I have taken an introductory course into Harmonic analysis and in the course, we went over square functions
$s_\phi f(x) =\left(\int_0^\infty |f *\phi_t (x)|^2 \frac{dt}{t}\right)^{1/2}$
where $\phi \in \mathcal{S}$ is Schwartz with average zero: $\int \phi = 0$.
Essentially by Plancharel's theorem, we get $\| s_\phi(f)\|_{L^2}^2 \lesssim \| f\|_{L^2}^2$ and then one can show this also holds with regards to $L^p$ norm.
I am curious if this result can be generalized in the following way: instead of defining the square function with regards to an $L^2$ convolution integral, can we do it against an $L^p$ convolution integral instead? In other words, what happens if we look at
$$\left(\int_0^\infty |f * \phi_t(x)|^p \, \frac{dt}{t}\right)^{1/p}$$
for $p>1$? Can we establish a similar estimate on the "p-square" function?