Heat kernels, upper bounds and Hardy spaces associated to the generalized Schrödinger operators

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• 作者：Liangchuan Wu ^{wulchuan@mail2.sysu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Lixin Yan ; ^{mcsylx@mail.sysu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：42B15 ; 42B20 ; 47F05
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：15 May 2016
• 年：2016
• 卷：270
• 期：10
• 页码：3709-3749
• 全文大小：587 K

文摘

Let L=−Δ+μ$\mathcal{L}=-\mathrm{\Delta }+\mu$ be the generalized Schrödinger operator on Rⁿ${\mathbb{R}}^n$, n≥3$n\ge 3$, where μ≢0$\mu \not\equiv 0$ is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. Based on Shen's work for the fundamental solution of L$\mathcal{L}$ in [23], we establish the following upper bound for semigroup kernels ${\mathcal{K}}_t(x,y)$, associated to e^−tL$e^{-t\mathcal{L}}$, where h_t(x)=(4πt)^−n/2e^−|x|2/(4t)$h_t(x)={(4\pi t)}^{-n/2}{e}^{-|x{|}^2/(4t)}$, and d_μ(x,y,t)$d_{\mu }(x,y,t)$ is some parabolic type distance function associated with μ. As a consequence,where m(x,μ)$m(x,\mu )$ is some auxiliary function associated with μ  . We then study a Hardy space a0788802163cd293a8af073">$H_{\mathcal{L}}^1$ by means of a maximal function associated with the heat semigroup e^−tL$e^{-t\mathcal{L}}$ generated by −L$-\mathcal{L}$ to obtain its characterizations via atomic decomposition and Riesz transforms. Also the dual space BMO_L${\mathrm{BMO}}_{\mathcal{L}}$ of a0788802163cd293a8af073">$H_{\mathcal{L}}^1$ is studied in this paper.