On the norm of the trace functions and applications (Q2866262)

Let \(p\) be a prime number, \(\mathbb Q_p\) the \(p\)-adic closure of the rationals, and \(\mathbb Z_p\) its ring of integers. Let \(\overline{\mathbb Q}_p\) be a fixed algebraic closure of \(\mathbb Q_p\), and let \(\mathbb C_p\) denote the completion of \(\overline{\mathbb Q}_p\) with respect to the extension of the \(p\)-adic absolute value. The pro-finite Galois group \(G\) of \(\overline{\mathbb Q}_p/\mathbb Q_p\) acts continuously on \(\overline{\mathbb Q}_p\), and actually \(G\) is canonically isomorphic to the group of continuous automorphisms of \(\mathbb C_p\) over \(\mathbb Q_p\). Let \(O(x)\) denote the orbit of an element \(x\in\mathbb C_p\) with respect to \(G\). Finally, let \(\mu\) be a strongly Lipschitz distribution defined on the orbit \(O(x)\). Then the author considers the function NEWLINE\[NEWLINEF_{s,\mu}(x)=\int_{O(x)}(z-t)^{-s}\, d\, \mu(t)NEWLINE\]NEWLINE (see \textit{M. Vâjâitu} and \textit{A. Zaharescu} [``Non-Archimedean Integration and Applications'', Timişoara: Romanian Academy (2007; Zbl 1300.11122)]. The main result of this article is the computation of the sup norm of \(F\) on \(E(x,\varepsilon)=\{y\in\mathbb C_p\cup\{\infty\}:| y-x|<\varepsilon\}\). In the last section, this result is applied to a class of transcendental functions.