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Register a new userWeighted embeddings for function spaces associated with Hermite expansions
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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.
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In this paper, we present explicit and computable error bounds for the asymptotic expansions of Hermite polynomials with Plancherel-Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory interval, or it is the turning point, are considered respectively. We introduce the branch cut technique to express the error term as an integral on the contour taking as the one-sided limit of curves approaching the branch cut. This new technique enables us to derive simple formulas for the error bounds in terms of elementary functions.
This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.
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.
We prove Carleson embeddings for Bergman-Orlicz spaces of the unit ball that extend the lower triangle estimates for the usual Bergman spaces.
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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