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Related: On a deceptively tricky calculus problem.

The way that Leonard Gross proves the log Sobolev inequality is in the following stages:

  1. He proves that for any operator $B$ that satisfies the log Sobolev inequality $\int f^2 \log \frac{f}{\lVert f\rVert_2}d\mu\leq \langle Bf,f\rangle$, the operator $e^{-tB}$ is hypercontractive.
  2. He then proves the log Sobolev inequality for carefully chosen operators $B_i$ on the discrete probability space $\{-1,1\}$ with $\{1/2,1/2\}$ as the corresponding probabilities.
  3. He then uses Segal's lemma [Lemma 1.4 from Segal - Construction of Non-Linear Local Quantum Processes: I] to show that if $e^{-tB_i}$ are hypercontractive, then $e^{-t(B_1+\dots B_n)}$ is also hypercontractive, and hence $(B_1+\dotsb+ B_n)$ satisfies the log Sobolev inequality. He then uses the central limit theorem to take the continuous limit and prove the log Sobolev inequality for the Gaussian measure.

I have operators $B'_i$ which satisfy the inequality $$\int f^2 \left(\log \frac{f}{\lVert f\rVert_2}\right)^2 d\mu\leq \langle B_i'f,f\rangle$$ on discrete probability spaces $\{-1,1\}$ with probabilities $\{1/2,1/2\}$. However in order to prove that $B'_1+\dotsb+ B'_n$ also satisfies the same inequality (and then use the central limit theorem to get the continuous limit), I was trying to prove that perhaps $e^{-tB_i'}$ are also hypercontractive perhaps with respect to an Orlicz norm. I would then use a variant of Segal's lemma. However, there are some obstructions to this.

  1. The kernel of the operator $e^{-tB'}$ is not positive, which is a necessary condition to use Segal's lemma. Note that $B'$ is not even a Markov generator (don't know if that is important).
  2. I have been unable to find any Orlicz norm with respect to which $e^{-tB'}$ is hypercontractive. Perhaps trying to prove hypercontractivity is the wrong path here?

Again, I am trying to prove that if $B_i'$ satisfy the above inequality, then $(B_1'+\dotsb+ B_n')$ also satisfies this inequality.

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    $\begingroup$ TeX notes: for norms, you want \lVert \rVert. Compare, for example, $||f||$ ||f|| to $\lVert f\rVert$ \lVert f\rVert. (\| \| is better but can fail, e.g., $\|-f\|$ \|-f\| vs. $\lVert-f\rVert$ \lVert-f\rVert.) Also, \dots can only work their magic with sufficient context; compare, e.g., $B_1' + \dots B_n'$ B_1' + \dots B_n' to $B_1' + \dots + B_n'$ B_1' + \dots + B_n'. You can explicitly tell the compiler you want "dots for binary operators" using \dotsb, as, for example, $B_1' + \dotsb B_n'$ B_1' + \dotsb B_n', if that is what you want. I edited both of these. $\endgroup$
    – LSpice
    Commented Jan 21 at 15:15
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    $\begingroup$ @LSpice- Thank you, appreciate the edits. $\endgroup$
    – matilda
    Commented Jan 21 at 16:50

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