We give conditions on pairs of weights which are necessary and sufficient for the operator $T\left(f\right)=K*f$ to be a weak type mapping of one weighted Lorentz space in another one. The kernel $K$ is an anisotropic radial decreasing function.

Let $S\subset {ℝ}^{n+1}$ be the graph of the function $\varphi :{[-1,1]}^n\to ℝ$ defined by $\varphi (x_1,\cdots ,x_n)={\sum }_{j=1}^n{\left|x_j\right|}^{{\beta }_j},$ with 1<${\beta }_1\le \cdots \le {\beta }_n,$ and let $\mu$ the measure on ${ℝ}^{n+1}$ induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with $\mu$ is $L^p$-$L^q$ bounded.

Let $b_1,b_2\in \mathrm{BMO}\left({ℝ}^n\right)$ and $T_{\sigma }$ be a bilinear Fourier multiplier operator with associated multiplier $\sigma$ satisfying the Sobolev regularity that ${sup}_{\kappa \in \mathbb{Z}}{\parallel {\sigma }_{\kappa }\parallel }_{W^{s_1,s_2}\left({ℝ}^{2n}\right)}<\infty$ for some $s_1,s_2\in (n/2,n]$. In this paper, the behavior on $L^{p_1}\left({ℝ}^n\right)\times L^{p_2}\left({ℝ}^n\right)$ $(p_1,p_2\in (1,\infty ))$, on $H^1\left({ℝ}^n\right)\times L^{p_2}\left({ℝ}^n\right)$ $(p_2\in [2,\infty ))$, and on $H^1\left({ℝ}^n\right)\times H^1\left({ℝ}^n\right)$, is considered for the commutator $T_{\sigma ,\overrightarrow{b}}$ defined by $$\begin{array}{ccc}\hfill T_{\sigma ,\overrightarrow{b}}(f_1,f_2)\left(x\right)=& b_1\left(x\right)T_{\sigma }(f_1,f_2)\left(x\right)-T_{\sigma }(b_1{f}_1,f_2)\left(x\right)\hfill & \hfill +b_2\left(x\right)T_{\sigma }(f_1,f_2)\left(x\right)-T_{\sigma }(f_1,b_2{f}_2)\left(x\right).\end{array}$$ By kernel estimates of the bilinear Fourier multiplier operators and employing some techniques in the theory of bilinear singular integral operators, it is proved that these mapping properties are very similar to those...