Let u , w 0 u , w 0 be measurable functions on a
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Let u,w⩾0 be measurable functions on a σ-finite measure space (X,A,μ).
(i) Show that tμ{u⩾t}⩽∫{u⩾t}wdμ for all t>0 implies that
∫updμ⩽pp−1∫up−1wdμ∀p>1
(ii) Assume that u,w∈Lp. Conclude from (i) that ∥u∥p⩽(p/(p−1))∥w∥p for p>1.
[use the technique of the proof of Theorem 25.12 ; for (ii) use Hölder's inequality.]
Data from theorem 25.12
Transcribed Image Text:
Theorem 25.12 (Doob's maximal IP-inequality) Let (X, A., H) be a o-finite filtered measure space, 1
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