In this paper we study integral operators of the form $$Tf\left(x\right)=\int |x-a_1{y|}^{-{\alpha }_1}\cdots {|x-a_{m}y|}^{-{\alpha }_m}f\left(y\right)\mathrm{d}y,$$ ${\alpha }_1+\cdots +{\alpha }_m=n$. We obtain the $L^p\left(w\right)$ boundedness for them, and a weighted $(1,1)$ inequality for weights $w$ in $A_p$ satisfying that there exists $c\ge 1$ such that $w\left(a_{i}x\right)\le cw\left(x\right)$ for a.e. $x\in {ℝ}^n$, $1\le i\le m$. Moreover, we prove ${\parallel Tf\parallel }_{\mathrm{B}MO}\le c{\parallel f\parallel }_{\infty }$ for a wide family of functions $f\in L^{\infty }\left({ℝ}^n\right)$.

This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation $${\displaystyle \frac{\partial u}{\partial t}}-(\lambda +\mathrm{i}\alpha )\Delta u+(\kappa +\mathrm{i}\beta ){\left|u\right|}^{q-1}u-\gamma u=0$$ in ${ℝ}^N\times (0,\infty )$ with $L^p$-initial data $u_0$ in the subcritical case ($1\le q<1+2p/N$), where $u$ is a complex-valued unknown function, $\alpha$, $\beta$, $\gamma$, $\kappa \in ℝ$, $\lambda >0$, $p>1$, $\mathrm{i}=\sqrt{-1}$ and $N\in \mathbb{N}$. The proof is based on the $L^p$-$L^q$ estimates of the linear semigroup $\{exp(t(\lambda +\mathrm{i}\alpha )\Delta \left)\right\}$ and usual fixed-point argument.

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.