# Embeddings of Sobolev-Lorentz space

The classical Sobolev embedding theorem asserts that, under suitable conditions on the exponents $s,p$ and $n$, the Sobolev space $W^{s,p}(\mathbb{R}^n)$ embeds into an Holder space $C^{r,\alpha}(\mathbb{R}^n)$. Suppose now to work with the more general Sobolev-Lorentz space $$W^{s,(p,q)}(\mathbb{R}^n):=\{f\in L^{p,q}(\mathbb{R}^n)\,s.t.\nabla^sf\in L^{p,q}(\mathbb{R}^n)\}$$ I wonder if is it possible to have (when $q<p$) some local logarithmic refinement of the Holder estimate, namely $$f\in W^{s,(p,q)}(\mathbb{R}^n)\Rightarrow \big(D^{\mu}f(x)-D^{\mu}f(y)\big)\frac{\ln^{\beta}{|x-y|}}{|x-y|^{\alpha}}\lesssim 1$$ for every multi-index $|\mu|=r$, for $|x-y|\leq C<1$ and for some positive parameter $\beta$.

• That implication is off. What is $D^\mu(x)$ supposed to mean? And where does $f$ come into play on the RHS? – Johannes Hahn Sep 19 '17 at 19:00
• I add f in the RHS. $D^{\mu}f$ is the derivative of $f$ with respect to the multiindex $\mu$. – Capublanca Sep 23 '17 at 22:33