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26077 results in Abstract analysis

Page 93 of 1304

  • JMJ volume 21 issue 6 Cover and Back matter

  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 21 / Issue 6 / November 2022
    Published online by Cambridge University Press:
    29 November 2022, pp. b1-b2
    Print publication:
    November 2022
    • Article
      • You have access
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  • JMJ volume 21 issue 6 Cover and Front matter

  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 21 / Issue 6 / November 2022
    Published online by Cambridge University Press:
    29 November 2022, pp. f1-f2
    Print publication:
    November 2022
    • Article
      • You have access
    • PDF
    • Export citation

  • Hardy–Littlewood–Sobolev inequality and existence of the extremal functions with extended kernel

  • Part of
  • Zhao Liu
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 5 / October 2023
    Published online by Cambridge University Press:
    31 October 2022, pp. 1683-1705
    Print publication:
    October 2023

  • In this paper, we consider the following Hardy–Littlewood–Sobolev inequality with extended kernel(0.1)

    \begin{equation} \int_{\mathbb{R}_+^{n}}\int_{\partial\mathbb{R}^{n}_+} \frac{x_n^{\beta}}{|x-y|^{n-\alpha}}f(y)g(x) {\rm d}y{\rm d}x\leq C_{n,\alpha,\beta,p}\|f\|_{L^{p}(\partial\mathbb{R}_+^{n})} \|g\|_{L^{q'}(\mathbb{R}_+^{n})}, \end{equation}
    for any nonnegative functions $f\in L^{p}(\partial \mathbb {R}_+^{n})$, $g\in L^{q'}(\mathbb {R}_+^{n})$ and $p,\,\ q'\in (1,\,\infty )$, $\beta \geq 0$, $\alpha +\beta >1$ such that $\frac {n-1}{n}\frac {1}{p}+\frac {1}{q'}-\frac {\alpha +\beta -1}{n}=1$.

    We prove the existence of all extremal functions for (0.1). We show that if $f$ and $g$ are extremal functions for (0.1) then both of $f$ and $g$ are radially decreasing. Moreover, we apply the regularity lifting method to obtain the smoothness of extremal functions. Finally, we derive the sufficient and necessary condition of the existence of any nonnegative nontrivial solutions for the Euler–Lagrange equations by using Pohozaev identity.

  • Null, recursively starlike-equivalent decompositions shrink

  • Part of
  • Jeffrey Meier, Patrick Orson, Arunima Ray
  • Journal:
    Glasgow Mathematical Journal / Volume 65 / Issue 2 / May 2023
    Published online by Cambridge University Press:
    28 October 2022, pp. 328-336

  • A subset E of a metric space X is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $\mathbb{R}^n$ for some n, sending E to a starlike set. A subset $E\subset X$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $\{E_i\}_{i=0}^{N+1}$ such that $E_{i}/E_{i+1}\subset X/E_{i+1}$ is starlike-equivalent for each i and $E_{N+1}$ is a point. A decomposition $\mathcal{D}$ of a metric space X is said to be recursively starlike-equivalent, if there exists $N\geq 0$ such that each element of $\mathcal{D}$ is recursively starlike-equivalent of filtration length N. We prove that any null, recursively starlike-equivalent decomposition $\mathcal{D}$ of a compact metric space X shrinks, that is, the quotient map $X\to X/\mathcal{D}$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman–Starbird and Freedman and is applicable to the proof of Freedman’s celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological 4-manifolds, including the four-dimensional Poincaré conjecture.

Page 93 of 1304