On my high end cookware instructions say not to heat much above medium. Many recipes call for high heat. What to do?

Medium high in very conductive pans is probably the equivalent of high

I forgot to mention that if your pans are indeed stainless with a copper core that the copper is an excellent conductor of heat, so no hot spots and you can get by with a less than high temperature.

For high heat I turn to my cast iron or carbon steel skillets. I don't cook many things on high anyway although I do brown meats in enameled cast iron pots. Stainless with an aluminum or copper core can be a bitch to clean which I suspect is what the manufacturer's instructions are referring to. But you can't expect your everyday pans to look pristine anyway unless they are just hanging there for show.

What kind and brand of cookware is it?