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 R = Q circle plus N, where Q is the maximal torus subalgebra of R, N is the nilradical of R and dim Q = dim N/N-2. We prove that any local derivation of such solvable Lie algebra R is a derivation. Further, we present two examples of solvable Lie algebras which satisfy the condition dim Q < dim N/N-2 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.