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...

We introduce a new type of variable exponent function spaces $M{\dot{K}}_{q,p(·)}^{\alpha (·),\lambda }\left({ℝ}^n\right)$ of Morrey-Herz type where the two main indices are variable exponents, and give some propositions of the introduced spaces. Under the assumption that the exponents $\alpha$ and $p$ are subject to the log-decay continuity both at the origin and at infinity, we prove the boundedness of a wide class of sublinear operators satisfying a proper size condition which include maximal, potential and Calderón-Zygmund operators and their commutators...

In this paper, we are going to characterize the space $\mathrm{BMO}\left({ℝ}^n\right)$ through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space $\mathrm{BMO}\left({ℝ}^n\right)$ by using various function spaces. For example, Ho obtained a characterization of $\mathrm{BMO}\left({ℝ}^n\right)$ with respect to rearrangement invariant spaces. However, variable Lebesgue spaces and Morrey spaces do not appear in the characterization. One of the reasons is that these spaces are not rearrangement invariant. We also obtain an analogue...

In this paper the author proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with variable exponent. As an application he proved the boundedness of certain sublinear operators on the weighted variable Lebesgue space. The proof of the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent does not contain any mistakes. But in the proof of the boundedness of certain sublinear operators on the...