Return to Answer

No. Take $\Omega = \mathbf{R}$, $$f(x) = \left\\{\begin{array}{cc} x^{-1/2} & |x| < 1\\\\ 0 & |x| \geq 1 \end{array}\right.$$

$x_i = 3^i$, and $r_i = x_i/2$ $$\int_{B(x_i,2r_i)} |f| = \int_0^{1} x^{-1/2} dx = 2$$

Assume there exists a $g$ in $L^1(\mathbf{R})$ $$\int_{\Omega} |g| \geq \sum_i \int_{B_i} |g| \geq C^{-1} \sum_i \int_{B(x_i,2r_i)} |f| \geq C^{-1} \sum_i 2 = \infty$$ which contradicts the hypothesis.

No. Take $\Omega = \mathbf{R}$, $$ f(x) = \left\{\begin{array}{cc} x^{-1/2} & |x| < 1\\\\ 0 & |x| \geq 1 \end{array}\right. $$

$x_i = 3^i$, and $r_i = x_i/2$ $$ \int_{B(x_i,2r_i)} |f| = \int_0^{1} x^{-1/2} dx = 2. $$

Assume there exists a $g$ in $L^1(\mathbf{R})$ $$\int_{\Omega} |g| \geq \sum_i \int_{B_i} |g| \geq C^{-1} \sum_i \int_{B(x_i,2r_i)} |f| \geq C^{-1} \sum_i 2 = \infty$$ which contradicts the hypothesis.

Take $\Omega = \mathbf{R}$, $f(x) = x^{-1/2}$,  $x_i = 3^i$, and $r_i = x_i/2$ $$\int_{B(x_i,2r_i)} |f| = \int_0^{2.3^i} x^{-1/2} dx = 2\sqrt{2}\times3^{i/2}$$

Assume there exists a $g$ in $L^1(\mathbf{R})$ $$\int_{\Omega} |g| \geq \sum_i \int_{B_i} |g| \geq C^{-1} \sum_i \int_{B(x_i,2r_i)} |f| \geq C^{-1} 2\sqrt{2}\sum_i 3^{i/2} = \infty$$ which contradicts the hypothesis.

No. Take $\Omega = \mathbf{R}$, $$f(x) = \left\\{\begin{array}{cc} x^{-1/2} & |x| < 1\\\\ 0 & |x| \geq 1 \end{array}\right.$$

$x_i = 3^i$, and $r_i = x_i/2$ $$\int_{B(x_i,2r_i)} |f| = \int_0^{1} x^{-1/2} dx = 2$$

Assume there exists a $g$ in $L^1(\mathbf{R})$ $$\int_{\Omega} |g| \geq \sum_i \int_{B_i} |g| \geq C^{-1} \sum_i \int_{B(x_i,2r_i)} |f| \geq C^{-1} \sum_i 2 = \infty$$ which contradicts the hypothesis.

I may be missing something, but set

$$g_{|B(x_i,r_i)} = \frac{\int_{B(x_i,2r_i)} |f|}{\int_{B(x_i,r_i)} 1}$$ and 0 everywhere else

$$\int_{\Omega} |g| = \sum_i \int_{B(x_i,2r_i)} |f| \leq 2^n\int_{\Omega} |f| < \infty$$

Take $\Omega = \mathbf{R}$, $f(x) = x^{-1/2}$, $x_i = 3^i$, and $r_i = x_i/2$ $$\int_{B(x_i,2r_i)} |f| = \int_0^{2.3^i} x^{-1/2} dx = 2\sqrt{2}\times3^{i/2}$$

Assume there exists a $g$ in $L^1(\mathbf{R})$ $$\int_{\Omega} |g| \geq \sum_i \int_{B_i} |g| \geq C^{-1} \sum_i \int_{B(x_i,2r_i)} |f| \geq C^{-1} 2\sqrt{2}\sum_i 3^{i/2} = \infty$$ which contradicts the hypothesis.