Let be a measurable function on an open set with values in . Denote by the Banach space of measurable functions on such that for some , , with norm . In the case , one defines as above. In general, is not a norm, but quasi-norm. The authors’ main tool (theorem) in this paper is the following: Given a family of ordered pairs of non-negative, measurable functions , suppose that for some , , and for every weight (Muckenhoupt’s weight class), it holds , , where depends only on and the constant of . Furthermore, let satisfy and for the conjugate exponent the Hardy-Littlewood maximal operator is bounded on . Then, for all with , , where is independent of the pair .
They give a generalization of this to the case of . They also discuss the case of or in place of and give vector-valued inequalities. As applications of these main results, they discuss a vector-valued inequality for the Hardy-Littlewood maximal operator, the sharp maximal functions, singular integral operators, commutators of a singular integral operator and a BMO function, Fourier multipliers, Littlewood-Paley’s functions, fractional integrals, the Calderón-Zygmund inequality for the solutions of Poisson’s equation, and the Calderón extension theorem for variable Sobolev spaces.