Let $1<p<\infty$. Suppose $a_i\geq 0$, and $\{B_{r_i}(x_i)\}$ is a sequence of open balls in $\mathbb{R}^n$ centered at $x_i$ with radius $r_i$.

Let $g\in L^q(\mathbb{R}^n)$ where $\frac 1p+\frac 1q=1$ and define the maximal function $$g^*(y)=\sup\{\frac{1}{|B|}\int_B|g|\,dx:B\ \text{is any open ball containing}\ y\}.$$

For convenience, denote $B_i=B_{r_i}(x_i)$ and $3B_i=B_{3r_i}(x_i)$.

Question 1) Prove that there exists $C_0>0$ such that $$\int_{\mathbb{R}^n}\sum_{i}a_i\chi_{3B_i}(x)|g(x)|\,dx\leq\int_{\mathbb{R}^n}C_0\sum_ia_i\chi_{B_i}(x)g^*(x)\,dx.$$

Question 2) Hence prove that there exists $C>0$ (independent of $a_i$) such that $$\|\sum_{i}a_i\chi_{3B_i}\|_{L^p}\leq C\|\sum_ia_i\chi_{B_i}\|L^p$$

Thanks for any help.

My attempt:

Upon seeing the "$3B_i$", I would think that Vitali Covering Lemma would be useful, since that is the only other place I have seen "$3B_i"$.

I tried proving a simplified version, where the index set only has one element, i.e. $\{a_i\}=\{a_1\}$ and $\{B_{r_i}(x_i)\}=\{B_{r_1}(x_1)\}$. However, it didn't quite work out:

\begin{align*} LHS&=\int_{\mathbb{R}^n}a_1\chi_{3B_1}(x)|g(x)|\,dx\\ &=\int_{3B_i}a_1|g(x)|\,dx\\ &=3^n|B_1|\frac{1}{|3B_1|}\int_{3B_1}a_1|g(x)|\,dx\\ &\leq 3^n|B_1|a_1 g^*(x) \end{align*}

Something is not quite right at the end...

Some other miscellaneous thoughts: Holder's inequality should be useful, since there is $\frac 1p+\frac 1q=1$.

  • 1
    $\begingroup$ I might be wrong but you established $LHS\le3^{n}\left\lvert B_{r_{1}}\left(x_{1}\right)\right\rvert a_{1}\left\{\frac{1}{\left\lvert 3B_{r_{1}}\left(x_{1}\right)\right\rvert}\int_{3B_{r_{1}}(x_{1})}\left\lvert g(x)\right\rvert dx\right\}$ and so for all $y\in B_{r_{1}}\left(x_{1}\right)$ the previous shows that $LHS\le3^{n}\left\lvert B_{r_{1}}\left(x_{1}\right)\right\rvert a_{1}g^{*}(y)$. This implies that: $\endgroup$ – user71352 Dec 14 '16 at 0:36
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    $\begingroup$ $LHS\le3^{n}\left\lvert B_{r_{1}}\left(x_{1}\right)\right\rvert a_{1}\inf_{y\in B_{r_{1}}\left(x_{1}\right)} g^{*}(y)=3^{n}a_{1}\int_{B_{r_{1}}\left(x_{1}\right)}\left(\inf_{y\in B_{r_{1}}\left(x_{1}\right)} g^{*}(y) \right)dz\le3^{n}a_{1}\int_{B_{r_{1}}\left(x_{1}\right)}g^{*}(z)dz$ $\endgroup$ – user71352 Dec 14 '16 at 0:36
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    $\begingroup$ For Question 2 I think you need that for $1<p<\infty$ that we have an inequality of the following form $\left\lvert\left\lvert f^{*}\right\rvert\right\rvert_{L^{p}\left(\mathbb{R}^{n}\right)}\le C\left\lvert\left\lvert f\right\rvert\right\rvert_{L^{p}\left(\mathbb{R}^{n}\right)}$ as well as the fact that for $1<p<\infty$ we have $\left\lvert\left\lvert f\right\rvert\right\rvert_{L^{p}\left(\mathbb{R}^{n}\right)}=\sup_{\left\lvert\left\lvert g\right\rvert\right\rvert_{L^{q}}\le1}\left\{\left\lvert\int_{\mathbb{R}^{n}}f(x)g(x)dx\right\rvert\right\}$ where $\frac{1}{p}+\frac{1}{q}=1$. $\endgroup$ – user71352 Dec 14 '16 at 1:39
  • $\begingroup$ @user71352 Thanks. Looks very promising indeed. Hardy-Littlewood Lemma and converse of Holder's inequality. $\endgroup$ – yoyostein Dec 14 '16 at 1:45
  • 1
    $\begingroup$ @user71352 It is from an old graduate analysis exam question. $\endgroup$ – yoyostein Dec 14 '16 at 1:50

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