A Lower Estimate for Entropy Numbers

AbstractThe behaviour of the entropy numbers ek(id:lnp→lnq), 0<p<q⩽∞, is well known (up to multiplicative constants independent of n and k), except in the quasi-Banach case 0<p<1 for “medium size” k, i.e., when logn⩽k⩽n, where only an upper estimate is available so far. We close this gap by proving the lower estimate ek(id:lnp→lnq)⩾c(log(n/k+1)/k)1/p−1/q for all 0<p<q⩽∞ and logn⩽k⩽n, with some constant c>0 depending only on p