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...

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...

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 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 discuss the convergence of approximate identities in Musielak-Orlicz spaces extending the results given by Cruz-Uribe and Fiorenza (2007) and the authors F.-Y. Maeda, Y. Mizuta and T. Ohno (2010). As in these papers, we treat the case where the approximate identity is of potential type and the case where the approximate identity is defined by a function of compact support. We also give a Young type inequality for convolution with respect to the norm in Musielak-Orlicz spaces. ...