On the Continuous embeddings between the fractional Hajłasz-Orlicz-Sobolev spaces

Let $G$ be an Orlicz function and let $\alpha ,\beta ,s$ be positive real numbers. Under certain conditions on the Orlicz function $G$, we establish some continuous embeddings results between the fractional order Orlicz-Sobolev spaces defined on metric-measure spaces $W_s^{\alpha ,G}(X,d,\mu )$ and the fractional Hajłasz-Orlicz-Sobolev spaces $M^{\beta ,G}(X,d,\mu )$.