Let u , w 0 u , w 0 be measurable functions on a

Question:

Let u,w0u,w0 be measurable functions on a σσ-finite measure space (X,A,μ)(X,A,μ).

(i) Show that tμ{ut}{ut}wdμtμ{ut}{ut}wdμ for all t>0t>0 implies that

updμpp1up1wdμp>1updμpp1up1wdμp>1

(ii) Assume that u,wLpu,wLp. Conclude from (i) that up(p/(p1))wpup(p/(p1))wp for p>1p>1.

[use the technique of the proof of Theorem 25.12 ; for (ii) use Hölder's inequality.]

Data from theorem 25.12 

Theorem 25.12 (Doob's maximal IP-inequality) Let (X, A., H) be a o-finite filtered measure space, 1

If ||UN||=o, the inequality is trivial; if uy ECP (u), then ,...,uy-1 are in LP () since (14) MEN is a



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