Limiting weak–type behavior for singular integral and maximal operators

Abstract

Abstract. The following limit result holds for the weak-type (1,1) constant of dilation-commuting singular integral operatorFor the maximal operator M , the corresponding result is

Optimal weak estimates for Riesz potentials

“…(3.5) is true for centered Hardy-Littlewood maximal function. On the other hand, using the limiting weak type behavior for maximum function in [7], (3.3) is also true for centered Hardy-Littlewood maximal function.…”

Limiting weak-type behaviors for singular integrals with rough $L\log L(\mathbb{S}^n)$ kernels

“…As we mentioned before [31,32], Ω ∈ L log L(S n−1 ) is a sufficient condition to guarantee the weak-type (1, 1) boundedness of T Ω . However, both the results of Janakiraman [23], Ding and Lai [11] were obtained under certain smoothness conditions. Therefore, one may ask, whether these smoothness conditions can be removed.…”

“…In 2004, Janakiraman [22] considered the Riesz transform and the singular integral operator T Ω . It was shown that the constant To explore the lower bounds of T Ω L 1 →L 1,∞ , in 2006, under the same kernel conditions, Janakiraman [23] gave the following limiting weak-type behavior for T Ω :…”

On the Bounds of Weak $(1,1)$ Norm of Hardy-Littlewood Maximal Operator with $L\log L({\mathbb S^{n-1}})$ Kernels

“…To explore the lower bounds of M L 1 →L 1,∞ , in 2006, Janakiraman [12] considered the following limiting weak-type behavior for the Hardy-Littlewood maximal operator M :…”