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Authors
Koichi Taniguchi
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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.
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The purpose of this paper is to establish bilinear estimates in Besov spaces generated by the Dirichlet Laplacian on a domain of Euclidian spaces. These estimates are proved by using the gradient estimates for heat semigroup together with the Bony paraproduct formula and the boundedness of spectral multipliers.
We prove thatthe Banach space $(oplus_{n=1}^infty ell_p^n)_{ell_q}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever $1leq p leqinfty$ and $1<q<infty$. Furthermore, the Banach spaces $(oplus_{n=1}^infty ell_p^n)_{ell_1}$, with $1<ple infty$, and $(oplus_{n=1}^infty ell_p^n)_{c_0}$, with $1le p<infty$ do not have a greedy bases. We prove as well that the space $(oplus_{n=1}^infty ell_p^n)_{ell_q}$ has a 1-greedy basis if and only if $1leq p=qle infty$.
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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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