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On approximation of maximal operators. (English) Zbl 1224.42058
M. de Guzmán [Real variable methods in Fourier analysis. Amsterdam - New York - Oxford: North-Holland Publishing (1981; Zbl 0449.42001)] showed that the uniform weak type $$(1,1)$$ boundedness of the Hardy-Littlewood maximal operator $$M$$ for linear combinations of Dirac deltas with positive integer coefficients implies the weak $$(1,1)$$ boundedness of $$M$$. The authors generalize this result as follows. Let $$(X,d,\mu)$$ be a space of homogeneous type. Assume that $$\omega$$ is a Borel measure such that $$d\omega = w \, d\mu$$ where $$w$$ is a locally integrable non-negative function on $$X$$. Let $$\{ k_{\ell} : \ell \in \mathbb{N} \}$$ be a sequence of continuous kernels with compact support on $$X \times X$$. Define $$K_{\ell} f(x) = \int_X k_{\ell}(x,y)f(y)\,dy$$, and $$K^{*}f(x) = \sup_{\ell} | K_{\ell}f(x) |$$. Let $$\{ (X_j , \omega_j ): j \in \mathbb{N} \}$$ be a sequence of measure spaces such that
(1) each $$X_j$$ is a Borel subset of $$X$$;
(2) $$X_j \subset X_{j+1}$$;
(3) $$\bigcup_{j \in \mathbb{N}} X_j$$ is dense in $$X$$;
(4) {supp} $$\omega_j \subset X_j$$;
(5) $$\omega_j \to \omega$$ in the weak star convergence.
If there exists a constant $$C$$ such that for every $$\lambda >0$$ and every finite set $$\{ x_1^j, x_2^j , \ldots , x_H^j\} \subset X_j$$, $\omega_j \Big( \Big\{ x^j \in X_j : \sup_{\ell \in \mathbb{N}} | \sum_{i=1}^{H} k_{\ell} (x^j, x_i^j) | > \lambda \Big\} \Big) \leq \frac{CH}{\lambda},$ then $$K^{*}$$ is of weak type $$(1,1)$$.
M. Carena [Rev. Unión Mat. Argent. 50, No. 1, 145–159 (2009; Zbl 1187.42014)] proved this theorem when $$\omega_j = \omega$$ and $$X_j = X$$.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory
##### Citations:
Zbl 0449.42001; Zbl 1187.42014