First, we establish necessary and sufficient conditions for embeddings of Bessel potential spaces
\({H^{\sigma}X(\mathbb R^n)}\) with order of smoothness less than one, modelled upon rearrangement invariant Banach function spaces
\({X(\mathbb R^n)}\), into generalized Hölder spaces. To this end, we derive a sharp estimate of modulus of smoothness of the convolution of a function
\({f\in X(\mathbb R^n)}\) with the Bessel potential kernel
g σ , 0 <
σ < 1. Such an estimate states that if
\({g_{\sigma}}\) belongs to the associate space of
X, then
$\omega(f*g_{\sigma},t)\precsim \int\limits_0^{t^n}s^{\frac{\sigma}{n}-1}f^*(s)\,ds \quad {\rm for\,all} \quad t\in(0,1) \quad {\rm and\,every}\quad f\in X(\mathbb R^n).$
Second, we characterize compact subsets of generalized Hölder spaces and then we derive necessary and sufficient conditions for compact embeddings of Bessel potential spaces
\({H^{\sigma}X(\mathbb R^n)}\) into generalized Hölder spaces. We apply our results to the case when
\({X(\mathbb R^n)}\) is the Lorentz–Karamata space
\({L_{p,q;b}(\mathbb R^n)}\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces
\({H^{\sigma}L_{p,q;b}(\mathbb R^n)}\) into generalized Hölder spaces and also compact embeddings of spaces in question. Applications cover both superlimiting and limiting cases.