Some new perspectives on Sard's theorem (Q2708164)

Let \(E\) be a finite dimensional vector space, \(F\) a normed space, \(k\geq 1\) an integer, \(\alpha\in [0,1]\) and \(f:E\to F\) a \(C^{k+\alpha}\) function (i.e., a \(k\)-times differentiable function such that \(D^kf\) satisfies a Hölder condition of exponent \(\alpha)\).NEWLINENEWLINENEWLINEThe main result of this paper is the following generalization of the classical Morse-Sard theorem: Under the above conditions, if \(t>0\) is a real number, \(r<t\) is an integer, \(A\subset E\) is a \(t\)-finite set such that \(\text{rank} (Df)(x)\leq r\), for all \(x\in A\) and \(d=r+(t-r)/(k+ \alpha)\), then:NEWLINENEWLINENEWLINE(i) if \(\alpha=0\), then \(f(A)\) is \(d\)-null: NEWLINENEWLINENEWLINE(ii) if \(\|f|_A\|_C k+\alpha <\infty\), then there is a constant \(C\) depending on \(\|f\|_{C^{k+\alpha}}, t,r,k,\alpha\) such that \({\mathcal H}^d (f(A))\leq C{\mathcal H}^t(A)\); NEWLINENEWLINENEWLINE(iii) if \(t\) is integer and \(A\) is \(t\)-rectifiable, then \(f(A)\) is \(d\)-nullNEWLINENEWLINENEWLINEFor differentiable mappings defined on infinite -- dimensional spaces the above results are true if \(k+\alpha\leq 3\).