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.


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
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