Optimality of embeddings of Bessel-potential-type spaces into generalized Hölder spaces

We establish the sharpness of embedding theorems for Bessel-potential spaces modelled upon Lorentz-Karamata spaces and we prove the non compactness of such embeddings. Target spaces in our embeddings are generalized Hölder spaces. As consequences of our results, we get continuous envelopes of Bessel-potential spaces modelled upon Lorentz-Karamata spaces