Assume that $g(x,y)$ and $h(x,y)$ are two positive functions such that $0<g<h$ and assume that $$T_g, T_h : L^2(B^n,R)\to L^2(B^n,R)$$ are integral operators defined by $$T_k[f](x)=\int_{B^n} f(x,y) k(y)dx.$$ Here $B^n$ is the unit ball and $R$ the real line. Can we state in general that $$\|T_g\|\le \|T_h\|$$ or $$\|T_g\|< \|T_h\|?$$
2026-03-25 03:03:16.1774407796
Does an inequality between kernels imply an inequality between the norms of integral operators?
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