On the Divergence of the Two-Dimensional Dyadic Difference of Dyadic Integrals Original Research Article

In 1989 F. Schipp and W. R. Wade (Appl. Anal.34, 203–218) proved for functions in L(I2) log+ L(I2) (I2 is the unit square) that the dyadic difference of the dyadic integral dn(If) converges to f a.e. in the Pringsheim sense (that is, min(n1, n2)→∞, n=(n1, n2)∈P2). We prove that this result cannot be sharpened. Namely, we prove that for all measurable functions δ: [0, +∞)→[0, +∞), limt→∞ δ(t)=0 we have a function f∈L log+ Lδ(L) such as dn(If) does not converge to f a.e. (in the Pringsheim sense).