Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 23J (Probability and Measure)

For $1\le p<\infty$, $L^p(E,\mathcal{E},\mu )$ consists of measurable functions with ${\int }_E|f{|}^{p}d\mu <\infty$, identifying functions equal almost everywhere. Its Lp norm is $\Vert f{\Vert }_p=(\int |f{|}^p{)}^{1/p}$. Thus membership is a condition on an almost-everywhere equivalence class, not on individual point values.

Hölder's inequality states that for conjugate exponents $1/p+1/q=1$, $f\in L^p$ and $g\in L^q$ imply $fg\in L^1$ andIt includes $p=1,q=\infty$ (and the reversed case), with the essential-supremum norm for $L^{\infty }$.

By the Cauchy-Schwarz inequality, the integral is bounded by $\Vert f{\Vert }_2\Vert k_x{\Vert }_2$, where $k_x(y)=|x-y{|}^{-\alpha }$. For $0\le x\le 1$,For $x$ outside the interval, the integral is bounded by the value at the nearest endpoint; the endpoint value is finite as well. Since $2\alpha <1$, the kernel is square-integrable for every real $x$, proving absolute convergence for every $x$.

Subtract the two absolutely convergent integrals and apply the Cauchy-Schwarz inequality to $f$ and the difference of the kernels. This gives precisely

For $h\ne 0$, enlarge the nonnegative integral $I(h)$ to all of $\mathbb{R}$, and substitute $y=x+|h|t$. Reflection if necessary givesNear $0$ and $1$, its integrand is bounded by a constant times $1+|t{|}^{-2\alpha }$ or $1+|t-1{|}^{-2\alpha }$, which is integrable. For large $|t|$, the mean value theorem bounds the difference by a constant times $|t{|}^{-\alpha -1}$, whose square is integrable. Thus $C_{\alpha }<\infty$. The preceding estimate yieldsIn particular $g$ is continuous on all of $\mathbb{R}$; the proof establishes Hölder continuity of a one-dimensional fractional kernel integral of exponent $1/2-\alpha$ uniformly in $x$.