Weak boundedness properties for maximal operators defined on weighted spaces
Let \(T_jf(x)=\int_{\mathbb R^n}K_j(x,y)f(y) dy\) and \(T^\ast f(x)=\sup_{j}|T_jf(x)|\), where \(\{K_j(x,y)\}_1^\infty\) is a family of measurable functions defined on \(\mathbb R^n\times \mathbb R^n\) satisfying, for weights \(u\), \(w\), \(K_j(x,\cdot)\in L^1_w(\mathbb R^n)\), \(K_j(\cdot, y)\in L_u^1(\mathbb R^n)\), and for every \(\varepsilon>0\), every ball \(B\) of \(\mathbb R^n\), and \(j\in \mathbb N\), there exists \(\eta(\varepsilon,j,B)\) such that \[ \int_B|K_j(x,y_2)-K_j(x,y_1)|u(x) dx\leq \varepsilon\qquad (y_1,\;y_2\in \mathbb R^n,\;|y_1-y_2|<\eta). \] The author's result is: \(T^\ast\) is of weak type \((1,q)\), \(q\geq 1\), with respect to the pair \((u, w)\) over simple functions, if and only if there is a \(C>0\) such that for every finite set of Lebesgue points of \(w\), \(\{a_j\}_{j=1}^N\subset \mathbb R^n\), and \(\lambda>0\), \[ u\Biggl( \bigg\{x;\sup_j |\sum_{k=1}^N K_j(x,a_k)|>\lambda\biggr\} \Biggr)\leq C\lambda^{-q}\Biggl( \sum_{k=1}^N w(a_k)\Biggr)^q. \] Using this, the author characterizes a pair of weights belonging to the Muckenhoupt's \(A_1\) by means of the fractional maximal operator. Analogous results are given for the one-sided fractional maximal operator. These are generalizations of the corresponding weak type \((1,1)\) results by \textit{M. de Guzmán} [``Real variable methods in Fourier analysis'' (1981; Zbl 0449.42001)] and by \textit{D. Termini} and \textit{C. Vitanza} [Bull. Lond. Math. Soc. 22, No. 4, 367-374 (1990; Zbl 0726.42013)].
scientific article; zbMATH DE number 1089785
scientific article