Prompt

Create an image that represents the concept of the Hardy-Littlewood maximal function, a fundamental idea in mathematical analysis, particularly in harmonic analysis and real analysis. The image should depict a graph or a visual representation that illustrates the process of taking a locally integrable function f on R^n and determining its maximal function Mf(x) as the supremum of the average of |f| over all balls B containing x. The style should be abstract yet mathematically insightful, incorporating elements that suggest the properties of being a sublinear operator, bounded on L^p for p > 1, and of weak type (1,1). Include symbolic representations or equations in the background that allude to its applications in differentiation theory, singular integrals, maximal inequalities, and the Lebesgue differentiation theorem. The overall aesthetic should convey the abstract mathematical nature of the concept while providing a clear visual intuition of its function and significance.

Image by stabilityai/stable-diffusion-3.5-large-turbo