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Consider an integral operator $J$ with kernel $k(x,y)$ (assuming its properties are nice), can we describe the operator $J$ in $$Jf(x) := \int_0^1 k(x,y) f(y) dy$$ as an isomorphism(i.e. $J$ has a bounded inverse) from one Sobolev space $H^s$ to the other $H^{s+t}$? Having tried but I found few in literature. Examples and references are both welcome.

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