Local derivations on solvable Lie algebras of maximal rank

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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 where is the maximal torus subalgebra of is the nilradical of and We prove that any local derivation of such solvable Lie algebra is a derivation. Further, we present two examples of solvable Lie algebras which satisfy the condition 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.