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References in periodicals archive ?
Moreover, by [[integral].sub.[OMEGA]] [a.sup.-1/(p-1)](x)dx < c, using (29) and Lebesgue dominated convergence theorem,
Multiplying through by [mathematical expression not reproducible] and using the dominated convergence theorem we obtain the identity
Since the function f is integrable, we can change the order of integration by using Lebesgue's dominated convergence theorem. Hence
Similarly, using the continuity and the growth assumptions in (A2) and (A3) and the fact that [mathematical expression not reproducible] in E, P-a.s, once again it follows from Lebesgue dominated convergence theorem that
invoking once again the Lebesgue dominated convergence theorem, and passing to the limit as n [right arrow] +[infinity], we find that
Let u = x - t + y/2 and v = x - y/2 in the last term, by Lemma, Fubini's Theorem and the Lebesgue dominated convergence theorem,
In particular, from real analysis: the Cauehy-Sehwarz inequality and the Lebesgue dominated convergence theorem; from Fourier analysis: Plancherel's theorem and [L.sup.2] Fourier inversion; and the inequalities of Bernstein and Nikol'skii are used in the proof of Lemma 1.
Then [for all]j [member of] {1, 2}, by the dominated convergence theorem
Since [v.sub.t] [member of] [L.sup.2]([Q.sub.T]), by the Lebesgue dominated convergence theorem, we have
Here the first term decays exponentially, and since [([rho](x)/[rho]).sup.-N] [right arrow] 0 for all x [member of] (0,1), by Lebesgue's Dominated Convergence Theorem the second integral also converges to zero.
Now, by Morera's theorem with aids of Holder's inequality and the dominated convergence theorem, we have (30) for [lambda] [member of] [C.sub.+].
Using the properties (5.8) and the fact that z [member of] [B.sup.a.sub.[infinity]] (I, E) [subset] [L.sup.a.sub.2] (I,E), and Lebesgue dominated convergence theorem, one can easily verify that