Assume  that  $M$ is  a  compact  parallelizable  manifold. Is there an upper bound  for the   absolute value of Fredholm index of all operators in the form  $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$  where $\{X_1,X_2,\ldots,X_n\}$ is  a global smooth frame?