ESTIMATES FOR THE NORM OF THE SPHERICAL MAXIMAL OPERATOR ON FINITE GRAPHS

For a simple, finite, and connected graph G, the spherical maximal operator is defined as M-G degrees h(t) = sup(r >= 0) 1/|S(t, r)| Sigma(u is an element of S(t,r)) |h(u)|, where S(t, r) = {w is an element of V | d(G) (w, t ) = r} is the sphere with center at t and having radius r. In this paper, we consider the spherical maximal operator M-G degrees on l(p) spaces and calculate the ||M-G degrees||(lp) for 0 < p <= 1 and estimate the ||M-G degrees||(lp) for 1 < p < infinity, when G is K-m. Furthermore, We establish the maximum and minimum bounds for the spherical maximum operator on finite graphs and indicate the graphs that achieve these bounds.