Bordering functions for differential bases. (English. Russian original) Zbl 0836.42014

Solyanik, A. A.

Math. Notes 54, No. 6, 1241-1245 (1993); translation from Mat. Zametki 54, No. 6, 82-89 (1993).

Let \({\mathcal B} = \{\cup {\mathcal B} (x); x \in \mathbb{R}^n\}\) be a differential basis in \(\mathbb{R}^n\). For \(f \in L^1 (\mathbb{R}^n)\) we define the maximal function with respect to the basis \({\mathcal B}\) by \[ M_{\mathcal B} f(x) = \sup_{B \in {\mathcal B} (x)} {1 \over |B |} \int_B |f |. \] The bordering function for the basis \({\mathcal B}\) is defined by \[ \varphi_{\mathcal B} (t) = \sup \left\{ {1 \over |E |} \left |\left\{ M_{{\mathcal B} \chi_E} > {1 \over t} \right\} \right |; \;|E |> 0,\;E \text{ bounded } \right\} \] where \(|E |\) denotes the Lebesgue measure of the set \(E\) and \(\chi_E\) denotes the characteristic function of the set \(E\).
The author finds asymptotic behaviors of the functions \(\varphi_{\mathcal B} (t) - 1\), \(t \to 1^+\), for the bases of cubes, rectangles, and centered cubes. In the one-dimensional case he proves that \(\varphi_{\mathcal B} (t) = 2t - 1\) provided \({\mathcal B}\) is the basis of intervals.
Remarks of the reviewer: Instead of the term “bordering” function it is usually used the term “halo” function [see, e.g., M. de Guzmán, “Differentiation of integrals in \(\mathbb{R}^n\)” (1975; Zbl 0327.26010)].

Reviewer: P.Gurka (Praha)

MSC:

42B25

Maximal functions, Littlewood-Paley theory

Keywords:

halo function; differential basis; maximal function; bordering function

Citations:

Zbl 0327.26010

References:

[1]	M. Guzman, Differential Operators in Rn [Russian translation], Mir, Moscow (1978).
[2]	I. Stein and H. Weiss, Introduction to Harmonic Analysis on Euclidean Spaces [Russian translation], Mir, Moscow (1974).
[3]	M. Guzman, Real Variable Methods in Fourier Analysis, North-Holland, Amsterdam (Math. Stud. Vol. 46) (1981). · Zbl 0449.42001