) belongs to no Lε(0 < ε < 1). The proof was based on the theorem (due to F. Riesz) that if, for an ε > 0, we havethen f(reiθ) exists for almost every θ.
1962 ◽
Vol 14
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pp. 540-551
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Keyword(s):
1956 ◽
Vol 52
(1)
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pp. 49-60
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2018 ◽
Vol 97
(3)
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pp. 435-445
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1994 ◽
Vol 37
(1)
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pp. 73-89
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Keyword(s):
1964 ◽
Vol 16
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pp. 721-728
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1970 ◽
Vol 17
(1)
◽
pp. 23-36
Keyword(s):
Keyword(s):
1975 ◽
Vol 20
(1)
◽
pp. 46-53
1962 ◽
Vol 13
(2)
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pp. 173-174