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  "body": "\n## Summary\n\nA Muckenhoupt weight is a positive locally integrable function $w$ on $\\R^n$ that satisfies the $A_p$ condition.\nThis condition is a uniform bound, over all cubes, on products of averages of $w$ and of a negative power of $w$.\nThe class $A_p$, for $1 \u003c p \u003c \\infty$, is exactly the class of weights $w$ for which the [[Hardy–Littlewood maximal function|VX1A56P5KQ]] is bounded on $L^p(w \\, dx)$.\nThe same weights make the Hilbert transform and other Calderón–Zygmund operators bounded on $L^p(w \\, dx)$.\nThey are the central objects of weighted norm inequalities.\n\n## Explanations\n\n### L1 human\n\nA weight changes how much each part of space counts in an $L^p$ norm.\nThe $A_p$ condition says that on every cube the weight is not too unbalanced.\nIt cannot be very large on average while also being very small on a large part of the cube.\nPower weights $\\abs{x}^a$ are the main examples; on the line, $\\abs{x}^a$ is in $A_2$ exactly when $-1 \u003c a \u003c 1$.\nFor $a \\le -1$ the weight is not even integrable near $0$, and for $a \\ge 1$ its reciprocal is not.\n",
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