In the univariate case, linear integral equations have the form (0): $$ f(x) = \lambda \phi(x) - \int_a^b K(x,y) \phi(y) dy $$ where $ a < x,y < b $ and $K:[a,b]\times[a,b] \to \mathbb R$ is the integral kernel

Could anyone indicate good references (accessible textbooks or papers) discussing the practical and theoretical issues presented by multivariate generalizations? For example (1): $$ f(\mathbf x) = \int_a^b K(\mathbf x,y) \phi(y) dy $$ with $x \in \mathcal D\subset \mathbb R^n,a<y<b $ and kernel $K:\mathcal D\times[a,b] \to \mathbb R$ (note that the second-kind equation would not make sense since $\phi$ is univariate)

or (2):

$$ f(\mathbf x) = \lambda \phi(\mathbf x) - \int_\mathcal D K(\mathbf x,\mathbf y) \phi(\mathbf y) d\mathbf y $$ with $\mathbf {x,y} \in \mathcal D $ and kernel $K:\mathcal D\times\mathcal D \to \mathbb R$

My personal understanding is that case (2) shares the same Fredholm-Riesz theory as the univariate case (0). I am not so sure about case (1)?


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