Local derivations on solvable Lie algebras of maximal rank

The present paper is devoted to the description of local derivations on solvable Lie algebras of maximal rank. Namely, we consider a solvable Lie algebra of the form (Formula presented.) where (Formula presented.) is the maximal torus subalgebra of (Formula presented.) (Formula presented.) is the nilradical of (Formula presented.) and (Formula presented.) We prove that any local derivation of such solvable Lie algebra (Formula presented.) is a derivation. Further, we present two examples of solvable Lie algebras which satisfy the condition (Formula presented.) and the first algebra admit a local derivation which is not a derivation, while for the second algebra we prove that any local derivation is a derivation. We also apply the main result of the paper to the description of local derivations on so-called standard Borel subalgebras of complex simple Lie algebras.