Restricted weak type on maximal linear and multilinear integral maps

It is shown that multilinear operators of the form of restricted weak type are always of weak type whenever the map is a locally integrable -valued function.

     

Weighted estimates for maximal multilinear commutators

Let T be a Calderón–Zygmund operator with regular kernel K and T * _b be the maximal multilinear commutator defined by . In this paper, the following weighted estimates for T * _b are discussed. Precisely, for 0 < p < ∞, ω ∈ A _∞ and b _j ∈ Osc equation/tex2gif-inf-5.gif, r_j ≥ 1, j = 1, … , m , there exists a positive constant C such that . For p = 1 and ω ∈ A ₁, the weighted weak L (log L )^1/r ‐type estimates are also established. Our theorems are parallel to the ones of the multilinear commutators of Calderón–Zygmund operators obtained in [18] and extend the main result in [14] essentially. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimates for the maximal multilinear singular integral operators

In this paper, some mapping properties are considered for the maximal multilinear singular integral operator whose kernel satisfies certain minimum regularity condition. It is proved that certain uniform local estimate for doubly truncated operators implies the Lp(Rn) (1 < p < ∞) boundedness and a weak type L log L estimate for the corresponding maximal operator.     

Pointwise estimates for the maximal multilinear singular integral operators

Two pointwise estimates relating the maximal multilinear singular integral operators and some classical maximal operators are established. These pointwise estimates imply the rearrangement estimate and the BLO(Rn) estimate for the maximal multilinear singular integral operators.     

A BMO estimate for the multilinear singular integral operator

The behavior on the space L∞((R)n) for the multilinear singular integral operator defined by TAf(x)=∫Rn Ω(x-y)/|x-y|n 1(A(x)-A(y)-(△)A(y)(x-y))f(y)dy is considered, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, A has derivatives of order one in BMO((R)n). It is proved that if Ω satisfies some minimum size condition and an L1-Dini type regularity condition, then for f ∈ L∞((R)n), TAf is either infinite almost everywhere or finite almost everywhere, and in the latter case, TAf ∈ BMO((R)n).     

Estimates for maximal multilinear commutators on non-homogeneous spaces

Under the assumption that μ is a non-doubling measure on Rd, the authors obtain some estimates for the maximal multilinear commutators generated by Calderón-Zygmund operator T and RBMO(μ) functions associated to this measures.     

Sharp maximal function estimate and weighted inequalities for maximal multilinear singular integrals

For maximal multilinear Calderón-Zygmund singular integral operators, the sharp maximal function estimate and some weighted norm inequalities are obtained.     

Lipschitz estimates for multilinear commutator of Littlewood-Paley operator

In this paper, we will study the continuity of multilinear commutator generated by Littlewood-Paley operator and Lipschitz functions on Triebel-Lizorkin space, Hardy space and Herz-Hardy space.      

An endpoint estimate for maximal multilinear singular integral operators

A weak type endpoint estimate for the maximal multilinear singular integral operator T*Af(x)=supε＞0|(f)(x-y)＞ε (Ω(x-y)/(|x-y|(n 1)))(A(x)-A(y)-▽A(y)(x-y))f(y)dy| is established, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, and A has derivatives of order one in BMO(Rn). A regularity condition on Ω which implies an LlogL type estimate of T*A is given.     

Multiple weighted estimates for commutators of multilinear maximal function

Let M be the multilinear maximal function and b =(b1,..., bm) be a collection of locally integrable functions. Denote by M b and [ b, M] the maximal commutator and the commutator of M with b, respectively. In this paper, the multiple weighted strong and weak type estimates for operators M b and [ b, M] are studied. Some characterizations of the class of functions bj are given, for which these operators satisfy some strong or weak type estimates.     

On a multilinear singular integral operator

In this paper, we prove that the maximal operatorsatisfiesis homogeneous of degree 0, has vanishing moment up to order M and satisfies Lq-Dini condition for some     

Sharp weighted inequality for a multilinear commutator of the Marcinkiewicz operator

A sharp inequality for a multilinear commutator related to the Marcinkiewicz operator is proved. As a consequence, weighted L^p-norm inequality for the multilinear commutator for 1 < p < ∞ is obtained. Bibliography: 12 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 86–97.     

Weighted inequalities for the maximal geometric mean operator

For nonnegative Borel measures on and for the maximal geometric mean operator , we characterize the weight pairs for which is of weak type and of strong type , . No doubling conditions are needed. We also note that a previously published different characterization for the strong type inequality for has an incorrect proof.

     

A multilinear operator for almost product evaluation of Hankel determinants

In a recent paper we have presented a method to evaluate certain Hankel determinants as almost products; i.e. as a sum of a small number of products. The technique to find the explicit form of the almost product relies on differential-convolution equations and trace calculations. In the trace calculations a number of intermediate nonlinear terms involving determinants occur, but only to cancel out in the end.In this paper, we introduce a class of multilinear operators γ acting on tuples of matrices as an alternative to the trace method. These operators do not produce extraneous nonlinear terms, and can be combined easily with differentiation.The paper is self contained. An example of an almost product evaluation using γ-operators is worked out in detail and tables of the γ-operator values on various forms of matrices are provided. We also present an explicit evaluation of a new class of Hankel determinants and conjectures.     

Sharp weighted inequalities for multilinear fractional maximal operators and fractional integrals

In this paper, we study weighted inequalities for multilinear fractional maximal operators and fractional integrals. We prove sharp weighted Lebesgue space estimates for both operators when the vector of weights belongs to . In addition we prove sharp two weight mixed estimates for multilinear operators in the spirit of the linear estimates given in 3 .     

Sharp maximal function estimates for multilinear singular integrals with non-smooth kernels

In this article we obtain weighted norm estimates for multilinear singular integrals with non-smooth kernels and the boundedness of certain multilinear commutators by making use of a sharp maximal function.     

On the centered Hardy-Littlewood maximal operator

We will study the centered Hardy-Littlewood maximal operator acting on positive linear combinations of Dirac deltas. We will use this to obtain improvements in both the lower and upper bounds or the best constant in the weak inequality for this operator. In fact we will show that .

     

Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators

In this article we prove weighted norm inequalities and pointwise estimates between the multilinear fractional integral operator and the multilinear fractional maximal. As a consequence of these estimations we obtain weighted weak and strong inequalities for the multilinear fractional maximal operator or function. In particular, we extend some results given in Carro et al. (2005) [7] to the multilinear context. On the other hand we prove weighted pointwise estimates between the multilinear fractional maximal operator Mα,B associated to a Young function B and the multilinear maximal operators Mψ=M0,ψ, ψ(t)=B(t1−α/(nm))^nm/(nm−α). As an application of these estimate we obtain a direct proof of the Lp−Lq boundedness results of Mα,B for the case B(t)=t and Bk(t)=tk(1+log⁺t) when 1/q=1/p−α/n. We also give sufficient conditions on the weights involved in the boundedness results of Mα,B that generalizes those given in Moen (2009) [22] for B(t)=t. Finally, we prove some boundedness results in Banach function spaces for a generalized version of the multilinear fractional maximal operator.     

New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory

A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller than the m-fold product of the Hardy-Littlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of Calderón-Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calderón-Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.