Multilinear Fractional Integral Operators Research Articles

Boundedness Criterion for Multi-sublinear Operators and Their Commutators Generated by Multilinear Fractional Integral Operator on Product Generalized Mixed Morrey Spaces

Boundedness Criterion for Multi-sublinear Operators and Their Commutators Generated by Multilinear Fractional Integral Operator on Product Generalized Mixed Morrey Spaces

Two-weighted estimates of the multilinear fractional integral operator between weighted Lebesgue and Lipschitz spaces with optimal parameters

Two-weighted estimates of the multilinear fractional integral operator between weighted Lebesgue and Lipschitz spaces with optimal parameters

Characterization of Lipschitz functions via the commutators of multilinear fractional integral operators in variable Lebesgue spaces

AbstractThe main purpose of this article is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of commutator of multilinear fractional Calderón-Zygmund integral operators in the context of the variable exponent Lebesgue spaces. The authors do so by applying the techniques of Fourier series and multilinear fractional integral operator, as well as some pointwise estimates for the commutators. The key tool in obtaining such a pointwise estimate is a certain generalization of the classical sharp maximal operator.

Sharp bounds for multilinear fractional integral operators on Morrey type spaces: the endpoint cases

In this paper, the Strichartz’s result of the exponential integrability of fractional integral operators is improved. Also, we establish the endpoint boundedness of the multilinear fractional integrals acting on the multi-Morrey spaces. The conclusions relax the restriction that $$p_{i}\ne 1$$ for all $$i=1,\ldots ,m$$ and extend some known results.

Optimal parameters related with continuity properties of the multilinear fractional integral operator between Lebesgue and Lipschitz spaces

We deal with the boundedness of the multilinear fractional integral operator $$I_{\gamma ,m}$$ from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. We characterize the classes of weights for which the problem described above holds and show the optimal range of the parameters involved. The optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region. We further exhibit examples of weights for the class which cover the mentioned area.

Weak factorizations of the Hardy space in terms of multilinear fractional integral operator

We give a constructive proof of the factorization theorem for the classical Hardy space in terms of fractional integral operator. Moreover, the result is extended to the multilinear case and weighted case. As an application, we obtain the characterization of BMO via the weighted boundedness of commutators of the multilinear fractional integral operator, without individual conditions on the weights class.

Some Weighted Estimates on Gaussian Measure Spaces

In this paper, we obtain the weighted boundedness for the local multi(sub)linear Hardy-Littlewood maximal operators and local multilinear fractional integral operators associated with the local Muckenhoupt weights on Gaussian measure spaces. We deal with these problems by introducing a new pointwise equivalent "radial" definitions of these local operators. Moreover using a similar approach, we also get the weighted boundedness for the local fractional maximal operators with rough kernel and local fractional integral operators with rough kernel on Gaussian measure spaces.

Extension of multilinear fractional integral operators to linear operators on mixed-norm Lebesgue spaces

In Kenig and Stein (Math Res Lett 6(1):1–15, 1999, https://doi.org/10.4310/MRL.1999.v6.n1.a1 ), the following type of multilinear fractional integral $$\begin{aligned} \int _{{\mathbb {R}}^{mn}} \frac{f_1(l_1(x_1,\ldots ,x_m,x))\cdots f_{m+1}(l_{m+1}(x_1,\ldots ,x_m,x))}{(|x_1|+\cdots +|x_m|)^{\lambda }} dx_1\ldots dx_m \end{aligned}$$ was studied, where $$l_i$$ are linear maps from $${\mathbb {R}}^{(m+1)n}$$ to $${\mathbb {R}}^n$$ satisfying certain conditions. They proved the boundedness of such multilinear fractional integral from $$L^{p_1}\times \cdots \times L^{p_{m+1}}$$ to $$L^q$$ when the indices satisfy the homogeneity condition. In this paper, we show that the above multilinear fractional integral extends to a linear operator for functions in the mixed-norm Lebesgue space $$L^{{\mathbf {p}}}$$ which contains $$L^{p_1}\times \cdots \times L^{p_{m+1}}$$ as a subset. Under less restrictions on the linear maps $$l_i$$ , we give a complete characterization of the indices $${\mathbf {p}}$$ , q and $$\lambda $$ for which such an operator is bounded from $$L^{{\mathbf {p}}}$$ to $$L^q$$ . And for $$m=1$$ or $$n=1$$ , we give necessary and sufficient conditions on $$(l_1, \ldots , l_{m+1})$$ , $${\mathbf {p}}=(p_1,\ldots , p_{m+1})$$ , q and $$\lambda $$ such that the operator is bounded.

On the behaviors of rough multilinear fractional integral and multi-sublinear fractional maximal operators both on productLpand weightedLpspaces

AbstractThe aim of this paper is to get the productLp${L}^{p}$-estimates, weighted estimates and two-weighted estimates for rough multilinear fractional integral operators and rough multi-sublinear fractional maximal operators, respectively. The author also studies two-weighted weak type estimate on productLp(ℝn)${L}^{p}\left({\mathrm{\mathbb{R}}}^{n}\right)$for rough multi-sublinear fractional maximal operators. In fact, this article is the rough kernel versions of [C. E. Kenig and E. M. Stein, “Multilinear estimates and fractional integration,”Math. Res. Lett., vol. 6, pp. 1–15, 1999, Y. Shi and X. X. Tao, “WeightedLp${L}_{p}$boundedness for multilinear fractional integral on product spaces,”Anal. Theory Appl., vol. 24, no. 3, pp. 280–291, 2008]'s results.

Multilinear fractional integral operators: a counter-example

By means of a counter-example we show that the multilinear fractional operator is not bounded from a product of Hardy spaces into a Hardy space.

Weighted estimates of multilinear fractional integral operators for radial functions

Weighted estimates of multilinear fractional integral operators for radial functions

Multilinear fractional integral operators on central Morrey spaces with variable exponent

In this paper, we obtain the boundedness of the multilinear fractional integral operators and their commutators on central Morrey spaces with variable exponent.

Endpoint estimates for multilinear fractional integral operators on metric measure spaces

Let (X,d,μ) be a metric measure space such that, for any fixed x∈X, μ(B(x,r)) is a continuous function with respect to r∈(0,∞). In this paper, we prove endpoint estimates for the multilinear fractional integral operators Im,α from the product of Lebesgue spaces L1(μ)×⋯×L1(μ)×Lpk+1(μ)×⋯×Lpm(μ) into the Lebesgue space Lq(μ), where k∈[1,m)∩N, α∈[k,m), pi∈(1,∞) for i∈{k+1,…,m} and 1/q=k+∑i=k+1m1/pi−α. We furthermore prove that Im,α is bounded from Lp1(μ)×⋯×Lpm(μ) into L∞(μ), where pi∈(1,∞) for i∈{1,…,m} and ∑i=1m1/pi=α∈[1,m).

English

The spaces of multi-Morrey type for positive Radon measures satisfying a growth condition on ℝd are introduced. After defining the spaces, we investigate the multilinear maximal function, the multilinear fractional integral operator and the multilinear Calderon-Zygmund operators, respectively, from multi-Morrey spaces to Morrey spaces.

On the Boundedness of Multilinear Fractional Integral Operators

Necessary and sufficient condition governing the boundedness of the multilinear fractional integral operator $$T_{\gamma , \mu }$$ defined with respect to a measure $$\mu $$ on a $$\sigma $$-algebra of Borel sets of quasi-metric space X from the product $$L^{p_1}(X, \mu )\times \cdots \times L^{p_m}(X, \mu )$$ to $$L^q(X, \mu )$$ is established. The related weak type inequality is also obtained. The derived results are used to get appropriate boundedness of $$T_{\gamma , \mu }$$ in Morrey spaces defined with respect to a measure $$\mu $$.

Multilinear BMO Estimates for the Commutators of Multilinear Fractional Maximal and Integral Operators on the Product Generalized Morrey Spaces

In this paper, we establish multilinear BMO estimates for commutators of multilinear fractional maximal and integral operators both on product generalized Morrey spaces and product generalized vanishing Morrey spaces, respectively. Similar results are still valid for commutators of multilinear maximal and singular integral operators.

On the Multilinear Fractional Integral Operators with Correlation Kernels

In this paper, we study a class of multilinear fractional integral operators which have correlation kernels $\prod_{1\leq i<j \leq k}|x_i-x_j|^{-\alpha_{ij}}$. The necessary and sufficient conditions are obtained under which these oprators are bounded from $L^{p_1}\times \cdots \times L^{p_k}$ into $L^q$. As a consequence, we also get the endpoint estimates from $L^{p_1}\times \cdots \times L^{p_k}$ to $BMO$ of these operators.

On multilinear fractional strong maximal operator associated with rectangles and multiple weights

In this paper, the multilinear fractional strong maximal operator $\mathcal{M}_{\mathcal{R},\alpha}$ associated with rectangles and corresponding multiple weights $A_{(\vec{p},q),\mathcal{R}}$ are introduced. Under the dyadic reverse doubling condition, a necessary and sufficient condition for two-weight inequalities is given. As consequences, we first obtain a necessary and sufficient condition for one-weight inequalities. Then, we give a new proof for the weighted estimates of multilinear fractional maximal operator $\mathcal{M}_\alpha$ associated with cubes and multilinear fractional integral operator $\mathcal{I}_{\alpha}$, which is quite different and simple from the proof known before.

Two-weight entropy boundedness of multilinear fractional type operators

This paper will be devoted to study the two-weight norm inequalities of the multilinear fractional maximal operator $\mathcal{M}_{\alpha}$ and the multilinear fractional integral operator $\mathcal{I}_{\alpha}$. The entropy conditions in the multilinear setting will be introduced and the entropy bounds for $\mathcal{M}_\alpha$ and $\mathcal{I}_\alpha$ will be given.

Notes on endpoint estimates for multilinear fractional integral operators

We prove the strong type estimates for the multilinear fractional integral operators of Kenig and Stein type at critical indices. Our results are optimal.