Liguang Liu

The Multilinear Strong Maximal Function

A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear maximal functions associated with bases of open sets are studied too. Bilinear interpolation results between distributional estimates, such as those satisfied by the multivariable strong maximal function, are also proved.

Multiple-weighted norm inequalities for maximal multi-linear singular integrals with non-smooth kernels

We obtain weighted norm inequalities for maximal truncated operators of multi-linear singular integrals with non-smooth kernels in the sense of Duong et al . This class of operators extends the class of multi-linear Calderon-Zygmund operators introduced by Coifman and Meyer and includes the higher-order commutators of Calderon. The weighted norm inequalities obtained in this work are with respect to the new class of multiple weights of Lerner et al . The key ingredient in the proof is the introduction of a new multi-sublinear maximal operator that plays the role of the Hardy-Littlewood maximal function in a version of Cotlars inequality. As applications of these results, new weighted estimates for the m th order Calderon commutators and their maximal counterparts are deduced.

Besov-type and Triebel–Lizorkin-type spaces associated with heat kernels

Let

Gaussian BV -Capacity

(M, \rho ,\mu )

Pointwise multipliers for Campanato spaces on Gauss measure spaces

(M,ρ,μ) be a space of homogeneous type satisfying the reverse doubling condition and the non-collapsing condition. In this paper, the authors introduce Besov-type spaces

Gaussian p-Capacity

B_{p,q}^{s,\tau }(M)