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

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

Zbl 0327.26010
Full Text:

### 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
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