Sharp Maximal Estimates Research Articles

Multilinear theory of strongly singular Calderón–Zygmund operators and applications

The main purpose of this paper is to study the multilinear strongly singular Calderón–Zygmund operator whose kernel is more singular near the diagonal than that of the standard multilinear Calderón–Zygmund operator.The sharp maximal estimate for this class of multilinear singular integrals is established, and as applications, its boundedness on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces is obtained, respectively. Moreover, the endpoint estimates of the types L∞×⋯×L∞→BMO, BMO×⋯×BMO→BMO, and LMO×⋯×LMO→LMO are established for the multilinear strongly singular Calderón–Zygmund operator.These results improve the corresponding known ones for the standard multilinear Calderón–Zygmund operator. Furthermore, we remove the size condition assumption for the kernel of the multilinear strongly singular Calderón–Zygmund operator which has been imposed in the literature for the case of the standard multilinear Calderón–Zygmund operator to get the sharp maximal estimates (see Remarks 1.1-1.4). Extra care is needed to deal with the mean oscillation over balls with small radius to overcome the stronger singularity in establishing such sharp maximal estimates and endpoint estimates.

Variation Inequalities for One-Sided Singular Integrals and Related Commutators

We establish one-sided weighted endpoint estimates for the ϱ -variation ( ϱ > 2 ) operators of one-sided singular integrals under certain priori assumption by applying one-sided Calderón–Zygmund argument. Using one-sided sharp maximal estimates, we further prove that the ϱ -variation operators of related commutators are bounded on one-sided weighted Lebesgue and Morrey spaces. In addition, we also show that these operators are bounded from one-sided weighted Morrey spaces to one-sided weighted Campanato spaces. As applications, we obtain some results for the λ -jump operators and the numbers of up-crossings. Our main results represent one-sided extensions of many previously known ones.

Sharp maximal estimates for multilinear commutators of multilinear strongly singular Calderón–Zygmund operators and applications

Abstract In this paper, we aim to establish the sharp maximal pointwise estimates for the multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions or Lipschitz functions, respectively. As applications, the boundedness of these multilinear commutators on product of weighted Lebesgue spaces are obtained. It is interesting to note that there is no size condition assumption for the kernel of the multilinear strongly singular Calderón–Zygmund operator. Due to the stronger singularity for the kernel of the multilinear strongly singular Calderón–Zygmund operator, we need to be more careful in estimating the mean oscillation over the small balls to get the sharp maximal function estimates.

Sharp maximal and weighted estimates for multilinear iterated commutators of multilinear integrals with generalized kernels

In this paper, the authors establish the sharp maximal estimates for the multilinear iterated commutators generated by BMO functions and multilinear singular integral operators with generalized kernels. As applications, the boundedness of this kind of multilinear iterated commutators on the product of weighted Lebesgue spaces and the product of variable exponent Lebesgue spaces can be obtained, respectively.

Multilinear singular integral operators with generalized kernels and their multilinear commutators

In this paper, the authors study a class of multilinear singular integral operators with generalized kernels and their multilinear commutators with BMO functions. By establishing the sharp maximal estimates, the boundedness on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces is obtained, respectively. Moreover, the endpoint estimate of this class of mutilinear singular integral operators is also established. These results can improve the corresponding known results of classical multilinear Calderon–Zygmund operators and multilinear Calderon–Zygmund operators with Dini type kernels.

MAXIMAL INEQUALITIES FOR FUNCTIONS OF BOUNDED LOWER OSCILLATION

We introduce a method which can be used to establish sharp maximal estimates for functions of bounded lower oscillation. The technique allows to deduce such estimates from the existence of certain special functions, and can be regarded as a version of Bellman function method, which has gained considerable interest in the recent literature. As an application, we establish a sharp exponential bound, which can be regarded as a version of integral John-Nirenberg inequality for BLO functions.