We characterize the class of weights, invariant under dilations, for which a modified fractional integral operator $I_{\alpha }$ maps weak weighted Orlicz$-\phi$ spaces into appropriate weighted versions of the spaces $BM{O}_{\psi }$, where $\psi \left(t\right)=t^{\alpha /n}{\phi }^{-1}(1/t)$. This generalizes known results about boundedness of $I_{\alpha }$ from weak $L^p$ into Lipschitz spaces for $p>n/\alpha$ and from weak $L^{n/\alpha }$ into $BMO$. It turns out that the class of weights corresponding to $I_{\alpha }$ acting on weak$-L_{\phi }$ for $\phi$ of lower type equal or greater than $n/\alpha$, is the same as the one solving the problem...

We prove sharp weighted inequalities of the form $${\int }_{{\mathbf{R}}^n}|f\left(x\right){|}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}|q\left(D\right)\left(f\right)\left(x\right){|}^{p}N\left(v\right)\left(x\right)dx$$ where $q\left(D\right)$ is a differential operator and $N$ is a combination of maximal type operator related to $q\left(D\right)$ and to $p$.

In this paper we study the mapping properties of the one-sided fractional integrals in the Calderón-Hardy spaces ${\mathscr{H}}_{q,\alpha }^{p,+}\left(\omega \right)$ for $0<p\le 1$, $0<\alpha <\infty$ and $1<q<\infty$. Specifically, we show that, for suitable values of $p,q,\gamma ,\alpha$ and $s$, if $\omega \in A_s^{+}$ (Sawyer’s classes of weights) then the one-sided fractional integral $I_{\gamma }^{+}$ can be extended to a bounded operator from ${\mathscr{H}}_{q,\alpha }^{p,+}\left(\omega \right)$ to ${\mathscr{H}}_{q,\alpha +\gamma }^{p,+}\left(\omega \right)$. The result is a consequence of the pointwise inequality $$N_{q,\alpha +\gamma }^{+}\left(I_{\gamma }^{+}F;x\right)\le C_{\alpha ,\gamma }{N}_{q,\alpha }^{+}\left(F;x\right),$$ where $N_{q,\alpha }^{+}(F;x)$ denotes the Calderón maximal function.