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above paper is in fact not the first reference on this lemma, at least another mathematcian proved this lemma in an unpublished notes (noticed by my supervisor as well).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

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    $\begingroup$ Unpublished notes is usually not a reason to credit the result. And it is not clear what exactly you are asking. Did your adviser tell you who was the author of those unpublished notes? $\endgroup$ – Alexandre Eremenko Dec 21 '13 at 21:27
  • $\begingroup$ To Alexandre Eremenko: Yes. We know the unpublished note is earier than the cited reference by Bojarski. But we do not know whether actually someone proves this fact even earlier. I think the name should give to the first person who has proved this. $\endgroup$ – Changyu Guo Dec 21 '13 at 22:30
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    $\begingroup$ And my opinion that the person who published first must be credited. $\endgroup$ – Alexandre Eremenko Dec 22 '13 at 0:09
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    $\begingroup$ Looking over Bojarski's paper that you have referenced, it seems the lemma was stated earlier (without proof) in: Boman, J. (1984). Lp-estimates for very strongly elliptic systems. Department of mathematics, University of Stockholm. $\endgroup$ – Benjamin Dickman Dec 22 '13 at 5:37
  • $\begingroup$ To Benjamin Dickman: thank you very much and I will check that paper. $\endgroup$ – Changyu Guo Dec 22 '13 at 6:49

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.

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    $\begingroup$ Note that Bojarski's inequality covers the case $p=1$. The proof of the lemma is actually very easy. It follows directly from the Hardy-Littlewood inequality and the duality argument. I will sketch it since the paper is difficult to find. $\endgroup$ – Piotr Hajlasz Jan 13 at 15:01
  • $\begingroup$ The $p=1$ case also follows from this argument (pick an $r$ greater than 1, and set $q=r$). In any case the $p=1$ case can also be proven by direct computation (Fubini theorem). $\endgroup$ – Terry Tao Jan 13 at 16:45

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 $L^{p'}$.


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