On the Differentiation of Integrals of Functions From Lφ(L)

The following alternative is established for any translation invariant differentiation basis B of rectangles with sides parallel to the coordinate axes: either B differentiates the integral of every summable function, or for every class Lφ(L) with φ(t) = o(lnt) as t→∞ there is a function whose integral is not differentiated by B. A geometric characteristic is introduced which permits to decide which class, L or Llog+ L, is precisely differentiated by a given basis. Also, a scale of non-translation invariant bases of rectangles with sides parallel to the axes is con­structed which differentiate precisely the classes Lφ(L) intermediate between L and Llog+ L. The results obtained, together with the theorems of Lebesgue and Jessen, Marcinkiewicz and Zygmund, yield a complete description of the behaviour of differentiation bases of rectangles with sides parallel to the axes. Applications to the theory of multiple Fourier series and extensions from R² to the multidimensional case are also given.