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Geometric approach to Sobolev spaces and badly degenerated elliptic equations. (English) Zbl 0877.46024
Kenmochi, N. (ed.) et al., Proceedings of the Banach Center minisemester on nonlinear analysis and applications, Warsaw, Poland, November 14 – December 15, 1994. Tokyo: Gakkōtosho Co., Ltd. GAKUTO Int. Ser., Math. Sci. Appl. 7, 141-168 (1995).
The author considers the geometric properties of the functions from the Sobolev space \[ W^{m,p}= \{u\in L^p(\Omega): D^\alpha u\in L^p(\Omega), |\alpha|\leq m\}, \] where \(\Omega\subset\mathbb{R}^n\) is an open set.
The structure of the paper is the following. In Section 1, the author considers “pointwise inequalities” for Sobolev functions with first-order derivatives. In Section 2, the author obtains relations of diffusions on fractals, to the analysis on infinite graphs, and to the infinite resistive networks, moreover this section contains a survey of the theory of strongly degenerated elliptic equations related to Hörmander’s vector fields.
In Section 3, the author considers applications of the Sobolev spaces on metric spaces to the theory of fractal type sets.
Finally in Section 4, the author investigates to extend the pointwise inequalities to the higher-order derivatives moreover, the author considers some applications of these higher-order inequalities.
For the entire collection see [Zbl 0853.00039].

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems