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In this paper, we establish several different characterizations of the vanishing mean oscillation space associated with Neumann Laplacian $Delta_N$, written ${rm VMO}_{Delta_N}(mathbb{R}^n)$. We first describe it with the classical ${rm VMO}(mathbb{R}^n)$ and certain ${rm VMO}$ on the half-spaces. Then we demonstrate that ${rm VMO}_{Delta_N}(mathbb{R}^n)$ is actually ${rm BMO}_{Delta_N}(mathbb{R}^n)$-closure of the space of the smooth functions with compact supports. Beyond that, it can be characterized in terms of compact commutators of Riesz transforms and fractional integral operators associated to the Neumann Laplacian. Additionally, by means of the functional analysis, we obtain the duality between certain ${rm VMO}$ and the corresponding Hardy spaces on the half-spaces. Finally, we present an useful approximation for ${rm BMO}$ functions on the space of homogeneous type, which can be applied to our argument and otherwhere.
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The purpose of this paper is to give a definition and prove the fundamental properties of Besov spaces generated by the Neumann Laplacian. As a by-product of these results, the fractional Leibniz rule in these Besov spaces is obtained.
We study weighted Besov and Triebel--Lizorkin spaces associated with Hermite expansions and obtain (i) frame decompositions, and (ii) characterizations of continuous Sobolev-type embeddings. The weights we consider generalize the Muckhenhoupt weights.
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 $H^{p(cdot)}(mathbb{R}^n)$.
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 $p(cdot): mathbb R^nto(0,1]$ 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.
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