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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 \(X_j = X\).

42B25 Maximal functions, Littlewood-Paley theory