An inequality of $L^p$ norms of linear combinations of characteristic functions of balls
Let $1<p<\infty$. Let $(a_n)_{n=1}^\infty$ be a sequence of nonnegative real numbers and $\{B_{r_i}(x_i)\}_{i=1}^\infty$ be a sequence of open balls in $\mathbb{R}^n$. Prove that there exists $C>0$ such that \begin{equation*} \Big\|\sum_i a_i\chi_{3B_i}\Big\|_p\leq C\Big\|\sum_ia_i \chi _{B_i}\Big\|_p. \end{equation*} Here $B_i=B_{r_i}(x_i)$, $3B_i=B_{3r_i}(x_i)$. Moreover, $C$ does not depend on the choice of $(a_n)_{n=1}^\infty$.
Hint: Let $g\in L^q(\mathbb{R}^n)$ for $1/p+1/q=1$. Let \begin{equation*} g^*(x)=\sup_{x\in B}\dfrac{\int_B|g| d\mu}{Vol(B)}. \end{equation*} Show that there exists $C_0>0$ such that \begin{equation*} \int_{\mathbb{R}^n}\sum_i a_i\chi_{3b_i}(x)|g(x)|d\mu\leq C_0\int_{\mathbb{R}^n}\sum_i a_i\chi_{B_i}(x)g^*(x)d\mu. \end{equation*}
How to solve this? How to prove the hints and how to use the hints to prove the result? I get totally lost. Thanks.
Solution 1:
Recall that the $L^p$ norm of a function $h$ is equal to $$\sup_{g\in L^q\setminus \{0\}}\frac{\int |hg|}{\|g\|_q}$$ where $q=p'$. So, it suffices to show $$ \int_{\mathbb{R}^n}\sum_i a_i\chi_{3B_i}(x)|g(x)|d\mu\leq C \|\sum_ia_i \chi _{B_i}\|_p \|g\|_q \tag{1}$$ The advantage of (1) over the original formulation is that the left side of (1) is linear in the function we care about; this allows us to split the integral of sum as sum of integrals.
First step to proving (1) is $$ \int_{3B_i} |g| \le C\int_{B_i} g^* \tag2 $$ where $g^*$ is the Hardy-Littlewood maximal function. The reason for (2) is that for every $x\in B_i$, the ball centered at $x$ with radius $4r_i$ covers $3B_i$. Thus, $$ g^*(x) \ge \frac{1}{\mu(B_{4r_i} (x))} \int_{B_{4r_i}(x)}|g| \ge \frac{1}{\mu(B_{4r_i} (x))} \int_{3B_i}|g| \tag3 $$ Integrating over $B_i$ we pick up the factor of $\mu(B_i)$ which essentially cancels the denominator in (3). This proves (2) and with it, the hint.
To use the hint, apply Hölder's inequality:
$$ \int_{\mathbb{R}^n}\sum_i a_i\chi_{B_i}(x)g^*(x)d\mu
\le \|\sum_ia_i \chi _{B_i}\|_p \|g^*\|_q
$$ and then use the strong $(q,q)$ inequality for the maximal function: $\|g^*\|_q\le C\|g\|_q$. You'll get (1).
By the way, the history of this result (sometimes called Bojarski's lemma) was discussed at MO.