Approximately differentiable transformations and change of variables on nilpotent groups
The paper studies some problems of analysis on nilpotent Lie groups of a special kind (on stratified homogeneous groups or, in other terminology, Carnot groups). Let \(G\) and \(\tilde G\) be Carnot groups, \(E\) be a measurable subset of \(G\), \(f: E\mapsto \tilde G\) be a mapping. Following \textit{P. Pansu} [Ann. Math. (2) 129, 1--60 (1989; Zbl 0678.53042)] the concepts of \(\mathcal P\)-differentiability and approximate \(\mathcal P\)-differentiability of \(f\) are introduced. In part one of the paper sufficient conditions are obtained under which the mapping \(f\) is \(\mathcal P\)-differentiable or approximate \(\mathcal P\)-differentiable almost everywhere. Part two is devoted to proving some change-of-variables formulas like \[ \int_A(u\circ f) |{\mathcal J}_f| \,dx= \int_Gu(y) N_f(y,A) \,dy \] for the mappings \(f: E\mapsto G\), where \(u: G\mapsto{\mathbb R}\) is an arbitrary measurable function, \(A\) is a measurable subset of \(E\), \(\mathcal J_f(x)\) is the Jacobian of \(f\), \(N_f(y,A)= \text{card}(f^{-1}(y)\cap A)\) is the multiplicity function.
change of variables for the Lebesgue integral
scientific article; zbMATH DE number 1023983
scientific article