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Register a new userVariable Hardy Spaces Associated with Operators Satisfying Davies-Gaffney Estimates
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Let $L$ be a one-to-one operator of type $omega$ in $L^2(mathbb{R}^n)$, with $omegain[0,,pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(cdot): mathbb{R}^nto(0,,1]$ be a variable exponent function satisfying the globally log-H{o}lder 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 $H^{p(cdot)}_L(mathbb{R}^n)$ is the BMO-type space ${rm BMO}_{p(cdot),,L^ast}(mathbb{R}^n)$, where $L^ast$ denotes the adjoint operator of $L$. In particular, when $L$ is the second order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of $H^{p(cdot)}_L(mathbb{R}^n)$ and show that the fractional integral $L^{-alpha}$ for $alphain(0,,frac12]$ is bounded from $H_L^{p(cdot)}(mathbb{R}^n)$ to $H_L^{q(cdot)}(mathbb{R}^n)$ with $frac1{p(cdot)}-frac1{q(cdot)}=frac{2alpha}{n}$ and the Riesz transform $ abla L^{-1/2}$ is bounded from $H^{p(cdot)}_L(mathbb{R}^n)$ to the variable Hardy space $H^{p(cdot)}(mathbb{R}^n)$.
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Let $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally log-Holder continuous condition and $L$ a one to one operator of type $omega$ in $L^2({mathbb R}^n)$, with $omegain[0,,pi/2)$, 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 $W!T^{p(cdot)}(mathbb R^n)$ which is also obtained in this article. In particular, when $L$ is non-negative and self-adjoint, the authors obtain the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$. As an application of the molecular characterization, when $L$ is the second-order divergence form elliptic operator with complex bounded measurable coefficient, the authors prove that the associated Riesz transform $ abla L^{-1/2}$ is bounded from $W!H_L^{p(cdot)}(mathbb R^n)$ to the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$. Moreover, when $L$ is non-negative and self-adjoint with the kernels of ${e^{-tL}}_{t>0}$ satisfying the Gauss upper bound estimates, the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$ is further used to characterize the space via non-tangential maximal functions.
Let $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally $log$-Holder continuous condition and $L$ a non-negative self-adjoint operator on $L^2(mathbb R^n)$ whose heat kernels satisfying the Gaussian upper bound estimates. Let $H_L^{p(cdot)}(mathbb R^n)$ be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels ${e^{-t^2L}}_{tin (0,infty)}$. In this article, the authors first establish the atomic characterization of $H_L^{p(cdot)}(mathbb R^n)$; using this, the authors then obtain its non-tangential maximal function characterization which, when $p(cdot)$ is a constant in $(0,1]$, coincides with a recent result by Song and Yan [Adv. Math. 287 (2016), 463-484] and further induces the radial maximal function characterization of $H_L^{p(cdot)}(mathbb R^n)$ under an additional assumption that the heat kernels of $L$ have the Holder regularity.
Let $L$ be a linear operator on $L^2(mathbb R^n)$ generating an analytic semigroup ${e^{-tL}}_{tge0}$ with kernels having pointwise upper bounds and $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors introduce the variable exponent Hardy space associated with the operator $L$, denoted by $H_L^{p(cdot)}(mathbb R^n)$, and the BMO-type space ${mathrm{BMO}}_{p(cdot),L}(mathbb R^n)$. By means of tent spaces with variable exponents, the authors then establish the molecular characterization of $H_L^{p(cdot)}(mathbb R^n)$ and a duality theorem between such a Hardy space and a BMO-type space. As applications, the authors study the boundedness of the fractional integral on these Hardy spaces and the coincidence between $H_L^{p(cdot)}(mathbb R^n)$ and the variable exponent Hardy spaces $H^{p(cdot)}(mathbb R^n)$.
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first introduce the variable weak Hardy space on $mathbb R^n$, $W!H^{p(cdot)}(mathbb R^n)$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $W!H^{p(cdot)}(mathbb R^n)$, respectively, by means of atoms, molecules, the Lusin area function, the Littlewood-Paley $g$-function or $g_{lambda}^ast$-function. As an application, the authors establish the boundedness of convolutional $delta$-type and non-convolutional $gamma$-order Calderon-Zygmund operators from $H^{p(cdot)}(mathbb R^n)$ to $W!H^{p(cdot)}(mathbb R^n)$ including the critical case $p_-={n}/{(n+delta)}$, where $p_-:=mathopmathrm{ess,inf}_{xin rn}p(x).$
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying that there exists a constant $p_0in(0,p_-)$, where $p_-:=mathop{mathrm {ess,inf}}_{xin mathbb R^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space $L^{p(cdot)/p_0}(mathbb R^n)$. In this article, via investigating relations between boundary valued of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $H^{p(cdot)}(mathbb R^n)$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A. D. Wang, the authors characterize $H^{p(cdot)}(mathbb R^n)$ via the first order Riesz transforms when $p_-in (frac{n-1}n,infty)$, and via compositions of all the first order Riesz transforms when $p_-in(0,frac{n-1}n)$.
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