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Weighted Hardy spaces and BMO spaces associated with Schrödinger operators. (English) Zbl 1266.42060

Let $ℒ:=-{\Delta }+V$ be a Schrödinger operator on ${ℝ}^{d}$, $d\ge 3$, where $V¬\equiv 0$ is a fixed non-negative potential and belongs to the reverse Hölder class $R{H}_{s}\left({ℝ}^{d}\right)$ for some $s\ge \frac{d}{2}$; that is, there exists a positive constant $C:=C\left(s,V\right)$ such that ${\left(\frac{1}{|B|}{\int }_{B}{\left[V\left(x\right)\right]}^{s}\phantom{\rule{0.166667em}{0ex}}dx\right)}^{\frac{1}{s}}\le \frac{C}{|B|}{\int }_{B}V\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx$ for every ball $B\subset {ℝ}^{d}$. Moreover, the measure $V\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx$ satisfies the doubling condition: there exists a positive constant $C$ such that ${\int }_{B\left(y,2r\right)}V\left(x\right)dx\le C{\int }_{B\left(y,r\right)}V\left(x\right)dx$ for all $y\in {ℝ}^{d}$ and $r>0$. Let ${\left\{{T}_{t}\right\}}_{t>0}$ be the semigroup of linear operators generated by $ℒ$ and ${T}_{t}\left(x,y\right)$ be their kernels, that is, for all $x\in {ℝ}^{d}$,

${T}_{t}f\left(x\right):={e}^{-tℒ}f\left(x\right):={\int }_{{ℝ}^{d}}{T}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.166667em}{0ex}}dy\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}t>0\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}f\in {L}^{2}\left({ℝ}^{d}\right)·$

A weighted Hardy-type space related to $ℒ$ is defined by

${H}_{ℒ}^{1}\left(\omega \right):=\left\{f\in {L}^{1}\left(\omega \right):\phantom{\rule{4pt}{0ex}}{𝒯}^{*}f\left(x\right)\in {L}^{1}\left(\omega \right)\right\}{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\parallel f\parallel }_{{H}_{ℒ}^{1}\left(\omega \right)}:={\parallel {𝒯}^{*}f\parallel }_{{L}^{1}\left(\omega \right)},$

where ${𝒯}^{*}f\left(x\right):={sup}_{t>0}|{T}_{t}f\left(x\right)|$ for all $x\in {ℝ}^{d}$ and $\omega$ is a Muckenhoupt weight. A function $a$ is called an atom for the weighted Hardy space ${H}_{ℒ}^{1}\left(\omega \right)$ associated to a ball $B\left({x}_{0},r\right)$ for some ${x}_{0}\in {ℝ}^{d}$ and ${r}_{0}\in \left(0,\infty \right)$, if $\mathrm{supp}\phantom{\rule{0.166667em}{0ex}}a\subset B\left({x}_{0},r\right)$, ${\parallel a\parallel }_{{L}^{\infty }\left({ℝ}^{d}\right)}\le \frac{1}{\omega \left(B\left({x}_{0},r\right)\right)}$, if ${x}_{0}\in {ℬ}_{n}$, then $r\le {2}^{1-n/2}$ and, if ${x}_{0}\in {ℬ}_{n}$ and $r\le {2}^{-1-n/2}$, then ${\int }_{{ℝ}^{d}}a\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx=0$, where ${ℬ}_{n}:=\left\{x:\phantom{\rule{4pt}{0ex}}{2}^{-\left(n+1\right)/2}<\rho \left(x\right)\le {2}^{-n/2}\right\}$ for $n\in ℤ$ and $\rho \left(x\right):=\rho \left(x,V\right):=sup\left\{r>0:\phantom{\rule{4pt}{0ex}}\frac{1}{{r}^{d-2}}{\int }_{B\left(x,r\right)}V\left(y\right)\phantom{\rule{0.166667em}{0ex}}dy\le 1\right\}$. The atomic norm of $f\in {H}_{ℒ}^{1}\left(\omega \right)$ is defined by ${\parallel f\parallel }_{ℒ-at\left(\omega \right)}:=inf\left\{\sum |{c}_{j}|\right\}$, where the infimum is taken over all decompositions $f=\sum {c}_{j}{a}_{j}$, ${\left\{{c}_{j}\right\}}_{j}\subset ℂ$ and ${\left\{{a}_{j}\right\}}_{j}$ are ${H}_{ℒ}^{1}\left(\omega \right)$ atoms. The authors establish the following atomic characterization for ${H}_{ℒ}^{1}\left(\omega \right)$: Assume that $V\in R{H}_{d/2}\left({ℝ}^{d}\right)$ is a non-negative potential and $V¬\equiv 0$, then the norms ${\parallel f\parallel }_{{H}_{ℒ}^{1}\left(\omega \right)}$ and ${\parallel f\parallel }_{ℒ-at\left(\omega \right)}$ are equivalent. The authors also characterize ${H}_{ℒ}^{1}\left(\omega \right)$ by the Riesz transforms ${R}_{j}:=\frac{\partial }{\partial {x}_{j}}{ℒ}^{-1/2}$ for $j\in \left\{1,...,d\right\}$: if $V\in R{H}_{d}\left({ℝ}^{d}\right)$ is a non-negative potential, then there exists a positive constant $C$ such that

${C}^{-1}{\parallel f\parallel }_{{H}_{ℒ}^{1}\left(\omega \right)}\le {\parallel f\parallel }_{{L}^{1}\left(\omega \right)}+\sum _{j=1}^{d}\parallel {R}_{j}{f\parallel }_{{L}^{1}\left(\omega \right)}\le C{\parallel f\parallel }_{{H}_{ℒ}^{1}\left(\omega \right)}\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}f\in {H}_{ℒ}^{1}\left(\omega \right)·$

The authors also prove that the dual space of ${H}_{ℒ}^{1}\left(\omega \right)$ is ${\text{BMO}}_{ℒ}\left(\omega \right)$, where $f\in {\text{BMO}}_{ℒ}\left(\omega \right)$ means that $f\in \text{BMO}\left(\omega \right)$ and there exists a positive constant $C$ such that $\frac{1}{\omega \left(B\right)}{\int }_{B}|f\left(y\right)|\phantom{\rule{0.166667em}{0ex}}dy\le C$ for all $B:=B\left(x,R\right)$ with $x\in {ℝ}^{d}$ and $R>\rho \left(x\right)$. A positive measure $\mu$ on ${ℝ}_{+}^{d+1}:={ℝ}^{d}×\left(0,\infty \right)$ is called an $\omega -$Carleson measure if

${\parallel \mu \parallel }_{𝒞}:=\underset{x\in {ℝ}^{d},r>0}{sup}\frac{\mu \left(B\left(x,r\right)×\left(0,r\right)\right)}{\omega \left(\left(B\left(x,r\right)\right)}<\infty ·$

The authors prove that, if $f\in {\text{BMO}}_{ℒ}\left(\omega \right)$, $V¬\equiv 0$ is a non-negative potential in $R{H}_{s}\left({ℝ}^{d}\right)$ for some $s\ge \frac{d}{2}$ and $\omega$ a weight in ${A}_{1}$, then $d{\mu }_{f}\left(x,t\right):={|{Q}_{t}f\left(x\right)|}^{2}\frac{|B\left(x,r\right)|}{\omega \left(B\left(x,r\right)\right)}\phantom{\rule{0.166667em}{0ex}}\frac{dxdt}{t}$ is an $\omega -$Carleson measure, where

$\left({Q}_{t}f\right)\left(x\right):={t}^{2}\left({\frac{d{T}_{s}}{ds}|}_{s={t}^{2}}f\right)\left(x\right),\phantom{\rule{4pt}{0ex}}\left(x,t\right)\in {ℝ}_{+}^{d+1},$

and, conversely, if $f\in {L}^{1}\left(\left(1+{|x|\right)}^{-\left(d+1\right)}dx\right)$ and $d{\mu }_{f}\left(x,t\right)$ is a $\omega -$Carleson measure, then $f\in {\text{BMO}}_{ℒ}\left(\omega \right)$. Finally, the authors establish the boundedness of the Hardy-Littlewood maximal operator and the semigroup maximal operator on ${\text{BMO}}_{ℒ}\left(\omega \right)$.

MSC:
 42B35 Function spaces arising in harmonic analysis 42B30 ${H}^{p}$-spaces (Fourier analysis) 42B25 Maximal functions, Littlewood-Paley theory 42B20 Singular and oscillatory integrals, several variables 42B37 Harmonic analysis and PDE