On the existence of capacitary strong type estimates in Rn

Theorem 1. If re=positive integer= t})dt p <= Cllf[l~, for all fELp(R"). C is a constant depending only on n, p and m. Of course the "weak type" estimate R,,,p({x: ]Imf(x)l >= t}) <= Cllfllf, t-" is trivially valid (with C= 1) and so what is new is the existence of what we prefer to call a "strong type" capacitary inequality. These weak and strong type capacitary estimates oppose each other in much the same way as do the usual weak and strong Lp estimates. The author wishes to thank Professor Maz'ya for the interesting discussion and for pointing out this problem and his work on it.