Necessary and sufficient conditions for boundedness of commutators of the general fractional integral operators on weighted Morrey spaces

We prove that $b$ is in Lip_{\bz}(\bz) if and only if the commutator $[b,L^{-\alpha/2}]$ of the multiplication operator by $b$ and the general fractional integral operator $L^{-\alpha/2}$ is bounded from the weighed Morrey space $L^{p,k}(\omega)$ to $L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\omega)$, where $0<\beta<1$, $0<\alpha+\beta<n, 1<p<{n}/({\alpha+\beta})$, ${1}/{q}={1}/{p}-{(\alpha+\beta)}/{n},$ $0\leq k<{p}/{q},$ $\omega^{{q}/{p}}\in A_1$ and $r_\omega> \frac{1-k}{p/q-k},$ and here $r_\omega$ denotes the critical index of $\omega$ for the reverse H\"{o}lder condition.Comment: 12 pages; Classical Analysis and ODEs (math.CA), Functional Analysis (math.FA

On General multilinear square function with non-smooth kernels

In this paper, we obtain some boundedness of the following general multilinear square functions $T$ with non-smooth kernels, which extend some known results significantly. $$T(\vec{f})(x)=\big( \int_{0}^\infty \big|\int_{(\mathbb{R}^n)^m}K_v(x,y_1,\dots,y_m) \prod_{j=1}^mf_{j}(y_j)dy_1,\dots,dy_m\big|^2\frac{dv}{v}\big)^{\frac 12}.$$ The corresponding multilinear maximal square function $T^*$ was also introduced and weighted strong and weak type estimates for $T^*$ were given.Comment: 19 page

Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

Let $0<\alpha<n$ and $M_{\alpha}$ be the fractional maximal function. The nonlinear commutator of $M_{\alpha}$ and a locally integrable function $b$ is given by $[b,M_{\alpha}](f)=bM_{\alpha}(f)-M_{\alpha}(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_{\alpha}]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or BMO(\rn) spaces, by which some new characterizations for certain subclasses of Lipschitz and BMO(\rn) spaces are obtained.Comment: 20 page

Some Weighted Estimates for Multilinear Fourier Multiplier Operators

We first provide a weighted Fourier multiplier theorem for multilinear operators which extends Theorem 1.2 in Fujita and Tomita (2012) by using Lr-based Sobolev spaces (1<r≤2). Then, by using a different method, we obtain a result parallel to Theorem 6.2 which is an improvement of Theorem 1.2 under assumption (i) in Fujita and Tomita (2012)

Estimates for iterated commutators of multilinear square fucntions with Dini-type kernels

Abstract Let TΠb→ $T_{\Pi\vec {b}}$ be the commutator generated by a multilinear square function and Lipschitz functions with kernel satisfying Dini-type condition. We show that TΠb→ $T_{\Pi\vec {b}}$ is bounded from product Lebesgue spaces into Lebesgue spaces, Lipschitz spaces, and Triebel–Lizorkin spaces