On the Morse-Bott property of analytic functions on Banach spaces with Łojasiewicz exponent one half (Q1988501)

The famous Łojasiewicz gradient inequality says that if \(f:U\rightarrow\mathbb{K}\), \(0\in U\subset \mathbb{K}^{d}\) (\(\mathbb{K=R}\) or \(\mathbb{C}\)) is an analytic function and \(f(0)=0,\) \(f^{\prime }(0)=0\) then there exist constants \(C\in (0,+\infty )\) and \(\theta \in \lbrack 1/2,1)\) such that \[ \left\vert \left\vert f^{\prime }(x)\right\vert \right\vert \geq C\left\vert f(x)\right\vert ^{\theta } \] in a smaller neighbourhood of \(0.\) The main result is a characterization of those functions \(f\) for which the optimal (\(=\) infimum) \(\theta \) in the above inequality is equal to 1/2. Namely it holds if and only if \(f\) is a Morse-Bott function (which means \(\operatorname{Sing}f:=\{x:f^{\prime }(x)=0\}\) is a smooth, connected submanifold \(S\) of \(U\) such that for each \(x\in S\) the equality of linear subspaces of \(T_{x}U\) holds: \(T_{x}S=\ker f^{\prime\prime }(x)\), where \(f^{\prime \prime }(x)\) is treated as the linear mapping of the tangent space of \(U\) at \(x\) into the cotangent space at \(x\) via Hessian). The part ``if'' was proved by the author in [Geom. Topol. 23, No. 7, 3273--3313 (2019; Zbl 1439.32020)]. The results are presented in the context of Banach spaces.