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

Question

The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds $$\bigg\|\sum_j a_j\chi_{\lambda B_j}\bigg\|_p\leq C(n,p,\lambda)\bigg\|\sum_j a_j\chi_{B_j}\bigg\|_p.$$

I saw in some papers that people call this Bojarski's lemma since it appeared in the paper of Bojarski: Bojarski, B. Remarks on Sobolev imbedding inequalities. Complex analysis, Joensuu 1987, 52–68, Lecture Notes in Math., 1351, Springer, Berlin, 1988.

However, I was informed by people that the above paper is in fact not the first reference on this lemma, at least another mathematcian proved this lemma in an unpublished notes (noticed by my supervisor as well).

As far as I know, in mathematics, we give name to some lemmas to express our respect on the mathematician who proved the corresponding results. But usually for young mathematicians, we are not aware of all the results we cited, in particular if some one add a name of some results and we just follow their name without going to the first reference for the result, it is easy to give a wrong title for some results. This sometimes causes servious problems for some mathematicians since they think the result should "belong to them".

The proof of the lemma was based on a maximal function argument and I do not know whether there are more elementary proofs than the one appeared in Bojarski's paper. If there were, then I would expect that there will be earlier references for this result. Then I will correct the name of this lemma in my paper.

Answer 1

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 $L^{p'}$.

Answer 2

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.