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I'm trying to find a rigorous formulation of an impression/intuitive notion related to Hardy spaces. It seems to me that functions in Hardy spaces tend to have modulus functions which do not oscillate. This apparently generalizes a property of the exponential function. In particular: $$\exp(ikz) = \cos(kz) + i\sin(kz)$$ with $\sin(kz)$ the harmonic conjugate of $\cos(kz)$, and $$|\exp(ikz)|^2 = \cos^2(kz) + \sin^2(kz) = 1.$$

For many examples of functions $F$ in Hardy spaces, one has $$F(x) = u(x) + i v(x)$$ with $u(x)^2 + v(x)^2$ very apparently non-oscillatory. For certain specific examples, I am able to prove something. Bessel functions are a good example, I think.

The spherical Hankel function $h_n(z)$ of the first kind admits the representation
$$h_n(z) = \exp(iz) C_n \int_0^\infty \exp(-zt) P_n(1+it) dt$$ with $C_n$ a constant depending on $n$ and $P_n$ the Legendre function of the first kind of order $n$. This can can be verified for integers $n$ by direct substitution of the (finite) series expansions for each of these functions. This holds for noninteger orders as well, but proving it is a bit harder. It follows from the equivalent formula
$$h_n(z) = C_n \int_1^\infty \exp(izt) P_n(t) dt$$ that $h_n$ is in a Hardy space of functions analytic on the upper half of the complex plane (not $H^p$ for any $p>0$, though, since $P_n(t)$ belows up at infinity).

It can be shown by direct expansion of the relevant series that
$$|h_n(x)|^2 = 1+ \int_0^\infty \exp(-xt) \frac{d}{dt} P_n(1+t^2/2) dt.$$
The derivative of $P_n(1+t^2/2)$ is positive on $(0,\infty)$, so $|h_n(x)|^2$ is completely monotone on $(0,\infty)$. An equivalent formula is $$|h_n(x)|^2 = z \int_0^\infty \exp(-xt) P_n(1+t^2/2) dt.$$

Obviously, it is too much to ask for $|F(z)|^2$ to be completely monotone for all $F$ in a Hardy space. But is there a general principal here? It is not enough for $F$ to be in a Hardy space because there are obvious examples where $|F|$ is highly oscillatory. For example, take $$F(z) = \int_0^\infty \exp(izt) \phi(t) dt$$ with $\phi(t)$ a smooth function which is equal to $1$ on $[1,100000]$. The function $|F(x)|^2$ will oscillate rapidly on $(0,\infty)$.

I have the strong impression that this must related to a well-known, standard result, but I don't know where to look for it.

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A function $f\ge 0$ on the circle is the absolute value $f=|F|$ of an $F\in H^p$, $F\not\equiv 0$, if and only if $f\in L^p$ and $\log f\in L^1$, and obviously you can make such functions as oscillatory as you desire them to be, so there's no general principle at work here.

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