Weighted inequalities for a general commutator associated to a singular integral operator satisfying a variant of Hörmander’s condition

In this paper, weighted inequalities for a certain general commutator associated to a singular integral operator satisfying a variant of Ho¨ rmander’s condition on Lebesgue spaces are obtained. To do this, some weighted sharp maximal function inequalities for the commutator are proved.     

Adams inequalities on measure spaces

In 1988 Adams obtained sharp Moser–Trudinger inequalities on bounded domains of Rn. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams' results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser–Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser–Trudinger trace inequalities, sharp Adams/Moser–Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.     

Sharpness of some properties of weighted modulation spaces

We obtain some optimal properties on weighted modulation spaces. We find the necessary and sufficient conditions for product inequalities, convolution inequalities and embedding on weighted modulation spaces. Especially, we establish the analogue of the sharp Sobolev embedding theorem on weighted modulation spaces.     

Reverse triangle inequalities for potentials

We study the reverse triangle inequalities for suprema of logarithmic potentials on compact sets of the plane. This research is motivated by the inequalities for products of supremum norms of polynomials. We find sharp additive constants in the inequalities for potentials, and give applications of our results to the generalized polynomials.We also obtain sharp inequalities for products of norms of the weighted polynomials , and for sums of potentials with external fields. An important part of our work in the weighted case is a Riesz decomposition for the weighted farthest-point distance function.     

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.     

Multidimensional weighted Opial inequalities

In this article we produce Opial-type weighted multidimensional inequalities over balls and arbitrary smooth bounded domains. The inequalities are sharp. The functions under consideration vanish on the boundary.     

Dimensional variance inequalities of Brascamp–Lieb type and a local approach to dimensional Prékopaʼs theorem

We give a new approach, inspired by Hörmander?s L²$L^2$-method, to weighted variance inequalities which extend results obtained by Bobkov and Ledoux. It provides in particular a local proof of the dimensional functional forms of the Brunn–Minkowski inequalities. We also present several applications of these variance inequalities, including reverse Hölder inequalities for convex functions, weighted Brascamp–Lieb inequalities and sharp weighted Poincaré inequalities for generalized Cauchy measures.     

Multiple weighted estimates for maximal vector-valued commutator of multilinear Calderón-Zygmund singular integrals

This paper concerns with multiple weighted norm inequalities for maximal vector-valued multilinear singular operator and maximal commutators. The Cotlar-type inequality of maximal vector-valued multilinear singular integrals operator is obtained. On the other hand, pointwise estimates for sharp maximal function of two kinds of maximal vector-valued multilinear singular integrals and maximal vector-valued commutators are also established. By the weighted estimates of a class of new variant maximal operator, Cotlar's inequality and the sharp maximal function estimates, multiple weighted strong estimates and weak estimates for maximal vector-valued singular integrals of multilinear operators and those for maximal vector-valued commutator of multilinear singular integrals are obtained.     

Multilinear commutators of BMO functions and multilinear singular integrals with non-smooth kernels

This paper is concerned with certain multilinear commutators of BMO functions and multilinear singular integral operators with non-smooth kernels. By the sharp maximal functions estimates, the weighted norm inequalities for this kind of commutators are established.     

Sharp constants in hardy type inequalities

We prove new weighted Hardy type inequalities with sharp constants and describe their applications to inequalities in multidimensional domains.     

两个严格的加权Ostrowski型不等式

使用Cauchy积分不等式和Grüss不等式的变式得到两个严格的加权Ostrowski型不等式.     

Weighted Norm Inequalities for Convolutions, Differential Operators, and Generalized Hypergeometric Functions

The recent work on discrete inequalities with the binomially made weights leads to a sharp weighted norm inequality for convolutions of complex-valued functions on a finite interval. This result and the Hadamard-type modifications of power series provide the basis for an integral approach to solution-kernel estimates for Volterra convolution equations and for various integro-differential and special function applications. The development of this approach and the new weighted norm inequalities for linear differential operators and generalized hypergeometric functions are presented in this paper.     

Some new and short proofs for a class of Caffarelli-Kohn-Nirenberg type inequalities

In this note we provide simple and short proofs for a class of inequalities of Caffarelli-Kohn-Nirenberg type with sharp constants. Our approach suggests some definitions of weighted Sobolev spaces and their embedding into weighted L² spaces. These may be useful in studying solvability of problems involving new singular PDEs.     

Monotonicity properties and functional inequalities for the Volterra and incomplete Volterra functions

In this paper we prove some monotonicity, log–convexity and log–concavity properties for the Volterra and incomplete Volterra functions. Moreover, as consequences of these results, we present some functional inequalities (like Turán type inequalities) as well as we determined sharp upper and lower bounds for the normalized incomplete Volterra functions in terms of weighted power means.     

L 1 Hardy Inequalities with Weights

We prove sharp homogeneous improvements to L ¹ weighted Hardy inequalities involving distance from the boundary. In the case of a smooth domain, we obtain lower and upper estimates for the best constant of the remainder term. These estimates are sharp in the sense that they coincide when the domain is a ball or an infinite strip. In the case of a ball, we also obtain further improvements.     

On Weighted Poincaré Inequalities

We are concerned with the problem of finding sharp summability conditions on the weights which render certain weighted inequalities of Poincaré - type true. The conditions we find consist of proper integral balances between the growths of the rearrangements of the weights.     

Sharp weighted bounds for fractional integral operators

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.     

Sharp weighted Hardy type inequalities and Hardy–Sobolev type inequalities on polarizable Carnot groups

In this Note, we establish sharp weighted Hardy type inequalities with a more general index p on polarizable Carnot groups, which include Kombe's recent results; then a weighted Hardy–Sobolev type inequality is obtained by using previous inequalities. To cite this article: J. Wang, P. Niu, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Weighted Dunkl transform inequalities and application on radial Besov spaces

In Dunkl theory on $\mathbb R ^d$ which generalizes classical Fourier analysis, we prove first weighted inequalities for certain Hardy-type averaging operators. In particular, we deduce for specific choices of the weights the $d$ -dimensional Hardy inequalities whose constants are sharp and independent of $d$ . Second, we use the weight characterization of the Hardy operator to prove weighted Dunkl transform inequalities. As consequence, we obtain Pitt’s inequality which gives an integrability theorem for this transform on radial Besov spaces.     

Weighted inequalities for multilinear commutators of some sublinear operators

We establish sharp estimates for some multilinear commutators related to the Littlewood-Paley and Marcinkiewicz operators. As an application, we obtain the weighted norm inequalities and L log L type estimate for the multilinear commutators.