A lower estimate for entropy numbers

The behaviour of the entropy numbers e(k)(id: l(p)(n) --> l(q)(n)). 0<p<q l ess than or equal to infinity, is well known (up to multiplicative constant s independent of n and k), except in the quasi Banach case 0 < p < 1 for "m edium size" k, i.e., when log n less than or equal to k less than or equal to n, where only an upper estimate is available so Far. We close this gap b y proving the lower estimate e(k)(id: l(p)(n) --> l(q)(n)) greater than or equal to c(log(n/k+ 1)/k)(1/p-1/q) for all 0 <p < q less than or equal to i nfinity and log n less than or equal to k less than or equal to n, with som e constant c >0 depending only on p. (C) 2001 Academic Press.