Norm inequalities for linear and multilinear singular integrals on weighted and variable exponent Hardy spaces
時 間:2023-12-13 14:00 (星期三) / 地 點:S101 / 茶 會:S205 (13:30)
I will discuss a (relatively) new approach to norm inequalities in the weighted and variable exponent Hardy spaces. The weighted Hardy spaces , , where is a Muckenhoupt weight, were first considered by Stromberg and Torchinsky in the 1980s. The variable Lebesgue space is, intuitively, a classical Lebesgue space with the constant exponent replaced by an exponent function . They have been studied extensively for the last 30 years. The corresponding variable Hardy spaces were introduced by me and Li-An Wang and independently by Nakai and Sawano.
We give inter-related conditions on a Calderon-Zygmund singular integral operator , a weight , and an exponent for to satisfy estimates of the form
Some of our results were known for convolution type singular integrals, but we give new and simpler proofs and give extensions to non-convolution type operators. Our proofs depend very heavily on three tools: atomic decompositions of the Hardy spaces, vector-valued inequalities, and the Rubio de Francia theory of extrapolation.
We will also discuss generalizations of these results to the bilinear setting, where we prove norm inequalities of the form
where is a bilinear Calder\’on-Zygmund singular integral operator, and
and the weights are Muckenhoupt weights. We also consider norm inequalities of the form
This is joint work with Kabe Moen and Hanh Nguyen of the University of Alabama.