In this paper we study the relationship between one-sided reverse Hölder classes $R{H}_r^{+}$ and the $A_p^{+}$ classes. We find the best possible range of $R{H}_r^{+}$ to which an $A_1^{+}$ weight belongs, in terms of the $A_1^{+}$ constant. Conversely, we also find the best range of $A_p^{+}$ to which a $R{H}_{\infty }^{+}$ weight belongs, in terms of the $R{H}_{\infty }^{+}$ constant. Similar problems for $A_p^{+}$, $1<p<\infty$ and $R{H}_r^{+}$, $1<r<\infty$ are solved using factorization.

In the paper we find conditions on the pair $({\omega }_1,{\omega }_2)$ which ensure the boundedness of the maximal operator and the Calderón-Zygmund singular integral operators from one generalized Morrey space ${ℳ}_{p,{\omega }_1}$ to another ${ℳ}_{p,{\omega }_2}$, $1<p<\infty$, and from the space ${ℳ}_{1,{\omega }_1}$ to the weak space $W{ℳ}_{1,{\omega }_2}$. As applications, we get some estimates for uniformly elliptic operators on generalized Morrey spaces.

We characterize the Choquet integrals associated to Bessel capacities in terms of the preduals of the Sobolev multiplier spaces. We make use of the boundedness of local Hardy-Littlewood maximal function on the preduals of the Sobolev multiplier spaces and the minimax theorem as the main tools for the characterizations.

Given a positive measure μ in ${ℝ}^n$, there is a natural variant of the noncentered Hardy-Littlewood maximal operator $M_{\mu }f\left(x\right)=su{p}_{x\in B}1/\mu \left(B\right){ʃ}_B\left|f\right|d\mu$, where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in ${ℝ}^n$. We give some necessary and sufficient conditions for $M_{\mu }$ to be bounded from $L^1\left(d\mu \right)$ to $L^{1,\infty }\left(d\mu \right)$.

For d > 1, let $\left(S_{d}f\right)(x,t)={ʃ}_{{ℝ}^n}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi$, $x\in {ℝ}^n$, where f̂ is the Fourier transform of $f\in S\left({ℝ}^n\right)$, and $(S_d*f)\left(x\right)=su{p}_{0<t<1}\left|\left(S_{d}f\right)(x,t)\right|$ its maximal operator. P. Sjölin ([11]) has shown that for radial f, the estimate (*) $({ʃ}_{\left|x\right|<R}|(S_d*f)\left(x\right){{|}^{p}dx)}^{1/p}\le C_R\parallel f{\parallel }_{H_{1/4}}$ holds for p = 4n/(2n-1) and fails for p > 4n/(2n-1). In this paper we show that for non-radial f, (*) fails for p > 2. A similar result is proved for a more general maximal operator.