Onsager's Conjecture for the Incompressible Euler Equations in the H\"{o}log Spaces $C^{0,\alpha}_\lambda(\bar{\Omega})$

In this note we extend a 2018 result of Bardos and Titi \cite{BT} to a new class of functional spaces $C^{0,\alpha}_\lambda(\bar{\Omega})$. It is shown that weak solutions $\,u\,$ satisfy the energy equality provided that $u\in L^3((0,T);C^{0,\alpha}_\lambda(\bar{\Omega}))$ with $\alpha\geq\frac{1}{3}$ and $\lambda>0$. The result is new for $\,\alpha = \,\frac{1}{3}\,.$ Actually, a quite stronger result holds. For convenience we start by a similar extension of a 1994 result of Constantin, E, and Titi, \cite{CET}, in the space periodic case. The proofs follow step by step those of the above authors. For the readers convenience, and completeness, proofs are presented in a quite complete form.