On the exact L2L2 Markov inequality on some unbounded domains in RdRd

We consider the Markov problem of finding the so-called Markov factor M(U,K):=supu∈U‖Du‖‖u‖, of the set of differentiable functions UU, where Du:=|∂u|ℓ2Du:=|∂u|ℓ2 stands for the ℓ2ℓ2-norm of the gradient vector of uu, and ‖⋅‖‖⋅‖ is the weighted L2L2 norm on the set K⊂RdK⊂Rd. In the univariate case exact L2L2-Markov inequalities are known for algebraic polynomials on the real line, half line and intervals. We outline a variational approach to the above problem and show how this leads either to certain partial differential equations, or to a system of homogeneous linear equations. This method will be illustrated by using it to solve the L2L2 Markov problem for the cases of dd-dimensional spaces and dd-dimensional hyperquadrants. In the case of dd-spaces the solution is given for homogeneous polynomials, as well.