In this paper, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and Lipschitz function b e $\dot \Lambda _{\beta _0 } $ (ℝn) is discussed from Lp(ℝn) to Lq(ℝn), $\tfrac{1}{q} = \tfrac{1}{p} - \tfrac{{\beta _0 }}{n}$, and from Lp(ℝn) to Triebel-Lizorkin space $\dot F_p^{\beta _0 ,\infty } $. We also obtain the boundedness of generalized Toeplitz operator Θα0b from Lp(ℝn) to Lq(ℝn), $\tfrac{1}{q} = \tfrac{1}{p} - \tfrac{{\alpha _0 + \beta _0 }}{n}$. All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and BMO function b is discussed on Lp(ℝn), 1 < p < ∞.

Toeplitz operators related to strongly singular Calderón-Zygmund operators

We obtain the Lipschitz boundedness for a class of fractional multilinear operators $I_{\Omega,\alpha}^{A,m}$ with rough kernels $\Omega\in L_{s}(\mathbb S^{n-1})$, $s>n/(n-\alpha)$ on the local generalized Morrey spaces $LM_{p,\varphi}^{\{x_0\}}$, generalized Morrey spaces $M_{p,\varphi}$ and vanishing generalized Morrey spaces $VM_{p,\varphi}$, where the functions $A$ belong to homogeneous Lipschitz space $\dot{\Lambda}_{\beta}$, $0<\beta<1$. We find the sufficient conditions on the pair $(\varphi_1,\varphi_2)$ which ensures the boundedness of the operators $I_{\Omega,\alpha}^{A,m}$ from $LM_{p,\varphi_1}^{\{x_0\}}$ to $LM_{q,\varphi_2}^{\{x_0\}}$, from $M_{p,\varphi_1}$ to $M_{q,\varphi_2}$ and from $VM_{p,\varphi_1}$ to $VM_{q,\varphi_2}$ for $1<p<q <\infty$ and $1/p-1/q=(\alpha+\beta)/n$. In all cases the conditions for the boundedness of the operator $I_{\Omega,\alpha}^{A,m}$ is given in terms of Zygmund-type integral inequalities on $(\varphi_1,\varphi_2)$, which do not assume any assumption on monotonicity of $\varphi_1(x,r), \varphi_2(x,r)$ in $r$.

Lipschitz estimates for rough fractional multilinear integral operators on local generalized Morrey spaces

In this paper, the author introduces some late results and puts forward a few problems on commutators of many important operators in harmonic analysis, included the Bochner-Riesz operator below the critical index, the strongly singular integral operator, the pseudo-differential operator, a class of convolution operators with oscillatory kernel, the Marcinkiewicz integral operator, and the fractional integral operator with rough kernel.

Some results and problems on commutators

We obtain the Lipschitz boundedness for a class of fractional multilinear operators with rough kernels on variable exponent Lebesgue spaces. Our results generalize the related conclusions on Lebesgue spaces with constant exponent.

/pdf/lipschitz-estimates-for-fractional-multilinear-singular-4a54zzqnk5.pdf

Lipschitz Estimates for Fractional Multilinear Singular Integral on Variable Exponent Lebesgue Spaces

In this paper, we show the strong and weak type boundedness of $T_{\Omega,\alpha}^A$ and $M_{\Omega,\alpha}^A$, the multilinear fractional integral operators and the corresponding fractional maximal operators, on the two weights weighted Morrey space. Furthermore, we also obtain the mulitilinear singular integral operators $T_{\Omega}^A$ and the corresponding maximal operators $M_{\Omega}^A$ are strong bounded operators on the same spaces.

The Boundedness of Multilinear operators with rough kernel on the weighted Morrey spaces

Let (0 1) boundedness, the weak (L1, Ln/(n–α–β)) boundedness and the (Lp, Ḟβ, ∞p) boundedness of are discussed, when DγA belongs to the Lipschitz function spaces.

Lipschitz estimates for generalized commutators of fractional integrals with rough kernel

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Lipschitz estimates for generalized commutators of fractional integrals with rough kernel