Martingale Interpretation of Weakly Cancelling Differential Operators
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MARTINGALE INTERPRETATION OF WEAKLY CANCELLING DIFFERENTIAL OPERATORS D. Stolyarov∗
UDC 517.5
We provide martingale analogs of weakly cancelling differential operators and prove a Sobolev-type embedding theorem for these operators in the martingale setting. Bibliography: 4 titles.
1. Preliminaries In [4], Van Schaftingen gives a characterization of linear homogeneous vector-valued elliptic differential operators A of order k in d > 1 variables such that the inequality ∇k−1 f L
d (R d−1
d)
Af L1 (Rd )
holds true for any smooth compactly supported function f .1 He calls such operators cancelling. Let k ≥ d and let l ∈ [1, . . . , d − 1]. It is also proved in [4] that the operator A is cancelling (assuming the ellipticity) if and only if ∇k−l f L
d d−l
Af L1 .
However, for the case l = d, the cancellation condition is only sufficient. In [3], Raita gives a necessary and sufficient condition on the operator A for the inequality ∇k−d f L∞ Af L1 to be true for any f ∈ C0∞ (Rd ). He calls such operators weakly cancelling operators. Paper [1] suggests a martingale interpretation of Van Schaftingen’s theorem. It appears that the cancellation condition has a direct analog in a probabilistic model earlier introduced in [2]. The present note provides an analog of Raita’s weak cancellation condition. We refer the reader to [1] for further historical details and motivation as well as for a more detailed description of notation. See Sec. 4 for comparison of our results with [3] and [4]. The author thanks Bogdan Raita for attracting his attention to the question under consideration. 2. Notation and statement Let m ≥ 2 be a natural number, let F = {Fn }n be an m-uniform filtration on a probability space. By this we mean that each atom of the algebra Fn is split into m atoms of Fn+1 having equal probability. The symbol AF n denotes the set of all atoms in Fn . For each ω ∈ AF n , we fix a map Jω : [1 . . . m] → {ω ∈ AF n+1 | ω ⊂ ω}. This fixes the tree structure on the set of all atoms. Each atom in AF n corresponds to a sequence of n integers in the interval [1, . . . , m], which we call digits. We may go further and consider the set T consisting of all infinite paths in the tree of atoms. Each path starts from the atom in F0 , then chooses one of its sons in F1 , then one of its sons in F2 , and so on. There ∗ Department of Mathematics and Computer Science, St.Petersburg State University and St.Petersburg Department of Steklov Institute of Mathematics, St.Petersburg, Russia, e-mail: [email protected]. 1The notation “X Y ” (as in the inequality above) means there exists a constant C such that X ≤ CY uniformly. The parameter with regard to which we apply the term “uniformly” is always clear from the context.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 480, 2019, pp. 191–198. Original article submitted May 28, 2019. 286 1072-3374/20/2512-0286 ©2020 Springer Science+Business Media, LLC
is a natural one-to-one correspondence between points in T, i.e., paths, and infinite sequences of digits in [
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