Theoretical analysis proves dimension-free maximal bounds for lattice balls and spheres across Lp spaces, establishing exact geometric constraints on high-dimensional discrete operators.
Key Points
Establish dimension-free maximal inequalities on discrete Lp spaces for averages taken over lattice power balls and exact spheres in high dimensions.
Formulated an approximation theorem connecting maximal bounds of approximating operators and summable remainders to general lattice sublevel sets.
Evaluated coordinate-separable functions across real power exponents to derive full-radius lattice ball bounds.
Transferred ball estimates to exact spheres using single-dimension spherical maximal estimates combined with integer representation counts.
Established dimension-free maximal bounds on Lp for full-radius lattice power balls across all real exponents with q between 1 and infinity.
Showed dimension-free bounds on exact lattice spheres hold strictly for integer exponents q at least 2, whereas for non-integer powers the spherical operator is unbounded in fixed dimensions.
Demonstrated that both ball and sphere maximal operators are non-expansive contractions on infinity-norm spaces for every exponent q.