Does anyone know what is the right reference for the following simple lemma from harmonic analysis?

This inequality is also a corollary of the main result of

Fefferman, Charles; Stein, Elias M., Some maximal inequalities, Am. J. Math. 93, 107-115 (1971). ZBL0222.26019.

which asserts that

$$ \| \sum_j |f_j^*|^r)^{1/r} \|_{L^q({\bf R}^n)} \leq C(n,q,r) \| (\sum_j |f_j|^r)^{1/r} \|_{L^q({\bf R}^n)}$$

whenever $1 < r,q < \infty$, where $f_j^*$ is the Hardy-Littlewood maximal function of $f_j$. Indeed, one takes an arbitrary $1 <r < \infty$ and applies this inequality with $q := pr$ and $f_j := a_j^{1/r} \chi_{B_j}$ (so that $f^*_j \gtrsim_{n,\lambda} a_j^{1/r} \chi_{\lambda B_j}$) to obtain the stated inequality of Bojarski.

The lemma is due to:

J. O. Strömberg, and A. Torchinsky. Weights, sharp maximal functions and Hardy spaces. Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 3, 1053–1056.

The lemma is stated there without proof, but the proof is in the paper by Boman:

J. Boman, $L^p$-estimates for very strongly elliptic systems, Report 29, Department of Mathematics, University of Stockholm, 1982.

Thanks to Dan Petersen the paper is available now: Famous but unavailable paper of Jan Boman.

Boman writes: I am indebted to Jan-Olof Strömberg for pointing out to me the usfulness of this lemma in this context and showing me the proof of the lemma that is given here.

The same proof is in the paper of Bojarski mentioned above. The lemma appears also as Lemma 4 on p. 115 in Weighted Hardy Spaces by Strömberg and Torchinsky, Lect Notes in Math. vol. 1381, 1989, but the proof given there seems quite different.

The proof is really easy, but tricky. There is no need to use the result of Fefferman and Stein. It goes as follows:

Let $\varphi\in L^{p'}$. Since $p'>1$ we can apply the Hardy-Littlewood maximal inequality to $\varphi$. $$ \left|\int_{\mathbb{R}^n}\sum_ia_i\chi_{\lambda B_i}\varphi\right|\leq \sum_ia_i|\lambda B_i|\left(\frac{1}{|\lambda B_i|}\int_{\lambda B_i}|\varphi|\right)\leq \lambda^n\sum_i a_i|B_i|C(n)\inf_{B_i}M\varphi $$ $$ \leq C(n)\lambda^n\sum_i \int_{B_i}a_iM\varphi= C(n)\lambda^n\int_{\mathbb{R}^n} \sum_ia_i\chi_{B_i} M\varphi $$ and the rest follows from the Holder inequality, the Hardy-Littlewood estimate for the maximal function and the duality between $L^p$ and $L^{p'}$.