{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "2RCTYXSJ67"
    }
  ],
  "body": "\n## Summary\n\nThe Newtonian space $N^{1,p}(X)$ is a version of the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}$ for functions on a metric measure space $X$.\nIt replaces the gradient by an upper gradient, a function that bounds the change of $u$ along every rectifiable curve.\nIt was introduced by Nageswari Shanmugalingam, and it is a [[Banach space|06GZ956657]] for $1 \\le p \u003c \\infty$.\nIt is the basic function space of nonlinear potential theory on metric spaces, where it carries $p$-capacities and [[p-harmonic functions|7BKQT71FYZ]].\n\n## Explanations\n\n### L1 human\n\nIn $\\R^n$ a Sobolev function is one whose gradient has finite size in an average sense.\nA general metric space has no gradient, but one can still measure how fast a function changes along curves.\nAn upper gradient of $u$ is a function $g$ with this property: along any curve, the change of $u$ between the ends is at most the integral of $g$.\nFor example, on the real line the function $u(x) = \\abs{x}$ has the upper gradient $g = 1$, because $\\abs{\\abs{b} - \\abs{a}} \\le \\abs{b - a}$.\nThe Newtonian space collects the functions that, together with some upper gradient, have finite $L^p$ norm.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "46E36",
      "31E05"
    ],
    "name": "Newtonian space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Sobolev_space"
      }
    ],
    "specializes": [
      "06GZ956657"
    ],
    "tag": "V90RGBV52W"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/V90RGBV52W/",
  "links": [
    {
      "field": "specializes",
      "target": "06GZ956657"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/V90RGBV52W.md",
  "msc": [
    "46E36",
    "31E05"
  ],
  "name": "Newtonian space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/V90/V90RGBV52W.md",
  "status": null,
  "tag": "V90RGBV52W",
  "uri": "m2://V90RGBV52W",
  "url": "https://tamnd.github.io/math2/wiki/newtonian-space/",
  "writing": {
    "tag": "V90RGBV52W",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}