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We prove two-weight norm inequalities in ℝⁿ for the minimal operator $f\left(x\right)=in{f}_{Q\ni x}1/\left|Q\right|{\int }_Q\left|f\right|dy$, extending to higher dimensions results obtained by Cruz-Uribe, Neugebauer and Olesen [8] on the real line. As an application we extend to ℝⁿ weighted norm inequalities for the geometric maximal operator $M₀f\left(x\right)={sup}_{Q\ni x}exp(1/|Q|{\int }_{Q}log\left|f\right|dx)$, proved by Yin and Muckenhoupt [27]. We also give norm inequalities for the centered minimal operator, study powers of doubling weights and give sufficient conditions for the geometric maximal operator to be equal to the closely...

We introduce the one-sided minimal operator, $m^{+}f$, which is analogous to the one-sided maximal operator. We determine the weight classes which govern its two-weight, strong and weak-type norm inequalities, and show that these two classes are the same. Then in the one-weight case we use this class to introduce a new one-sided reverse Hölder inequality which has several applications to one-sided $\left(A_p^{+}\right)$ weights.