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Does anyone know a reference on the result in the picture? I believe there should be one, at least for (p=q=2). Here, $$\|f\|_{p,q}:= \left(\int_X\left(\int_Y|f(x,y)|^q d \mu_Y(y)\right)^{p/q} d \mu_X(x)\right)^{1/p}.$$ enter image description here

I wrote down my proof on here, but I want a reference in the literature.

C. Ding
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  • Exercise 7 on page 177 of Conway's Functional Analysis consider these kinds of functions in the context of compact operators for $1<p<\infty$. A recent posting in MSE (with reference to Zaanen's Linear operators) uses these types of functions in the same context, a similar posting appeared here. ($\frac1-+\frac1q=1$) – Mittens Jun 16 '24 at 19:06
  • Notice that if $|f|{p,q}<\infty$ the $\mathbb{1}{{|f|>a}}\leq\frac{1}{a}|f|$ and so $|\mathbb{1}{{|f|>a}}|{p,q}\leq\frac{1}{a}|f|{p, q}$ In particular, the simple functions $s_n=\sum{|k|\leq n2^{n}}k2^{-n}\mathbb{1}{k2^{-n}\leq |f|\leq (k+1)2^{-n}}$ converge to $f$ in $|;|_{p,q}$. – Mittens Jun 16 '24 at 19:22

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