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On a class of integral operators

Published online by Cambridge University Press:  24 October 2008

S. Verblunsky
S. Verblunsky
Affiliation:
Queen's University, Belfast
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Extract

Introduction. The integral operators we shall be concerned with can be described with the help of the following:

Definition. A function F(u, v) will be called a primitive, if it is defined on the square 0 ≥ u, v ≥ 1, and satisfies the conditions (i) F(u, v) = 0 if u ≥ v; (ii) F(u, v) is a.c. (absolutely continuous) in each variable; (iii) there is a function p ∈ L(0, 1) such that |F1(u, v)| ≤ p(u), |F2(u, v)| ≤ p(v)a.e. (almost everywhere) on the square, where F1(u, v), F2(u, v) denote the partial derivatives of F(u, v) with respect to u, v respectively. It is not difficult to see that conditions (ii), (iii) imply that F(u, v) is a.c. in each variable uniformly with respect to the other variable or, as we say, is equi-a.c. in each variable. Thus, in the above definition, condition (iii) can be replaced by (iii), there is a p ∈ L(0, 1) such that for each v ∈ [0, 1], |F1(u, v)| ≤ p(u) a.e. in [0, 1] and for each v ∈ [0, 1], |F2(u, v)| ≤ p(u)a.e. in [0, 1].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 1 , January 1967 , pp. 133 - 138
Copyright
Copyright © Cambridge Philosophical Society 1967

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References

REFERENCES

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