Optimal embeddings of spaces of generalized smoothness in the critical case (Q650074)

Let \(\Psi (t)\) be a positive slowly varying function on \((0,1]\), e.g. \((1 + |\log t|)^{a}\), \(a \in \mathbb R\). Let \(\{ \varphi_j \}^\infty_{j=0}\) be the usual dyadic resolution of unity. Let \(0<p,q \leq \infty\) and \(s \in \mathbb R\). Then \(B^{(s,\Psi)}_{p,q} (\mathbb R^n)\) collects all \(f \in S' (\mathbb R^n)\) such that \[ \| f \, | B^{(s, \Psi)}_{p,q} (\mathbb R^n) \| = \Big( \sum^\infty_{j=0} 2^{jsq} \Psi (2^{-j} )^q \, \| (\varphi_j \widehat{f})^\vee \, | L_p (\mathbb R^n) \|^q \Big)^{1/q} \] is finite. There is an \(F\)-counterpart \(F^{(s,\Psi)}_{p,q} (\mathbb R^n)\). Of special interest are the limiting cases \(B^{(n/p, \Psi)}_{p,q} (\mathbb R^n)\) and \(F^{(n/p, \Psi)}_{p,q} (\mathbb R^n)\). It is the main aim of this paper to study the embeddings \[ B^{(n/p, \psi)}_{p,q} (\mathbb R^n) \hookrightarrow \Lambda^\mu_{\infty,r} (\mathbb R^n), \qquad F^{(n/p, \psi)}_{p,q} (\mathbb R^n) \hookrightarrow \Lambda^\mu_{\infty, r} (\mathbb R^n), \] where \(\Lambda^\mu_{\infty, r} (\mathbb R^n)\) are Hölder spaces normed by \[ \| f \, | L_\infty (\mathbb R^n) \| + \Big( \int^1_0 \frac{\omega^r (f,t)}{\mu (t)^r} \, \frac{dt}{t} \Big)^{1/r}, \qquad 0<r \leq \infty, \] for some \(\mu (t) >0\) and the usual modulus of continuity \(\omega (f,t)\).