By S. Prössdorf (auth.), V. G. Maz’ya, S. M. Nikol’skiĭ (eds.)

A linear vital equation is an equation of the shape XEX. (1) 2a(x)cp(x) - Ix k(x, y)cp(y)dv(y) = f(x), right here (X, v) is a degree area with a-finite degree v, 2 is a fancy parameter, and a, okay, f are given (complex-valued) features, that are often called the coefficient, the kernel, and the unfastened time period (or the right-hand aspect) of equation (1), respectively. the matter is composed in identifying the parameter 2 and the unknown functionality cp such that equation (1) is chuffed for the majority x E X (or even for all x E X if, for example, the critical is known within the experience of Riemann). within the case f = zero, the equation (1) is termed homogeneous, another way it's referred to as inhomogeneous. If a and okay are matrix services and, for this reason, cp and f are vector-valued services, then (1) is known as a procedure of necessary equations. fundamental equations of the shape (1) come up in reference to many boundary worth and eigenvalue difficulties of mathematical physics. 3 forms of linear critical equations are extraordinary: If 2 = zero, then (1) is named an equation of the 1st type; if 2a(x) i= zero for all x E X, then (1) is called an equation of the second one sort; and eventually, if a vanishes on a few subset of X yet 2 i= zero, then (1) is related to be of the 3rd kind.

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1. 1) Here (X, 11) is a measure space with a-finite measure 11 and k is a (fl x fl)measurable function on X x X, which is referred to as the kernel of the integral operator K. Dictated by the needs of several applications, the solution of the integral equation Kf = g is usually sought in the space of continuous functions or in the space Lp(X, fl). Therefore it is useful to have conditions ensuring that the given integral operator is bounded or even compact on these spaces. In this section we shall establish such boundedness and compactness criteria.

By applying Weyl's inequality (see Sec. 3 of Chap. 1), we get the following inequalities between the eigenvalues and singular values of operators K E Y'p,q: L nq/P-1IAn(KW ~ c N;,q(K) for 0 < p ~ 00,0 < q ~ 00, n for 0 < p ~ 00, q = 00, n where c = c(p, q) is some constant depending ,only on p and q, Hence, if K E Y'p,q then (An(K)) E lp,q' In case K belongs even to g;"q (0 < p < (0), we have IAn(K)1 = o(n- 1/ p). Thus, the problem of finding estimates for the eigenvalues and s-numbers has been reduced to establishing criteria for an operator to belong to a Triebel class.

By Fubini's theorem,