Shanzhen Lu

Singular integrals and related topics

Hardy-Littlewood Maxial Operator Singular Integral Operators Fractional Integral Operators Oscillatory Singular Integrals Littlewood-Paley Operator.

Singular integrals and commutators on homogeneous groups

Let G be a homogeneous group. In this paper, the authors establish several general theorems for the boundedness of sublinear operators and commutators generated by linear operators and BMO(G) functions on the weighted Lebesgue space on G. The conditions of these theorems are satisfied by many important operators in analysis and these operators satisfy only some weak conditions on the size of operators and are known to be bounded in the unweighted case.Some of these theorems are best possible even when G is the Euclidean space. The authors also give some applications of their theorems to the boundedness on weighted spaces of rough singular integrals, oscillatory integrals, parabolic singular integrals, their commutators and the maximal operators associated with them.AbstractПусть G — однородная группа. В работе устанавливается несколько обших теорем об ограниченности сублинейных операторов и коммутаторов, порожденных линейными операторами и функциями из BMO(G) на невесовом лебеговом пространстве над G. Условиям этих теорем удовлетворяют многие важные в анализе операторы. Для этих последних известно лишь, что они имеют слабый тип, при этом только в невесовом случае.Некоторые иэ полученных результатов являются неулучшаемыми даже в том случае, когда G — евклидово пространство. Даны некоторые приложения полученных теорем к исследованию ограниченности сильно сингулярных, осциллирующих и параболических осциллирующих интегралов, а также связанных с ними коммутаторов и максимальных операторов.

Marcinkiewicz integral on hardy spaces

In this paper we prove that the Marcinkiewicz integral μΩ is an operator of type (H1,L1) and of type (H1,∞,L1,∞). As a corollary of the results above, we obtain again the the weak type (1,1) boundedness of μΩ, but the smoothness condition assumed on Ω is weaker than Steins condition.

Herz-type Sobolev and Bessel potential spaces and their applications

The Herz-type Sobolev spaces are introduced and the Sobolev theorem is established. The Herz-type Bessel potential spaces and the relation between the Herz-type Sobolev spaces and Bessel potential spaces are discussed. As applications of these theories, some regularity results of nonlinear quantities appearing in the compensated compactness theory on Herz-type Hardy spaces are given.

Hardy-Littlewood-Sobolev theorems of fractional integration on Herz-type spaces and its applications

In this paper, the authors first establish the Hardy-Littlewood-Sobolev theorems of fractional integration on the Herz spaces and Herz-type Hardy spaces. Then the authors give some applications of these theorems to the Laplacian and wave equations.

Sharp bounds for Hardy type operators on higher-dimensional product spaces

A new class of Hardy type operator defined on a higher-dimensional product space is discussed. It includes two different kinds of the classical Hardy operators. In addition, we also consider the fractional Hardy operator Hβ. The bound of operator Hβ from Lp to Lq is explicitly worked out. Especially, the bound of operator Hβ from L1 to Lnn−β,∞ is sharp.MSC:26D10, 26D15, 42B99.

Endpoint estimates for n-dimensional Hardy operators and their commutators

In this paper, the sharp bound for the weak-type (1, 1) inequality for the n-dimensional Hardy operator is obtained. Moreover, the precise norms of generalized Hardy operators on the type of Campanato spaces are worked out. As applications, the corresponding norms of the Riemann-Liouville integral operator and the n-dimensional Hardy operator are deduced. It is also proved that the n-dimensional Hardy operator maps from the Hardy space into the Lebesgue space. The endpoint estimate for the commutator generated by the Hardy operator and the (central) BMO function is also discussed.

Inhomogeneous discrete calderón reproducing formulas for spaces of homogeneous type

In this article a Littlewood-Paley theorem for a new kind of Littlewood-Paley g-functions over spaces of homogeneous type is presented. Based on it the authors establish inhomogeneous discrete Calderón reproducing formulas for spaces of homogeneous type, making use of Calderón-Zygmund operators.

Boundedness of commutators on homogeneous Herz spaces

The boundedness on homogeneous Herz spaces is established for a large class of linear commutators generated by BMO(Rn) functions and linear operators of rough kernels which include the Calderón-Zygmund operators and the Ricci-Stein oRfiUatory singular integrals with rough kernels.

Boundedness of One-Sided Oscillatory Integral Operators on Weighted Lebesgue Spaces

We consider one-sided weight classes of Muckenhoupt type, but larger than the classical Muckenhoupt classes, and study the boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces using interpolation of operators with change of measures.