Let’s get the hype out of the way first. Louis CK is, according to Ricky Gervais, “the funniest stand-up on the circuit”. He has, according to his puzzlingly specific former employer Chris Rock, “the finest comic mind of the past 25 years”. And, sure enough, when he arrived on a British stage for the first time on Wednesday night, he was greeted with the ovation of a comedy legend.

Well, he’s pretty good. His delivery is quite something. Whether he’s quite the ginger Richard Pryor that his endorsements suggest, I’m not so sure. But his best material is personal and provocative – fatherhood, for example, quicksand for most stand-ups, is given a cheeringly unsentimental treament. Does he love both his daughters equally? Of course not, he says – rousing boos where gags about “faggots” and “the N-word” earn whoops of laughter. He’s aggressively liberal, lambasting white folk with insightful self-hatred, then smilingly judgmental about modern foibles such as hyperbole, ungraciousness and sanctified inarticulacy.

Well, nobody could accuse CK – the name a condensation of his original surname Szekely – of inarticulacy. He conveys his cheerfully cantankerous heresies with the unfailing slickness that typifies American comedians, whether hard-edged or soft-centred. With a booming voice he could use to sell life insurance, amplified to a level where you fear you’ll need it, he can zero in on an inconsistency of language or behaviour, including his own, and mock it until it bleeds.

The downside of that is that he can keep kicking at a notion when it’s already down. He looks like a blue-collar Bill Bailey after a haircut, but some of his more generic material plays like a less cloying Jeff Green. Talking about the difference between women and girls, he’s inspired; talking about the difference between men and women, he’s generic.

His fluency keeps him persuasive, even when his jokes aren’t as sharp as his energised delivery suggests. But even as he advertises his vulnerability, with material about his weight or knackered 40-year-old body, he’s so punchy that you’re dazzled by him more than connecting with him. I liked him. Isolated lines made me roar with laughter. But I couldn’t quite surrender to him.

Box office: 020-7478 0100, to Aug 23. Also at Pleasance, Edinburgh (0131-556 6550), Aug 15, 16

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