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SUMMARY:David Cruz-Uribe (University of Alabama)
DTSTART:20231213T060000Z
DTEND:20231213T070000Z
DTSTAMP:20260423T053015Z
UID:MAC8028/15
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MAC8028/15/"
 >Norm inequalities for linear and multilinear singular integrals on weight
 ed and variable exponent Hardy spaces</a>\nby David Cruz-Uribe (University
  of Alabama) as part of Trends in Mathematical Research\n\nLecture held in
  NTNU Gongguan S101.\n\nAbstract\nI will discuss a (relatively) new approa
 ch\nto norm inequalities in the weighted and variable exponent Hardy\nspac
 es.  The weighted Hardy spaces $H^p(w)$\, $p>0$\, where $w$ is a\nMuckenho
 upt weight\, were first considered by Stromberg and Torchinsky\nin the 198
 0s.  The variable Lebesgue space $L^{p(\\cdot)}$ is\,\nintuitively\, a cla
 ssical Lebesgue space with the constant exponent $p$\nreplaced by an expon
 ent function $p(\\cdot)$.  They have been studied\nextensively for the las
 t 30 years.  The corresponding variable Hardy\nspaces $H^{p(\\cdot)}$ were
  introduced by me and Li-An Wang and\nindependently by Nakai and Sawano.\n
 \nWe give inter-related conditions\non a Calderon-Zygmund singular integra
 l operator $T$\, a weight $w$\,\nand an exponent $p(\\cdot)$ for $T$ to sa
 tisfy estimates of the\nform$$ T : H^p(w) \\rightarrow L^p(w)\, \\qquad T:
  H^{p(\\cdot)}\n\\rightarrow L^{p(\\cdot)}.$$ Some of our results were kno
 wn for\nconvolution type singular integrals\, but we give new and simpler\
 nproofs and give extensions to non-convolution type operators.  Our\nproof
 s depend very heavily on three tools: atomic decompositions of\nthe Hardy 
 spaces\, vector-valued inequalities\, and the Rubio de Francia\ntheory of 
 extrapolation.\n\nWe will also discuss generalizations of these\nresults t
 o the bilinear setting\, where we prove norm inequalities of\nthe form\n$$
  T:  H^{p_1}(w_1) \\times H^{p_2}(w_2) \\rightarrow L^p(w)\, $$\nwhere $T$
  is a bilinear Calder\\'on-Zygmund singular integral operator\,\n$p\,\\\,p
 _1\,\\\,p_2>0$ and \n$$ \\frac{1}{p_1}+\\frac{1}{p_2} = \\frac{1}{p}\, $$\
 nand the weights $w\,\\\,w_1\,\\\,w_2$ are Muckenhoupt weights.\nWe also c
 onsider norm inequalities of the form\n$$ T:  H^{p_1(\\cdot)}\\times H^{p_
 2(\\cdot)}\\rightarrow L^{p(\\cdot)}. $$\n\nThis is joint work with Kabe M
 oen and Hanh Nguyen of the University of\nAlabama.\n
LOCATION:https://researchseminars.org/talk/MAC8028/15/
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