[專題演講] 【12月13日】David Cruz-Uribe / Norm inequalities for linear and multilinear singular integrals on weighted and variable exponent Hardy spaces

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)

David Cruz-Uribe
University of Alabama

I will discuss a (relatively) new approach to norm inequalities in the weighted and variable exponent Hardy spaces. The weighted Hardy spaces Hp(w)H^p(w), p>0p>0, where ww is a Muckenhoupt weight, were first considered by Stromberg and Torchinsky in the 1980s. The variable Lebesgue space Lp()L^{p(\cdot)} is, intuitively, a classical Lebesgue space with the constant exponent pp replaced by an exponent function p()p(\cdot). They have been studied extensively for the last 30 years. The corresponding variable Hardy spaces Hp()H^{p(\cdot)} 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 TT, a weight ww, and an exponent p()p(\cdot) for TT to satisfy estimates of the formT:Hp(w)Lp(w),T:Hp()Lp(). T : H^p(w) \rightarrow L^p(w), \qquad T: H^{p(\cdot)} \rightarrow L^{p(\cdot)}.

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
T:Hp1(w1)×Hp2(w2)Lp(w), T: H^{p_1}(w_1) \times H^{p_2}(w_2) \rightarrow L^p(w),  
where TT is a bilinear Calder\’on-Zygmund singular integral operator, p,p1,p2>0p,\,p_1,\,p_2>0 and
1p1+1p2=1p, \frac{1}{p_1}+\frac{1}{p_2} = \frac{1}{p},  
and the weights w,w1,w2w,\,w_1,\,w_2 are Muckenhoupt weights. We also consider norm inequalities of the form
T:Hp1()×Hp2()Lp(). T: H^{p_1(\cdot)}\times H^{p_2(\cdot)}\rightarrow L^{p(\cdot)}.

This is joint work with Kabe Moen and Hanh Nguyen of the University of Alabama.

更多資訊
https://arxiv.org/abs/1902.01519

個人網頁 
https://math.ua.edu/people/david-cruz-uribe/

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