Boundedness of operators and inequalities on Morrey-Banach spaces (Q2165819)

Let \(X\) be a Banach function space of Lebesgue measurable functions on \(\mathbb{R}^n\); see, e.g., [\textit{C. Bennett} and \textit{R. Sharpley}, Interpolation of operators. Boston, MA etc.: Academic Press, Inc. (1988; Zbl 0647.46057)]. Given a Lebesgue measurable function \(u(y,r): \mathbb{R}^n \times (0,\infty) \to (0,\infty)\), the corresponding Morrey-Banach space \(M^u_X\) consists of all measurable functions \(f\) on \(\mathbb{R}^n\) satisfying \[ ||f||_{M^u_X}=\sup\limits_{y\in \mathbb{R}^n, r>0} \, \frac{1}{u(y,r)}\, ||\chi_{B(y,r)} f||_X <\infty, \] \(\chi_{B(y,r)}\) being the characteristic function of the open ball with center \(y\in \mathbb{R}^n\) and radius \(r\). The authors obtain boundedness results and the relevant inequalities for a large number of operators acting on \(M^u_X\) and arising in harmonic analysis. These include spherical means, Bochner-Riesz operators, Rubio de Francia operators, Fourier integral operators, strongly singular integral operators, Coifman-Fefferman inequality, and some others.