Abstract
We develop a new method that enables us to solve the open problem of characterizing discrete inequalities for kernel operators involving suprema. More precisely, we establish necessary and sufficient conditions under which there exists a positive constant C such that (Sigma(n is an element of z)(Sigma(n)(t = -infinity) U(i, n)a(i))(q)w(n))(1/q) <= C(Sigma(n is an element of Z)a(n)(p)v(n))(1/p) holds for every sequence of nonnegative numbers where {a(n)}(nzZ) where U is a kernel satisfying certain regularity condition, 0 < p,q <= infinity and (u(n))(nzZ) and {w(n)}(nzZ) are fixed weight sequences.
We do the same for the inequality (Sigma(n is an element of z)w(n)(sup-infinity<i <= n U(i, n) Sigma(i)(j=-infinity) a(j)](q))(1/q) <= C(Sigma(n is an element of Z)a(n)(p)v(n))(1/p). We characterize these inequalities by conditions of both discrete and continuous nature.