{
  "id": "math/0010010",
  "version": "v1",
  "published": "2000-10-02T12:39:05.000Z",
  "updated": "2000-10-02T12:39:05.000Z",
  "title": "Recent Developments in the Theory of Lorentz Spaces and Weighted Inequalities",
  "authors": [
    "Maria J. Carro",
    "Jose A. Raposo",
    "Javier Soria"
  ],
  "comment": "viii+116 pp",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "The main objective of this work is to bring together two well known and, a priori, unrelated theories dealing with weighted inequalities for the Hardy-Littlewood maximal operator $M$, and thus, we consider the boundedness of $M$ in the weighted Lorentz space $\\Lambda^p_u(w)$. Two examples are historically relevant as a motivation: If $w=1$, this corresponds to the study of the boundedness $M:L^p(u)\\longrightarrow L^p(u),$ which was characterized by B. Muckenhoupt, giving rise to the so called $A_p$ weights. The second case is when we take $u=1$. This is a more recent theory, and was completely solved by M.A. Ari\\~no and B. Muckenhoupt in 1991. It turns out that the boundedness $M:\\llo\\longrightarrow\\llo,$ can be seen to be equivalent to the boundedness of the Hardy operator $A$ restricted to decreasing functions of $L^p(w)$. The class of weights satisfying this boundedness is known as $B_p$. Even though the $A_p$ and $B_p$ classes enjoy some similar features, they come from very different theories, and so are the techniques used on each case: Calder\\'on--Zygmund decompositions and covering lemmas for $A_p$, rearrangement invariant properties and positive integral operators for $B_p$. It is our aim to give a unified version of these two theories. Contrary to what one could expect, the solution is not given in terms of the limiting cases above considered (i.e., $u=1$ and $w=1$), but in a rather more complicated condition, which reflects the difficulty of estimating the distribution function of the Hardy-Littlewood maximal operator with respect to general measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-10-02T12:39:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "26D10",
      "26D15",
      "46E30",
      "47B38",
      "47G10"
    ],
    "keywords": [
      "weighted inequalities",
      "hardy-littlewood maximal operator",
      "boundedness",
      "developments",
      "rearrangement invariant properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 116,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math.....10010C"
    }
  }
}