{
  "id": "2411.13266",
  "version": "v1",
  "published": "2024-11-20T12:36:07.000Z",
  "updated": "2024-11-20T12:36:07.000Z",
  "title": "A new maximal regularity for parabolic equations and an application",
  "authors": [
    "Jinlong Wei",
    "Wei Wang",
    "Guangying Lv",
    "Jinqiao Duan"
  ],
  "comment": "45 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We introduce the Lebesgue--H\\\"{o}lder--Dini and Lebesgue--H\\\"{o}lder spaces $L^p(\\mathbb{R};{\\mathcal C}_{\\vartheta,\\varsigma}^{\\alpha,\\rho}({\\mathbb R}^n))$ ($\\vartheta\\in \\{l,b\\}, \\, \\varsigma\\in \\{d,s,c,w\\}$, $p\\in (1,+\\infty]$ and $\\alpha\\in [0,1)$), and then use a vector-valued Calder\\'{o}n--Zygmund theorem to establish the maximal Lebesgue--H\\\"{o}lder--Dini and Lebesgue--H\\\"{o}lder regularity for a class of parabolic equations. As an application, we obtain the unique strong solvability of the following stochastic differential equation \\begin{eqnarray*} X_{s,t}(x)=x+\\int\\limits_s^tb(r,X_{s,r}(x))dr+W_t-W_{s}, \\ \\ t\\in [s,T], \\ x\\in \\mathbb{R}^n, \\ s\\in [0,T], \\end{eqnarray*} for the low regularity growing drift in critical Lebesgue--H\\\"{o}lder--Dini spaces $L^p([0,T];{\\mathcal C}^{\\frac{2}{p}-1,\\rho}_{l,d}({\\mathbb R}^n;{\\mathbb R}^n))$ ($p\\in (1,2]$), where $\\{W_t\\}_{0\\leq t\\leq T}$ is a $n$-dimensional standard Wiener process. In particular, when $p=2$ we give a partially affirmative answer to a longstanding open problem, which was proposed by Krylov and R\\\"{o}ckner for $b\\in L^2([0,T];L^\\infty({\\mathbb R}^n;{\\mathbb R}^n))$ based upon their work ({\\em Probab. Theory Relat. Fields 131(2): 154--196, 2005}).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-20T12:36:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic equations",
      "maximal regularity",
      "application",
      "dimensional standard wiener process",
      "stochastic differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}