The Pointillist principle for variation operators and jump functions

Hughes, Kevin

Abstract

I extend the pointillist principles of Moon and Carrillo-de Guzmán to variational operators and jump functions. 1. The pointillist principle In [11], Moon observed that, for a sequence of sufficiently smooth convolution operators and any q ≥ 1, the weak (1, q) boundedness of their maximal operator is equivalent to restricted weak (1, q) boundedness of the maximal operator. In this paper, the goal is to extend this theorem to variational operators and to jump functions. I now recall a couple definitions in order to make this precise. For a sequence of operators (T m) m∈N , define their maximal function M (T m f (x) : m ∈ N) := sup m∈N |T m f (x)| for f : R d → C and x ∈ R d. Suppose that p, q ≥ 1. An operator T is weak-type (p, q) with norm C if it satisfies the inequality T f L q,∞ ≤ Cf L p for all f ∈ L p (1) where f L p := |f (x)| p dx 1/p and g L q,∞ := sup t>0 t|{x ∈ R d : |g(x)| ≥ t}| 1/q for functions f, g : R d → C with the usual modifications made when p or q is infinite. Here and throughout, C is non-negative. In this paper, we will restrict our functions to be defined on R d and will work with the Lebesgue measure thereon. So, I will rarely include this in the notation, and I will also let |X| denote the measure of a finite (Lebesgue) measurable set X in R d. Additionally, an operator T is said to be restricted weak-type (p, q) with norm C if (1) holds for each function f which is the characteristic function of a finite measurable set. Moon's theorem. Suppose that (T m) m∈N is a sequence of convolution operators given by T m f := f * g m with g m ∈ L 1 (R d) for each m ∈ N. For any q ≥ 1, M (T m f (x) : m ∈ N) is restricted weak-type (1, q) with norm C if and only if M (T m f (x) : m ∈ N) is weak-type (1, q) with norm C. The essential difference between the two distinct weak-types lies in the class of input functions used to define them. The class of all L p functions serve as input to the (unqualified) weak-type inequalities while its subclass of characteristic functions 2020 Mathematics Subject Classification. Primary 42B25.

Citation

Hughes, K. (online). The Pointillist principle for variation operators and jump functions. Revista de la Unión Matemática Argentina, https://doi.org/10.33044/revuma.4124

• Journal Article Type: Article
• Acceptance Date: Feb 6, 2024
• Online Publication Date: Aug 27, 2024
• Deposit Date: May 1, 2024
• Publicly Available Date: Aug 27, 2024
• Print ISSN: 0041-6932
• Electronic ISSN: 1669-9637
• Peer Reviewed: Peer Reviewed
• DOI: https://doi.org/10.33044/revuma.4124