Two consequences of Davies's Hardy inequality
英文摘要:
In this short note we would like to show that one can use Davies's Hardy inequality to rederive well-known results of Lieb and Rozenblum.
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Rupert L. Frank, Simon Larson
In this short note we would like to show that one can use Davies's Hardy inequality to rederive well-known results of Lieb and Rozenblum.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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Let $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $L$ a one to one operator of type $ω$ in $L^2({\mathbb R}^n)$, with $ω\in[0,\,π/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space $W\!H_L^{p(\cdot)}(\mathbb R^n)$ associated with $L$ via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space $W\!T^{p(…
Let $L$ be a one-to-one operator of type $ω$ in $L^2(\mathbb{R}^n)$, with $ω\in[0,\,π/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space $H^{p(\cdot)}_L(\mathbb{R}^n)$ associated with $L$. By means of variable tent spaces, the authors establish the molecular characterization of $H^{p(\cdot)}_L(\mathbb{R}^n)$. Then the authors show that the dual space of $…
Hardy Spaces $H_L^p({\mathbb R}^n)$ Associated to Operators Satisfying $k$-Davies-Gaffney Estimates
设$L$是一个$ω$型的一对一算子,具有有界的$H_\infty$函数演算,并满足$k$ -Davies-Gaffney估计。利用$\{e^{-t^{2k}L}\}_{t>0}$定义的平方函数,引入了Hardy空间$H_L^p(\mathbb{R}^n)$,并建立了它们的分子和广义平方函数刻画.这种算子的典型例子包括$2k$ -阶发散形式的齐次椭圆算子$L_1$和$2k$ -阶薛定谔型算子$L_2\equiv (-Δ)^k+V^k$,其中$Δ$是拉普拉斯算子,而$0\le V\in L^k_{\mathop\mathrm{loc}}(\mathbb{R}^n)$。作为应用,对$i\in\{1,\,2\}$,证明了相应的Riesz变换$\nabla^k(L_i^{-1/2})$对$p\in(n/(n+k),\,1]$是从$H_{L_i}^p(\mathbb{R}^n)$到$H^p(\mathbb{R}^n)$有界的,并建立了$H_{L_1}^p(\mathbb{R}^n)$对$ p\in(rn/(n+kr),\,1]$的Riesz变换特征,如果$\{e^{-tL_1}\}_{t>0}$满足PDMATHABNHPDMATHABMP-非对角估计.当$k\equiv1$和$L\equiv L_1$已知时,这些结果。
Weak Hardy Spaces $WH_L^p({\mathbb R}^n)$ Associated to Operators Satisfying $k$-Davies-Gaffney Estimates
设$L$是一个$ω$型的一对一算子,具有有界的$H_\infty$函数演算,并满足$k$-Davies-Gaffney估计。本文通过非切平方函数$S_L$引入了与$L$相关联的弱Hardy空间$WH_L^p(\mathbb{R}^n)$,建立了$WH_L^p(\mathbb{R}^n)$的弱分子刻画.这种算子的典型例子包括$2k$-阶发散形式的齐次椭圆算子$L_1:=(-1)^k\sum_{|α|=k=|β|}\partial^β(a_{α,β}\partial^α)$,其中$\{a_{α,β}\}_{|α|=k=|β|}$是复有界可测函数,以及$2k$-阶薛定谔型算子$L_2:= (-Δ)^k+V^k$,其中$Δ$是拉普拉斯算子和$0\le V\in L^k_{\mathop\mathrm{loc}}(\mathbb{R}^n)$.作为应用,对$i\in\{1,\,2\}$和$p\in(\frac{n}{n+k},\,1]$,证明了相应的Riesz变换$\nabla^k (L_i^{-1/2})$从$WH^p_{L_i}(\mathbb{R}^n)$到经典弱Hardy空间$WH^p(\mathbb{R}^n)$是有界的,对所有的$0<p<r\le1$和$α=n(\frac{1}{p}-\frac{1}{r})$,分数幂$L_i^{-\fracα{2k}}$从$WH_{L_i}^p…
Orlicz-Hardy Spaces Associated with Operators Satisfying Davies-Gaffney Estimates
设${\mathcal X}$是具有加倍测度的度量空间,$L$是$L^2({\mathcal X})$中满足Davies-Gaffney估计的非负自伴算子,$ω$是$(0,\infty)$上的凹函数,且$p_ω\in (0, 1]$和$ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$对所有的$H_{ω,L}({\mathcal X})$都是严格下型的.建立了$H_{ω,L}({\mathcal X})$与$\mathrm{BMO}_{ρ,L}({\mathcal X})$之间的对偶关系,作为推论,得到了空间${\mathop\mathrm{BMO}_{ρ,L}(\mathcal X)}$的$ρ$-Carleson测度刻画.给出了$H_{ω,L}({\mathcal X})$的一些特征,包括原子和分子的特征以及与Poisson半群相关的Lusin面积函数的特征.设${\mathcal X}={\mathbb R}^n$和$ L=-Δ+V$为薛定谔算子,其中$V\in L^1_{\mathrm{\,loc\,}}({\mathbb R}^n)$是非负势。作为应用,证明了Riesz变换$\nabla L^{-1/2}$是从$H_{ω,L}({{\mathbb R}^n})$到$L(ω)$有界的;此外,如果存在$q_1,\,q_2\in (0,\infty)$使得$q_1<1<q…
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