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Boundedness of Generalized Riesz Transforms on Orlicz–Hardy Spaces Associated to Operators

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Abstract

Let \({\Phi}\) be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index \({p_\Phi^- \in(0,\,1]}\). Let L be an injective operator of type ω having a bounded H ∞ functional calculus and satisfying the k-Davies–Gaffney estimates with \({k \in {\mathbb Z}_+}\). In this paper, the authors first introduce an Orlicz–Hardy space \({H^{\Phi}_{L}(\mathbb{R}^n)}\) in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform \({D_{\gamma}L^{-\delta/(2k)}}\) is bounded from the Orlicz–Hardy space \({H^{\Phi}_{L}(\mathbb{R}^n)}\) to the Orlicz space \({L^{\widetilde{\Phi}}(\mathbb{R}^n)}\) when \({p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}\), \({0 < \gamma \le \delta < \infty}\) and \({\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}\), or from \({H^{\Phi}_{L}(\mathbb{R}^n)}\) to the Orlicz–Hardy space \({H^{\widetilde \Phi}(\mathbb{R}^n)}\) when \({p_\Phi^-\in (\frac{n}{n + \delta+ \lfloor \gamma \rfloor- \gamma},\,\frac{n}{n+ \delta- \gamma}]}\), \({1\le \gamma \le \delta < \infty}\) and \({\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}\), or from \({H^{\Phi}_{L}(\mathbb{R}^n)}\) to the weak Orlicz–Hardy space \({WH^\Phi(\mathbb{R}^n)}\) when \({\gamma = \delta}\) and \({p_\Phi=n/(n + \lfloor \gamma \rfloor)}\) or \({p_\Phi^-=n/(n + \lfloor \gamma \rfloor)}\) with \({p_\Phi^-}\) attainable, where \({\widetilde{\Phi}}\) is an Orlicz function whose inverse function \({\widetilde{\Phi}^{-1}}\) is defined by \({\widetilde{\Phi}^{-1}(t):=\Phi^{-1}(t)t^{\frac{1}{n}(\gamma- \delta)}}\) for all \({t \in (0,\,\infty)}\) , \({p_\Phi}\) denotes the strictly critical lower type index of \({\Phi}\), \({\lfloor \gamma \rfloor}\) the maximal integer not more than \({\gamma}\) and \({(p_-(L),\,p_+(L))}\) the range of exponents \({p \in[1,\, \infty]}\) for which the semigroup \({\{e^{-tL}\}_{t >0 }}\) is bounded on \({L^p(\mathbb{R}^n)}\).

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China

    Jun Cao, Dachun Yang & Sibei Yang

  2. Department of Mathematics, Department of Computer Science, Georgetown University, Washington, DC, 20057, USA

    Der-Chen Chang

  3. Department of Mathematics, Fu Jen Catholic University, Taipei, 242, Taiwan

    Der-Chen Chang

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  1. Jun Cao
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  2. Der-Chen Chang
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  3. Dachun Yang
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Correspondence to Dachun Yang.

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Jun Cao is supported by the Fundamental Research Funds for the Central Universities (Grant No. 2012YBXS16). Der-Chen Chang is partially supported by an NSF grant DMS-1203845 and Hong Kong RGC competitive earmarked research grant #601410. Dachun Yang is supported by the National Natural Science Foundation of China (Grant No. 11171027) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003).

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Cao, J., Chang, DC., Yang, D. et al. Boundedness of Generalized Riesz Transforms on Orlicz–Hardy Spaces Associated to Operators. Integr. Equ. Oper. Theory 76, 225–283 (2013). https://doi.org/10.1007/s00020-013-2058-0

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