Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 7 / 4 / i / Solution

The defining convolution integral is absolutely convergent for every $x$ when $e\in L^1+L^2$. Indeed, writing $e=a+b$ with $a\in L^1$ and $b\in L^2$, the Cauchy-Schwarz inequality givesDisjoint supports imply ${\sum }_n|e_n(y)|=|e(y)|$ almost everywhere. The Tonelli theorem therefore yieldsWe may exchange the sum and integral; the resulting numerical series is absolutely convergent. Its triangle inequality givesHere $e$ belongs to the stated operator domain, as is implicit when writing $T(e)$. Disjoint supports also ensure that each summand belongs to that domain.