A real polynomial in one real variable is called hyperbolic if it has only real roots. The polynomial f is called a primitive of order ν of the polynomial g if f(ν) = g. A hyperbolic polynomial is called very hyperbolic if it has hyperbolic primitives of all orders. In the paper we prove some geometric properties of the set D of values of the parameters ai for which the polynomial xn + a1xn−1 + … + an is very hyperbolic. In particular, we prove the Whitney property (the curvilinear distance to be equivalent to the Euclidean one) of the set D ∩{a1 = 0, a2 ≥ −1}.

The purpose of this paper is to prove strong-type inequalities with one-sided weights for commutators (with symbol b ∈ BMO) of several one-sided operators, such as the one-sided discrete square function, the one-sided fractional operators, or one-sided maximal operators given by the convolution with a smooth function. We also prove that b ∈ BMO is a necessary condition for the boundedness of commutators of these one-sided operators.

Nets of Schrödinger C0-semigroups (Sε)ε with the polynomial growth with respect to ε are used for solving the Cauchy problem (∂t − Δ)U + VU = f(t, U), U(0, x) = U0(x) in a suitable generalized function algebra (or space), where V and U0 are singular generalized functions while f satisfies a Lipschitz-type condition. The existence of distribution solutions is proved in appropriate cases by the means of white noise calculus as well as classical energy estimates.