Sharp Estimates of m-Linear p-Adic Hardy and Hardy-Littlewood-Póya Operators

In recent years, p-adic numbers are widely used in theoretical and mathematical physics cf. 1–8 , such as string theory, statistical mechanics, turbulence theory, quantum mechanics, and so forth. For a prime number p, let Qp be the field of p-adic numbers. It is defined as the completion of the field of rational numbers Q with respect to the non-Archimedean p-adic norm | · |p. This norm is defined as follows: |0|p 0; If any nonzero rational number x is represented as x p m/n , where m and n are integers which are not divisible by p and γ is an integer, then |x|p p−γ . It is not difficult to show that the norm satisfies the following properties: