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The commutator of a singular integral operator with homogeneous kernel Ω(x)/|x|ⁿ is studied, where Ω is homogeneous of degree zero and has mean value zero on the unit sphere. It is proved that $\Omega \in L{\left(logL\right)}^{k+1}\left(S^{n-1}\right)$ is a sufficient condition for the kth order commutator to be bounded on $L^p\left(ℝⁿ\right)$ for all 1 < p < ∞. The corresponding maximal operator is also considered.

Some boundedness results are established for sublinear operators on the homogeneous Herz spaces. As applications, some new theorems about the boundedness on homogeneous Herz spaces for commutators of singular integral operators are obtained.

We establish a variant sharp estimate for multilinear singular integral operators. As applications, we obtain the weighted norm inequalities on general weights and certain $Llo{g}^{+}L$ type estimates for these multilinear operators.

Let μ be a nonnegative Radon measure on ${ℝ}^d$ which satisfies μ(B(x,r)) ≤ Crⁿ for any $x\in {ℝ}^d$ and r > 0 and some positive constants C and n ∈ (0,d]. In this paper, some weighted norm inequalities with $A_p^{\varrho }\left(\mu \right)$ weights of Muckenhoupt type are obtained for maximal singular integral operators with such a measure μ, via certain weighted estimates with $A_{\infty }^{\varrho }\left(\mu \right)$ weights of Muckenhoupt type involving the John-Strömberg maximal operator and the John-Strömberg sharp maximal operator, where ϱ,p ∈ [1,∞).

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 of the bilinear...

The $L^p\left(ℝⁿ\right)$ boundedness is established for commutators generated by BMO(ℝⁿ) functions and convolution operators whose kernels satisfy certain Fourier transform estimates. As an application, a new result about the $L^p\left(ℝⁿ\right)$ boundedness is obtained for commutators of homogeneous singular integral operators whose kernels satisfy the Grafakos-Stefanov condition.

Under the assumption that m is a non-doubling measure on Rd, the authors obtain the (Lp,Lq)-boundedness and the weak type endpoint estimate for the multilinear commutators generated by fractional integrals with RBMO (m) functions of Tolsa or with Osc exp Lr(m) functions for r greater than or equal to 1, where Osc exp Lr(m) is a space of Orlicz type satisfying that Osc exp Lr(m)=RBMO(m) if r=1 and Osc exp Lr(m) is a subset of RBMO(m) if r&gt;1.

Let $\mu$ be a nonnegative Radon measure on ${ℝ}^d$ which only satisfies $\mu \left(B(x,r)\right)\le C_0{r}^n$ for all $x\in {ℝ}^d$, $r>0$, with some fixed constants $C_0>0$ and $n\in (0,d].$ In this paper, a new characterization for the space $\mathrm{RBMO}\left(\mu \right)$ of Tolsa in terms of the John-Strömberg sharp maximal function is established.