Optimal embeddings and compact embeddings of Bessel-potential-type spaces

First, we establish necessary and sufficient conditions for embeddings of Bessel potential spaces H-sigma X (R-n) with order of smoothness less than one, modelled upon rearrangement invariant Banach function spaces X (R-n), into generalized Holder spaces. To this end, we derive a sharp estimate of modulus of smoothness of the convolution of a function f is an element of X (R-n) with the Bessel potential kernel g(sigma), 0 < sigma < 1. Such an estimate states that if g(sigma) belongs to the associate space of X, then omega(f * g(sigma), t) less than or similar to integral(tn)(0) s(sigma/n-1) f* (s) ds for all t is an element of (0, 1) and every f is an element of X (R-n). Second, we characterize compact subsets of generalized Holder spaces and then we derive necessary and sufficient conditions for compact embeddings of Bessel potential spaces H-sigma X (R-n) into generalized Holder spaces. We apply our results to the case when X (R-n) is the Lorentz-Karamata space L-p,L-q;b (R-n). In particular, we are able to characterize optimal embeddings of Bessel potential spaces H-sigma L-p,L-q;b (R-n) into generalized Holder spaces and also compact embeddings of spaces in question. Applications cover both superlimiting and limiting cases.