Solving for $Mf \notin L^1(\mathbb{R})$ with $f \in L^1 (\mathbb{R})$

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In summary, when $Mf \notin L^1(\mathbb{R})$, it means that the function $f$ cannot be integrated over the entire real line and the integral does not converge. This could be due to the function having large oscillations or decaying too slowly. $Mf$ refers to the maximal function of $f$, which measures the average value of $f$ over small intervals around a point $x$. $L^1(\mathbb{R})$ is a function space that consists of all measurable functions $f$ on the real line such that $\int_{-\infty}^{\infty} |f(x)|dx < \infty$. Solving for $Mf \
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mathmari
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Hello! :eek:

How can we find a $f \in L^1 (\mathbb{R})$ such taht $Mf \notin L^1 ( \mathbb{R})$, where $Mf$ is the maximum function ?? (Wondering)
 
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What is the maximum function?
 

FAQ: Solving for $Mf \notin L^1(\mathbb{R})$ with $f \in L^1 (\mathbb{R})$

What does it mean for $Mf \notin L^1(\mathbb{R})$?

When $Mf \notin L^1(\mathbb{R})$, it means that the function $f$ cannot be integrated over the entire real line and the integral does not converge. This could be due to the function having large oscillations or decaying too slowly.

What is $Mf$ in this context?

$Mf$ refers to the maximal function of $f$, which is defined as $Mf(x) = \sup_{r>0} \frac{1}{2r} \int_{x-r}^{x+r} |f(t)| dt$. Essentially, it measures the average value of $f$ over small intervals around a point $x$.

What is $L^1(\mathbb{R})$?

$L^1(\mathbb{R})$ is a function space that consists of all measurable functions $f$ on the real line such that $\int_{-\infty}^{\infty} |f(x)|dx < \infty$. In other words, $f$ is integrable over the entire real line.

Why is it important to solve for $Mf \notin L^1(\mathbb{R})$ with $f \in L^1 (\mathbb{R})$?

Solving for $Mf \notin L^1(\mathbb{R})$ with $f \in L^1 (\mathbb{R})$ is important in understanding the behavior of functions that are integrable but have a maximal function that does not converge. This could have implications in various fields, such as signal processing and analysis.

What are some possible methods for solving this problem?

Some possible methods for solving this problem include using techniques from harmonic analysis, such as Fourier series and Fourier transforms, as well as studying the properties of the maximal function itself. Other approaches may involve approximating the function $f$ or considering specific types of functions that have a maximal function that does not converge.

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