Commutators Of Fractional Integral Operators Research Articles

Compactness of commutators of fractional integral operators on ball Banach function spaces

<abstract><p>Let $ 0 < \alpha < n $ and $ b $ be a locally integrable function. In this paper, we obtain the characterization of compactness of the iterated commutator $ (T_{\Omega, \alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator with the homogeneous kernel $ T_{\Omega, \alpha} $ on ball Banach function spaces. As applications, we derive the characterization of compactness via the commutator $ (T_{\Omega, \alpha})_b^m $ on weighted Lebesgue spaces, and further obtain a necessary and sufficient condition for the compactness of the iterated commutator $ (T_{\alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on Morrey spaces. Moreover, we also show the necessary and sufficient condition for the compactness of the commutator $ [b, T_{\alpha}] $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on variable Lebesgue spaces and mixed Morrey spaces.</p></abstract>

A note on some Endpoint Estimates of Commutators of Fractional Integral Operators

A note on some Endpoint Estimates of Commutators of Fractional Integral Operators

Two matrix weighted inequalities for commutators with fractional integral operators

In this paper we prove two matrix weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a matrix symbol. More precisely, we extend the recent results of the second author, Pott, and Treil on two matrix weighted norm inequalities for commutators of Calderon-Zygmund operators and multiplication by a matrix symbol to the fractional integral operator setting. In particular, we completely extend the fractional Bloom theory of Holmes, Rahm, and Spencer to the two matrix weighted setting with a matrix symbol and also provide new two matrix weighted norm inequalities for commutators of fractional integral operators with a matrix symbol and two arbitrary matrix weights. These results are new even in the scalar one weighted setting with a scalar symbol.

Endpoint boundedness for commutators of singular integral operators on weighted generalized Morrey spaces

In this paper, we obtain the endpoint boundedness for the commutators of singular integral operators with BMO functions and the associated maximal operators on weighted generalized Morrey spaces. We also get similar results for the commutators of fractional integral operators with BMO functions and the associated maximal operators.

Maximal commutators and commutators of potential operators in new vanishing Morrey spaces

We study mapping properties of commutators in certain vanishing subspaces of Morrey spaces, which were recently used to solve the delicate problem of describing the closure of nice functions in Morrey norm. We show that the vanishing properties defining those subspaces are preserved under the action of maximal commutators and commutators of fractional integral operators.

Boundedness of commutators of fractional integral operators on mixed Morrey spaces

ABSTRACTIn this paper, we give the necessary and sufficient conditions for the boundedness of commutators of fractional integral operators on mixed Morrey spaces. We construct the predual spaces of mixed Morrey spaces. Furthermore, we show the sharp maximal inequality of the mixed Morrey norm. This inequality is one of the key inequalities of the main theorem.

Weighted estimates for a class of multilinear fractional type operators

In this paper, a class of multiple fractional type weights A ( p → , q ) is defined as sup Q ( 1 | Q | ∫ Q ν ω → q ) 1 q ∏ i = 1 m ( 1 | Q | ∫ Q ω i − p i ′ ) 1 p i ′ < ∞ , where ν ω → = ∏ i = 1 m ω i . And the necessary condition for the characterization of A ( p → , q ) is also obtained. Strong ( L p 1 ( ω 1 p 1 ) × ⋯ × L p m ( ω m p m ) , L q ( ν ω → q ) ) estimates when each p i > 1 and weighted endpoint estimates ( L p 1 ( ω 1 p 1 ) × ⋯ × L p m ( ω m p m ) , L q , ∞ ( ν ω → q ) ) when there exists p i = 1 for some multilinear fractional type operators (e.g. fractional maximal operator, fractional integral, commutators of fractional integral operators) are obtained. As applications of these results, we give some weighted estimates for the above operators with rough homogeneous kernels when suitable conditions were assumed on the kernels. Weighted strong and L ( log L ) type endpoint estimates for commutators of multilinear fractional integral operators are also obtained. Similar results for multilinear Calderón–Zygmund singular integral can be found in A.K. Lerner et al. (in press) [12].

COMMUTATORS OF FRACTIONAL INTEGRAL OPERATOR ASSOCIATED TO A NONDOUBLING MEASURES ON METRIC SPACES

The main purpose of this paper is to prove the boundedness of commutators of fractional integral operators associated to a measure on metric space satisfying just a mild growth condition.

Commutators of fractional integral operators on Vanishing-Morrey spaces

In this note we prove a sufficient condition for commutators of fractional integral operators to belong to Vanishing Morrey spaces VL p,?. In doing this we use an extension on Morrey spaces of an inequality by Fefferman and Stein concerning the sharp maximal function and the fractional maximal function and related Morrey inequalities.

Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on classical Morrey spaces

We prove that $b$ is in $BMO(\R^n)$ if and only if the commutator $[b,I_{\alpha}]$ of the multiplication operator by $b$ and the fractional integral operator $I_{\alpha}$ is bounded from the classical Morrey space $L^{p,\lambda}(\R^n)$ to $L^{q,\mu}(\R^n)$, where $1<p< \infty$, $0<\alpha <n$, $0<\lambda <n- \alpha p$, $1/q=1/p-\alpha /n$ and $\lambda /p = \mu /q$. Also we will show that $b$ is in $\dot{\Lambda}_{\beta}(\R^n)$ if and only if the commutator $[b,I_{\alpha}]$ is bounded from the classical Morrey space $L^{p,\lambda}(\R^n)$ to $L^{q,\mu}(\R^n)$ or from $L^{p,\lambda}(\R^n)$ to $L^{q,\lambda}(\R^n)$, where $\alpha$ and $\beta$ satisfy some conditions.

Weak estimates for commutators of fractional integral operators

By introducing a kind of maximal operator of the fractional order associated with the mean Luxemburg norm and using the technique of the sharp function, the weak type L log L estimates for the commutators of the fractional integral operator and the related maximal operator are established.