We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic function on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasiewicz [62] proved by Simon as [75, Theorem 3]. We prove that the optimal exponent of the Lojasiewicz-Simon gradient inequality is obtained when the function is Morse-Bott, improving on similar results due to Chill [17, Corollary 3.12], [18, Corollary 4], Haraux and Jendoubi [44, Theorem 2.1], and Simon [77, Lemma 3.13.1]. In [33], we apply our abstract gradient inequalities to prove Lojasiewicz-Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Those inequalities generalize those of Kwon [58, Theorem 4.2], Liu and Yang [60, Lemma 3.3], Simon [75, Theorem 3], [76, Equation (4.27)], and Topping [83, Lemma 1]. In [32], we prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. Those inequalities generalize that of the pure Yang-Mills energy function due to the first author [26, Theorems 23.1 and 23.17] for base manifolds of arbitrary dimension and due to Råde [69, Proposition 7.2] for dimensions two and three.
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