×

Norm estimates and integral kernel estimates for a bounded operator in Sobolev spaces. (English) Zbl 1233.47031

Summary: We show that a bounded linear operator from the Sobolev space \(W^{-m}_{r}(\Omega)\) to \(W^{m}_{r}(\Omega)\) is a bounded operator from \(L_{p}(\Omega)\) to \(L_{q}(\Omega)\), and estimate the operator norm, if \(p,q,r\in [1,\infty]\) and a positive integer \(m\) satisfy certain conditions, where \(\Omega\) is a domain in \(\mathbf{R}^{n}\). We also deal with a bounded linear operator from \(W^{-m}_{p'}(\Omega)\) to \(W^{m}_{p}(\Omega)\) with \(p'=p/(p-1)\), which has a bounded and continuous integral kernel. The results for these operators are applied to strongly elliptic operators.

MSC:

47B38 Linear operators on function spaces (general)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J40 Boundary value problems for higher-order elliptic equations
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] K. Maruo and H. Tanabe, On the asymptotic distribution of eigenvalues of operators associated with strongly elliptic sesquilinear forms, Osaka J. Math. 8 (1971), 323-345. · Zbl 0227.35073
[2] Y. Miyazaki, The \(L^{p}\) theory of divergence form elliptic operators under the Dirichlet condition, J. Differential Equations 215 (2005), no. 2, 320-356. · Zbl 1081.47050 · doi:10.1016/j.jde.2004.10.014
[3] Y. Miyazaki, Higher order elliptic operators of divergence form in \(C^{1}\) or Lipschitz domains, J. Differential Equations 230 (2006), no. 1, 174-195. · Zbl 1143.35023 · doi:10.1016/j.jde.2006.07.024
[4] Y. Miyazaki, Heat asymptotics for Dirichlet elliptic operators with non-smooth coefficients, Asymptot. Anal. 72 (2011), 125-167. · Zbl 1247.47018
[5] T. Muramatu, On Besov spaces of functions defined in general regions, Publ. Res. Inst. Math. Sci. 6 (1970/71), 515-543. · Zbl 0225.46036 · doi:10.2977/prims/1195193919
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.