On more general Lipschitz spaces (Q5926115)

For \(1\leq p\leq\infty\), \(0< q\leq\infty\) and \(\alpha> 1/q\) the logarithmic Lipschitz space \(\text{Lip}^{(1,-\alpha)}_{p,q}(\mathbb{R}^n)\) consists of those \(f\in L_p(\mathbb{R}^n)\) for which NEWLINE\[NEWLINE\|f\|=\|f\|_p+ \Biggl(\int^{1/2}_0 \Biggl[{w(f,t)_p\over t|\log t|^\alpha}\Biggr]^q {dt\over t}\Biggr)^{1/q}NEWLINE\]NEWLINE is finite. Here \(w(f,t)+p= \sup_{|h|\leq t}\|\Delta_h f\|_p\) is the modulus of smoothness and the usual modification is made when \(q=\infty\). These spaces generalize the classical Lipschitz spaces where \(p=q=\infty\) and \(\alpha=0\). Embeddings are obtained of certain Besov spaces \(B^s_{p,q}\) and Triebel-Lizorkin spaces \(F^s_{p,q}\) in logarithmic Lipschitz spaces. As well comparisons are made with the work of others dealing with logarithmic smoothness.