AVERAGING WITH THE DIVISOR FUNCTION: l(p)-IMPROVING AND SPARSE BOUNDS

We study averages along the integers using the divisor function d(n), defined as KNf(x) = 1/D(N) Sigma(n <= N) d(n) f (x+n), where D(N) = Sigma(N) (n=1) d(n). We shall show that these averages satisfy a uniform, scale free lp-improving estimate for p is an element of (1, 2), that is (1/N Sigma vertical bar K(N)f broken vertical bar(p '))(1/p ') less than or similar to (1/N Sigma broken vertical bar f broken vertical bar(p))(1/p) as long as f is supported on [0, N]. We will also show that the associated maximal function K* f = sup(N) |K-N f | satisfies (p, p) sparse bounds for p is an element of(1, 2), which implies that K* is bounded on l(p) (w) for p is an element of (1,infinity), for all weights w in the Muckenhoupt A(p) class.