A characterization of some weighted norm inequalities for maximal operators
The main result of the paper concerns the boundedness of the generalized maximal operator \[ M_\varphi f(x)=\sup\limits_Q \frac{1}{\varphi(|Q|)}\int\limits_Q |f(y)|\,dy \] within the frameworks of the variable exponent Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R}^n),\) where the function \(\varphi\) is non-negative, continuous, increasing, with \(\varpi(0)=0\) and satisfying the subadditivity condition of the type \(\varphi\left(\sum t_i\right)\leq C \sum\varphi\left(t_i\right), \;\;\;t_i\in \mathbb{R}^1_+\). The main result is given in Theorem 4.7 which provides a certain necessary and sufficient condition on the exponent functions \(p(x), q(x) \) and \(\varphi(x)\) for the operator \(M_\varphi\) to be bounded from \(L^{p(\cdot)}(\mathbb{R}^n)\) to \(L^{q(\cdot)}(\mathbb{R}^n), \;\;p(x)\leq q(x)\). The proof is based on the previous results of the author, E.~Berezhnoi and L.~Diening and the results of the first sections of the paper on the validity of the generalized Minkowsky inequality for the variable exponent Lebesgue spaces which are of interest in themselves.
scientific article; zbMATH DE number 5816828