Criteria of a four-weight weak type inequality for one-sided maximal operators in Orlicz classes
The symbol \(\Phi\) denotes the set of all real-valued functions \(\varphi\) on the real line, which are nonnegative, even, increasing on \((0,\infty)\), and such that \(\lim_{t\to 0^+}\varphi(t) = 0\), \(\lim_{t\to\infty}\varphi(t) =\infty\). Let \(g\) be a positive locally integrable function on the real line, and the onesided maximal operator \(\mathcal M^+_g\) for \(f\in L^1_{\mathrm{loc}}(\mathbb R)\) is defined by \[\mathcal{M}^+_g f(x) = \sup_{h>0} \frac{1}{g(x, x + h)} \int^{x+h}_{x} |f(y)|g(y)dy,\] where \(g(x, x + h) =\int^{x+h}_{x} g(y)dy\).\N\NThe authors' main result is as follows: Let \(\varphi\in \Phi\) be a quasi-convex function and \(\omega_i\) \((i =1,2,3,4)\) be a weight, then there exists a constant \(C_1 > 0\), such that \[\int_{\{\mathcal{M}^+_g (f)>\lambda\}}\varphi(\lambda\omega_1(x)) \omega_2(x)g(x)dx \le C_1\int_{-\infty}^\infty \varphi(C_1 |f(x)|\omega_3(x))\omega_4(x)g(x)dx\] holds for all \(f\) and \(\lambda > 0\). This extends the corresponding result for the Hardy-Littlewood maximal operator, which is a multi-weight generalization of Muckenhoupt's result in Orlicz classes.\N\NAlso, some related results including some necessary and sufficient conditions are discussed.
scientific article; zbMATH DE number 7846852
scientific article