# Is this some kind of adjoint?

I have a bounded operator $T$ from $L^p$ to itself for $1 \leqslant p \leqslant \infty$. Furthermore, on $L^2$ we have that $T$ is self-adjoint.

Now I wish to relate $\|(Tf)g\|_{L^1}$ to $\|f(Tg)\|_{L^1}$ (equal up to a constant perhaps). What properties should I need for $T$ for this to hold?

The question is not really well-defined, but I don't know what property I should look for in my operator.

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Some thoughts: since you know $T$ is bounded on every $L^p$, I think it's enough to consider $f$ and $g$ continuous with compact support (the general case whould then follow by approximation, perhaps with the case $f\in L^1$, $g\in L^\infty$ needing separate treatment). Secondly, is your operator $T$ given formally by convolution with some kind of kernel function? –  user16299 Feb 8 '12 at 22:15