Therefore, using the almost everywhere convergence
of both [nabla][u.sub.n] and [u.sub.n], and applying Fatou's lemma, we get
Although several properties and applications of orthogonal series with respect to a vector measure are known ([13, 14, 21]), the question of the almost everywhere convergence of series defined by such functions has not been studied yet.
In this section we study the almost everywhere convergence of functional series defined by (real valued) functions that are weak m-orthogonal for a vector measure m.
Note that the almost everywhere convergence with respect to a measure defined by a positive element x' do not provide m-almost everywhere convergence, since such measures are not in general Rybakov measures.
The requirements on L in Theorem 4.2 show that the problem of the almost everywhere convergence of weak m-orthogonal series is closely related to the calculus of estimates of 2-summing norms for the operators [[sigma].sup.N.sub.a,L] : L [right arrow] [l.sup.[infinity]] for suitable sequence spaces L.
Using Proposition 2.2, and since almost everywhere convergence holds for functions on S(R), which is a dense subset of [L.sup.p] (R, [d[mu].sub.[alpha]]), (see ), we obtain
Prestini, Almost everywhere convergence of the spherical partial sums for radial functions, Mh.
For the almost everywhere convergence of the sum in (2) observe that [l.sub.q] [??] [l.sub.p] if q [less than or equal to] p and so
The almost everywhere convergence is treated in the next section.
Almost everywhere convergence
: proof of Theorem 1.1.
For the passage to the limit, we prove the strong converge of the truncation of [u.sub.[epsilon]] and the almost everywhere convergence of [nabla][u.sub.[epsilon]].
Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear analysis, T.M.A., 19 (1992), n 6, pp.