I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on Semigroups of Linear operators I found on many places properties of the Neumann Laplacean. 

In  W. Arendt, *Semigroups and evolution equations: functional calculus, regularity and kernel estimates*, Handbook of Differential Equations: Evolutionary Equations. Vol. 1. North-Holland, 2002, pages 1-85 (it can be seen [here](https://pdf.sciencedirectassets.com/275750/1-s2.0-S1874571704X80014/1-s2.0-S1874571704800033/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEHoaCXVzLWVhc3QtMSJGMEQCIGr7B1Ys0MtGCGlRjYjdliXo%2B%2BYZNaLUT5GZgfDgNeloAiArjt6HRoR5puTLQreuJufjgP%2BHghYmGilHIbK0f8VLGyqDBAij%2F%2F%2F%2F%2F%2F%2F%2F%2F%2F8BEAQaDDA1OTAwMzU0Njg2NSIMxFoN563xJJfTAHFyKtcDjZaYJo13RtY%2Bic1%2FU6AGKL1G%2BOgsnCAw1kIuVo3upp6LTB%2F1O5rKT35vmHUHlVhPRrITyX2%2BSAs0iCllHHUXaXgTfwPBCSCjVgyGeH9Z6PVsAyZe28kBWYl8P%2B8Rnlmj6GfEhv1zHTvdSxVs4390eqY%2FT67Mxh3xb2QYpgSbe3%2Fxe0U9SA8Fnkmr74kI8%2FKzDlwdxso0owdrkiF8HOtZf4ln3emQbYLQ%2FKnQrAbamDksJgJe6jTvx3VkGpDoNFgcjyW8c1KN%2B5YgfzbJlRYl9J2WxACfUx0lNMz0zh%2BFHye%2F0Mp8sUsxenO7AN%2Ff1FFvUPU%2F0lNkZlfZJ5UEJGn0%2FIpVVsC7OQZnX%2FSo8V64HVEbwA6TcD6Ol6bNx6dqTsRLeXlHPaXXezYYqdiYvBxMHp35YsCgNEQLj%2Bx2JD2xrlJZAvXMDATlTsrUhG6IQ8kbu1qs3oXomiCDAF2CBeqe%2FLucHnmd9gtTOWY5ShPEgG5%2F5x2009zC1JHZGgewzzzZ2%2FGitRu3gxtUDXF2CMN3vL%2FTM81w10Mc6IJlcM%2F%2FDKQB%2BFNMHdz2Z%2F7t5aQzgItoLD0oOmRlj9iA7P0hATqVJaDVo%2FWNnG68BXfLMBgw5io257IIDJM9MJ7Y%2BIgGOqYBwWzFrP5bKt64cy1cV%2FOfwY4XAze79x0RKsYwxFk2sjXy%2B2DDVdFNW2DiGdvyE1Mt6b6AUDcKJQgiu3Z1a377%2BpUQw71xbxF9T6UyG%2F8YX828RHudQdXka8FfobSrJ6R%2FL3%2BgiudgDRb6vtf7M3TKx%2B63Eh8vaRC2tmEBtqZRQq1WqSl8U8FOZpmnW38mJTTZ5lodAZYUGYTjB4tlgEeBZAt2%2BlHc9A%3D%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20210819T111640Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYXTP6BI6O%2F20210819%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=2e2f5db38ce340b9ccb54fdc9707dfa7c41e08ce4db583d70228765e4c54d162&hash=04113dd051fb45f73f1cda5e3e3cf9adf04ab704daeb12de710164836b83e713&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S1874571704800033&tid=spdf-9d3b6b70-5436-4746-b235-e7030149c934&sid=415ad98c96b3e44254187a148604ba4e4751gxrqb&type=client)),

on page 69 we found an assertion that the Neumann Laplacean generates an ultracontractive semigroup. This means that the below property is satisfied only for any $1\geq t>0$ (see the definition from page 65).

**Here is my question**: How can we prove that the property is true for any $t\in (0,\infty)$? And why the definition of ultracontractivity is only for $t\in (0,1]$?

>For an open, bounded, connected and with an uniform Lipschitz boundary $\Omega\subseteq\mathbb{R}^2$ consider the semigroup of linear operators $S(t)_{t\geq 0}$ generated by the **Neumann Laplacean**:

\begin{equation}
\Delta_N:D(\Delta_N)\to L^2(\Omega),\ D(\Delta_N)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}=0, \ \mathcal{H}^{1}\text{- a.e. on}\ \partial\Omega\right\}.
\end{equation}

>Then for any $1\leq p\leq q\leq +\infty$ there is a constant $c=c(\Omega,p,q)$ that possess the following property (called *ultracontractivity*):

\begin{equation}
\Vert S(t)\phi\Vert_{L^q(\Omega)}\leq c t^{-\frac{N}{2}\left (\frac{1}{p}-\frac{1}{q}\right )}\Vert\phi\Vert_{L^p(\Omega)},\ \forall\ \phi\in L^p(\Omega),\ \forall\ t\geq 0
\end{equation}

**P.S. It's a natural question, since in many other books like *Cazenave & Haraux - An introduction to semilinear evolution equations* (page 44) or *Barbu Viorel - Analysis and Control of Nonlinear Infinite Dimensional Systems* (page 31) the above property is proved for any $t>0$ in the case of Dirichlet Laplacean.**