In this talk, I will show that smooth functions on convex bodies in Euclidean space, whose sequence of derivatives is dominated by a suitable given weight sequence of positive real numbers, have many polynomial-like properties. Let us call them “controlled differentiable functions” for brevity. Functions in quasianalytic Denjoy--Carleman classes are examples, but sometimes the results also apply in the non-quasianalytic setting.
I will introduce an integer, depending on the given weight sequence, the diameter of the domain, and the sup-norm of the function, which, in analogy to the polynomial degree, allows to express the polynomial-like behavior quantitatively. For instance, I will present a bound on the codimension one Hausdorff measure of the zero set and show that it can be locally parameterized by Sobolev functions. Moreover, I will discuss a Remez-type inequality and several applications for controlled differentiable functions. Many of the results depend only on the derivatives up to some finite order, which can be determined explicitly.
The local parameterization of the zero set by $W^{1,p}$ Sobolev functions is based on joint work with Adam Parusinski in which, for any smooth family of monic polynomials, we determined the optimal order of summability $p \ge 1$ (solely in terms of the degree) such that there is a $W^{1,p}$ choice of the roots.