Global regularity and a priori estimation for elliptic operators on weighted Sobolev spaces
Let \(A\) be a differential operator of order \(m\geq 1\) in \({\mathbb R}^{n}\) defined by \[ A=\sum_{|\alpha |\leq m}a_{\alpha }(x)\partial ^{\alpha }, \] where the coefficients \(a_{\alpha }\) are from a weighted Sobolev space determined in the paper. The operator \(A\) is taken to be uniformly elliptic in the sense that there is a constant \(\Theta >0\) such that for all \(x\) and \( \xi \) in \({\mathbb R}^{n}\) \[ \bigl|\sum_{|\alpha |=m}a_{\alpha }(x)\xi ^{\alpha }\bigl|\geq \Theta |\xi |^{m}. \] Under weaker assumptions than in \textit{L. Nirenberg} and \textit{H. F. Walker} [J. Math. Anal. Appl. 42, 271-301 (1973; Zbl 0272.35029)] the following regularity result is given: if \(u\in W_{\text{loc}}^{p,l+m}({\mathbb R}^{n})\cap L_{a}^{p}({\mathbb R}^{n})\) and if \(Au\in W_{a+m}^{p,l}({\mathbb R}^{n})\) then \(u\in W_{a}^{p,l+m}({\mathbb R}^{n})\), where the previous weighted spaces are determined in the paper. Also, a corresponding a priori estimation is given: there is a constant \(C>0\) such that for each \(u\in W_{a}^{p,l+m}({\mathbb R}^{n}) \) we have \[ ||u||_{W_{a}^{p,l+m}({\mathbb R}^{n})}\leq C\{||Au||_{W_{a+m}^{p,l}({\mathbb R} ^{n})}+||u||_{L_{a}^{p}({\mathbb R}^{n})}\}. \] In order to prove the preceding results, a so-called ``scaling'' method is used. That method was exploited already in \textit{P. Bolley} and \textit{P. T. Lai} [J. Math. Pures Appl., IX. Sér. 72, No.1, 105-119 (1993; Zbl 0826.47008)] for the case of suitable weighted Hölder spaces.
scientific article; zbMATH DE number 1563230
scientific article