Dimension-free maximal inequalities for lattice power balls and exact spheres

We develop an approximation theorem for dimension-free maximal inequalities over lattice sublevel sets. Theorem X applies to general defining functions \(\Phi_d:\mathbb Z^d\to[0,\infty)\): maximal bounds for suitable approximating operators, together with summable remainders of arbitrary order, give bounds on every \(\ell^p\), \(1 3/2\). For functions of the form \(\Phi_d(y)=\sum_j\phi(y_j)\), Theorem A gives sufficient conditions for X and yields full-radius ball bounds for every fixed real \(1\le q<\infty\), with \(\phi(k)=|k|^q\). Theorem B transfers such ball estimates to exact lattice spheres using a spherical maximal estimate and representation-count bounds in one fixed dimension. It gives dimension-free sphere bounds for every finite integer \(q\ge2\) and every \(1 1\), the sphere maximal operator is unbounded in every fixed dimension on \(\ell^p\), \(1\le p<\infty\). Thus balls are bounded for all fixed \(1\le q\le\infty\) and \(1<p<\infty\), whereas the corresponding high-dimensional spherical bounds hold exactly for finite integers \(q\ge2\). All suprema include every radius with a nonempty lattice ball or sphere. At \(p=\infty\), both maximal operators are contractions for every \(q\).