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I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim:

Pick an arbitrary real number $s$, we have that the intersection of $\cap_{t \in \mathbb{R}}H^{t}$ is dense in $H^{s}$.

The author mentioned that it suffices to prove the following equivalence of norms:

$$c^{-1}\|f-P_{\leq k}f\|_{H^{s}}^2 \leq \sum_{l \geq k}2^{2ls}\|P_{l}f\|_{L^2}^{2} \leq c\|f-P_{\leq k}f\|_{H^{s}}^2 \quad (\forall \ f \in H^s)$$

where $c = c(s) > 0$ above is some constant dependent on $s$ and $P_{\leq k} = \sum_{j \leq k}P_{j}$ is the sum of a subset of Littlewood-Paley projection operators.

I can see how the dense property can be derived from the equivalence of norms exhibited above. However, I have tried using several properties of the Littlewood-Paley projection operators, but none of them helps me gain any progress in showing the inequality above...any hint/idea?

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    $\begingroup$ What kind of domain are we using here? Is it not enough in context to note that $C^\infty_c$ is contained in all of them? $\endgroup$
    – Nate Eldredge
    Commented May 17, 2021 at 0:44
  • $\begingroup$ Isn't this almost by definition? Where exactly are you stuck? You have $f - P_{\leq k f} = P_{>k} f + (P_k + P_{k-1})(f - P_{\leq k} f)$. Then you just use almost orthogonality. $\endgroup$
    – Willie Wong
    Commented May 17, 2021 at 3:24
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    $\begingroup$ For what it's worth, this is special case of a simple general result (simple in that it follows easily from the spectral theorem). Let $A$ be positive s.a. operator (we can assume that it is bounded below by the identity operator). Then it defines in a natural way a scale $H^t$ of Sobolev spaces (for $t>0$, $H^t$ is the domain of definition of $A^t$). Then $\bigcap H^t$ is dense in each component. The classical Sobolev spaces are generated in this way by the Laplace operator on $n$-space and various s.a. differential operators (Laplace-Beltrami, Schrödinger) of physics supply variants. $\endgroup$
    – bathalf15320
    Commented May 17, 2021 at 11:40
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    $\begingroup$ Does this answer your question? Littlewood-Paley theory and Dense property of Sobolev spaces $\endgroup$
    – Willie Wong
    Commented May 21, 2021 at 15:14
  • $\begingroup$ Yeah it does! Thank you so much! $\endgroup$
    – geooranalysis
    Commented Jun 1, 2021 at 1:44

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