Local and 2-local $\frac{1}{2}$-derivations of infinite-dimensional Lie algebras

In this work, we describe local and 2-local $\frac{1}{2}$-derivations of infinite-dimensional Lie algebras. We prove that all local and 2-local $\frac{1}{2}$-derivations of the Witt algebra as well as of the positive Witt algebra and the classical one-sided Witt algebra are $\frac{1}{2}$-derivations. We also give an example of an infinite-dimensional Lie algebra with a local (2-local) $\frac{1}{2}$-derivation which is not a $\frac{1}{2}$-derivation. Further we prove that all local (2-local) $\frac{1}{2}$-derivations on the $\mathcal{W}(a,b)$ algebra are $\frac{1}{2}$-derivations.