Dissipative Euler flows with Onsager-critical spatial regularity

For any $\epsilon >0$ we show the existence of continuous periodic weak solutions $v$ of the Euler equations which do not conserve the kinetic energy and belong to the space $L^1_t (C_x^{\frac{1}{3}-\epsilon})$, namely $x\mapsto v (x,t)$ is $(\frac{1}{3}-\epsilon)$-H\"older continuous in space at a.e. time $t$ and the integral $\int [v(\cdot, t)]_{\frac{1}{3}-\epsilon} dt$ is finite. A well-known open conjecture of L. Onsager claims that such solutions exist even in the class $L^\infty_t (C_x^{\frac{1}{3}-\epsilon})$.