We know that Morrey's inequality says $W^{1,p} \subset C^{0,\gamma}$ for $\gamma = 1 - n/p$ where $n$ is the dimension. However, in 1D, following the proof of Evans "Partial Differential Equations" (first edition, pp. 267), we have for any function $f$ in $W^{1,p}(0,1)$, and $x, y \in (0,1)$


$u(x) - u(y) = \int\_0^1 \frac{d}{dt}u(tx + (1-t)y) dt = \int\_y^x Du(s) (x-y) ds$.


Take absolute value on both sides and use the basic inequalities, we have


$|u(x)-u(y)| \le |Du|\_{L^p(0,1)} |x-y|$.


We obtain something more than Morrey's inequality would indicate.


Is there anything wrong?



In 1D, is a $W^{1,p}$ function always Lipschitz, for $p\ge1$? 
This does not look right. You have $u(x)-u(y) = \int\_y^x Du(s)(x-y)ds$ but the correct expression is clearly $\int\_y^x Du(s)ds$. I think a change of variables went wrong somewhere.


A counterexample: take $u(x) = x^{3/4}$ on $(0,1)$, with $p=2$. Then $Du(x) = \frac{3}{4}x^{-1/4} \in L^2(0,1)$, so $u \in W^{1,2}(0,1)$, but $u$ is not Lipschitz.


