Geodesic
Homogeneous Wiener--Hopf Double Integral Equation with Symmetric Kernel in the Conservative Case
Matematičeskie zametki, Tome 106 (2019) no. 1, pp. 3-12

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We establish nontrivial solvability conditions for the homogeneous double integral equation $$ S(x,y)=\int^\infty_0 \int^\infty_0 K(x-x',y-y')S(x',y')\,dx'\,dy',\qquad (x,y)\in\mathbb R_+\times \mathbb R_+, $$ where $\mathbb R_+\equiv[0,+\infty)$, under the assumption that the given function $K$ satisfies the conservativity conditions $$ 0\le K\in L_1,\qquad \iint_{\mathbb R^2}K(x,y)\,dx\,dy=1 $$ and some additional conditions on its first and second moments.
Keywords: Wiener–Hopf double integral equation, conservativity conditions, factorization of the integral operator. 
@article{MZM_2019_106_1_a0,
     author = {L. G. Arabadzhyan},
     title = {Homogeneous {Wiener--Hopf} {Double} {Integral} {Equation} with {Symmetric} {Kernel} in the {Conservative} {Case}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {3--12},
     publisher = {mathdoc},
     volume = {106},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_106_1_a0/}
}
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L. G. Arabadzhyan. Homogeneous Wiener--Hopf Double Integral Equation with Symmetric Kernel in the Conservative Case. Matematičeskie zametki, Tome 106 (2019) no. 1, pp. 3-12. http://geodesic.mathdoc.fr/item/MZM_2019_106_1_a0/