# Questions tagged [integral-operators]

This tag is for questions relating to integral operators, which are an important special class of linear operators that act on function spaces.

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### Solving an Integral Equation Using Neumann Series in $C([0,1])$

Question: Let $U, V \subset \mathbb{R}^d$ be compact, $K \in C(U \times V)$ and $T_K: C(V) \rightarrow C(U)$ given by $$T_K(u)(x)=\int_V K(x, y) u(y) d y$$ Let $U=V=[0,1] \subset \mathbb{R}$. Show ...
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### How to Prove $T_K\left(B_1(0)\right)$ is Precompact in $C(U)$ for a Kernel Operator $K$ in $\mathbb{R}^d$

Question: Let $U, V \subset \mathbb{R}^d$ be compact, $K \in C(U \times V)$ and $T_K: C(V) \rightarrow C(U)$ given by $$T_K(u)(x)=\int_V K(x, y) u(y) d y .$$ Prove that $T\left(B_1(0)\right)$ is ...
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### Boundedness of integral operator induced by kernel $K(x,y) := \frac{1}{x+y}$.

Let $t_0 > 0$, $p \in (1,+\infty)$ and define $K:(0,t_0)\times (0,t_0) \rightarrow \mathbb{R}$ by $K(x,y) := \frac{1}{x+y}$. Is it true that for all $f \in L^p\bigl((0,t_0);\mathbb{R}\bigr)$ the ...
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### Integral operator and the kernel function

Let $D$ be an open bounded set in $\mathbb{R}^{n}$. Let $p \in [1,\infty)$, and $q$ is the dual exponent to $p$. Assume that $K : \mathbb{R} \times D \rightarrow \mathbb{R}$ is a bounded continuous ...
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### Bounding the trace-class norm of integral operator on $L^2(\mathbb{R})$

Consider an integral operator on $L^2(\mathbb{R})$ with kernel $K : \mathbb{R}^2 \to \mathbb{C}$. My question is if it is possible to bound the trace-class norm of $T$ based on properties of $K$? This ...