In this paper, we analyze evolution problems associated to homogenous operators. We show that they have an homogenous associated semigroup of solutions that must satisfy some sharp estimates when acting on homogenous spaces and on the associated fractional power spaces. These sharp estimates are determined by the homogeneity alone. We also consider fractional diffusion problems and Schrödinger type problems as well. We apply these general results to broad classes of PDE problems including heat or higher order parabolic problems and the associated fractional and Schrödinger problems or Stokes equations. These equations are considered in Lebesgue or Morrey spaces.

中文翻译：

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齐次空间中齐次半群的尖锐估计。ℝN 中 PDE 和分数扩散的应用

在本文中，我们分析了与同质算子相关的演化问题。我们证明了它们有一个同质相关的半群解，当作用于同质空间和相关的分数幂空间时，它必须满足一些尖锐的估计。这些尖锐的估计仅由同质性决定。我们还考虑分数扩散问题和薛定谔类型问题。我们将这些一般结果应用于广泛的 PDE 问题，包括热或高阶抛物线问题以及相关的分数和薛定谔问题或斯托克斯方程。这些方程在 Lebesgue 或 Morrey 空间中被考虑。