Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 8 / 2

The layer cake representation, followed by the assumed tail bound and Tonelli theorem, givesAll integrands are nonnegative, so this use of Tonelli theorem is valid before finiteness of the left-hand side is known. Since $q=p/(p-1)$, the result is

Write $F=f_2$. Split the integral in the assumed tail bound according to whether $g\le a/2$ or $g>a/2$:The finite-superlevel hypothesis permits subtraction of the first term. With $a=2b$, this becomesThus $F/2$ satisfies part (i), which proves finiteness and the preliminary estimateNow integrate the original tail bound against $p{a}^{p-2}$. The layer cake representation and Tonelli theorem givewhere the second step is Hölder's inequality. The preliminary estimate justified the finiteness needed here. If $\Vert F{\Vert }_p=0$ the result is immediate; otherwise divide by its $(p-1)$st power. This proves the Lp bound from a tail domination inequality:The printed strict signs cannot hold when $g=F=0$: both sides are zero. The universally valid formulation therefore has non-strict inequalities. If $\Vert g{\Vert }_p>0$, the sharper estimate is actually strict, as follows.

Suppose equality held and $\Vert F{\Vert }_p>0$. Equality in Hölder's inequality and the norm relation force $g=F/q$ almost everywhere; equality in the preceding integrated tail inequality forces equality in that tail inequality for almost every $a>0$. Put $H(a)=\mu \{F>a\}$ and $J(a)={\int }_a^{\infty }H(t)dt$. The layer-cake identity on $\{F>a\}$ gives ${\int }_{\{F>a\}}F=aH(a)+J(a)$, so equality impliesBecause $F\in L^p$ and $p>1$, $J$ is finite for positive $a$, locally absolutely continuous, and $J^{\prime }=-H$ almost everywhere. It follows that $J^{\prime }=-(p-1)J/a$, whose solutions on $(0,\infty )$ are $J(a)=c{a}^{1-p}$. If $c>0$, then $H(a)=c(p-1)a^{-p}$ almost everywhere, and $p{\int }_0^{\infty }{a}^{p-1}H(a)da$ diverges. If $c=0$, then $F=0$ almost everywhere. Both contradict the assumed nonzero finite equality case. If $F=0$ but $g\ne 0$, strictness is immediate.

Thus both requested strict estimates hold when $\Vert g{\Vert }_p>0$: the coarse constant is strictly larger than the sharp one since $q^{1/q}\le e^{1/e}<2$, so $q^p<2^{p}q$. The zero-data exception is the only qualification needed here.

Let $(X_k{)}_{k=0}^n$ be a nonnegative submartingale and assume $X_n\in L^p$. Set $X^{\ast }={\max }_{0\le k\le n}{X}_k$. For $a>0$, let $A_k$ be the event that $k$ is the first index with $X_k>a$. These events are disjoint, $A_k\in {\mathcal{F}}_k$, and their union is $\{X^{\ast }>a\}$.

The submartingale property gives $\mathbb{E}[X_n\mid {\mathcal{F}}_k]\ge X_k$. HenceThus part (ii) applies on the probability space with $F=X^{\ast }$ and $g=X_n$, giving the Doob Lp maximal inequalityThe same split at $a/2$ makes $X^{\ast }/2$ satisfy part (i), recovering the preliminary constant $2^{p}q$. Probability spaces automatically have finite superlevel measures. For a right-continuous nonnegative submartingale on $[0,T]$, apply these finite-time bounds to increasing finite grids containing $T$ and a dense set of times; right continuity and monotone convergence theorem extend the estimate to ${\sup }_{t\le T}{X}_t$.