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.**



Contractivity of Neumann Laplacean
It might be helpful to point out the following conceptual reasons why ultracontractivity estimates are mainly interesting for times close to $0$.


Let us consider the following general setting: We have a finite measure space $(\Omega,\mu)$, two integrability indices $1 \le p < q \le \infty$ and a $C\_0$-semigroup $(S(t))\_{t \ge 0}$ on $L^p(\Omega,\mu)$.


Generally speaking, we are interested in situations where one of the operators $S(t)$ (or all of them for $t > 0$) map $L^p$ into $L^q$ (which can be interpreted as somekind of "smooting" property). Ultracontractivity means that we have specific estimates of the operator norm from $L^p$ to $L^q$ for all $t > 0$.


Now here are some useful observations:


**The long-time behaviour does not depend one the choice of spaces.**


If, for some $t\_0 > 0$, we have $S(t\_0)L^p \subseteq L^q$, then there are constants $c\_1,c\_2,c\_3,c\_4 > 0$ such that
\begin{align\*}
 \tag{1}
 \|S(t)\|\_{L^p \to L^p} & \le c\_1 \|S(t)\|\_{L^p \to L^q} \le c\_2 \|S(t-t\_0)\|\_{L^q \to L^q} \\
 & \le c\_3 \|S(t-2t\_0)\|\_{L^q \to L^p} \le c\_4 \|S(t-2t\_0)\|\_{L^p \to L^p}
\end{align\*}
for all $t > 2t\_0$. The first and the fourth inequality follow from the fact that $L^q$ embeds continuously into $L^p$ (since the measure space is finite), and the second and the third inequality follow from the semigroup law and the fact that $S(t\_0)$ is a bounded operator from $L^p$ to $L^q$ due to the closed graph theorem.


Inequality (1) shows that estimates for the norm $\|S(t)\|\_{L^p \to L^q}$ are not particularly interesting for large $t$, since they are not different from the estimates that one can prove for $\|S(t)\|\_{L^p \to L^p}$. For instance, if the semigroup is generated by the Dirichlet Laplacian, then $\|S(t)\|\_{L^p \to L^p}$ converges exponentially to $0$, and hence so does $\|S(t)\|\_{L^p \to L^q}$ (with the same rate).


It is also worthwhile to note that the long-term behaviour of $\|S(t)\|\_{L^p \to L^p}$ (and hence of $\|S(t)\|\_{L^p \to L^q}$) can be changed by rescaling the semigroup: if one substracts $a\operatorname{id}$ (for a real number $a$) from the semigroup generator, then the semigroup gets multiplied by $e^{-ta}$, so we are always able to enforce exponential decay of the $\|\cdot\|\_{L^p \to L^q}$-norm as $t \to \infty$, simply by a scalar shift of the generator.


**The short-time behaviour depends heavily on the choice of spaces.**


This is probably easiest to explain by considering an explicit example, say the Neumann-Laplace operator $\Delta\_N$ on a bounded domain, which generates a semigroup $(S(t))\_{t \ge 0}$.


The semigroup satisfies $\|S(t)\|\_{L^\infty \to L^\infty} \le 1$ for all $t \ge 0$ (this follows, for instance, from the positivity of the semigroup and the fact that the constant one-function is a fixed point of the semigroup) as well as $\|S(t)\|\_{L^1 \to L^1} \le 1$ for all $t \ge 0$ (this follows, for instance, by duality from the aforementioned $L^\infty$-estimate). Hence, by interpolation, $\|S(t)\|\_{L^p \to L^p} \le 1$ for all $p \in [1,\infty]$ and all $t \ge 0$. In other words, we have boundedness of the semigroup for all times when we consider the operator norm on a fixed $L^p$-space.


However, the situation changes significantly if we consider the norm $\|\cdot\|\_{L^p \to L^q}$ for $p > q$. To see this, fix a function $f$ which is in $L^p$ but not in $L^q$. By the strong continuity of the semigroup on $L^p$ we have $S(t)f \to f$ in $L^p$ as $t \downarrow 0$, but we must have $\|S(t)f\|\_{L^q} \to \infty$ since $f \not \in L^q$. This proves that $\|S(t)\|\_{L^p \to L^q} \to \infty$ as $t \downarrow 0$.


In other words, while the "smoothing effect" caused by the operator $S(t)$ is still present for small $t$ (i.e., $S(t)L^p \subseteq L^q$ for all $t > 0$), the effect becomes quantitively weaker in the sense that the operator norm from $L^p$ to $L^q$ explodes as $t \downarrow 0$.


So, *the interesting thing about an ultracontractivity estimate is that it gives us an explicit quantititve bound for the blow-up of $\|S(t)\|\_{L^p \to L^q}$ as $t \downarrow 0$.*


**Relation to Sobolev embeding theorems.**


I think the following relation to Sobolev embedding theorems is quite illuminating. As pointed out in the question, for the semigroup $(S(t))\_{t \ge 0}$ generated by the Neumann Laplace operator, one has the ultracontractivity estimate
$$
 \tag{2}
 \|S(t)\|\_{L^p \to L^q} \le c t^{-\frac{N}{2}(\frac{1}{p} - \frac{1}{q})} \quad \text{for } t \in (0,1],
$$
where $N$ is the dimension of the the underlying domain.


Now, if $\lambda > 0$ is a real number, then the resolvent of $\Delta\_N$ on $L^p$ is given by the integral
$$
 R(\lambda,\Delta\_N)f = \int\_0^\infty e^{-\lambda t} S(t) f \, dt,
$$
for each $f \in L^p$; the integral converges as a Bochner integral in $L^p$. Now let us analyse whether the integral converges also converges in $L^q$, even if $f$ is only in $L^p$. To this end, we split the integral into the integrals $\int\_0^1$ and $\int\_1^\infty$:


* It follows from $(1)$ that $\int\_1^\infty \|e^{-\lambda t} S(t)f\|\_{L^q} \, dt < \infty$ for each $f \in L^p$.
* It follows from the ultracontractivity estimate $(2)$ that $\int\_0^1 \|e^{-\lambda t} S(t)f\|\_{L^q} \, dt < \infty$ for each $f \in L^p$ if the additional condition $\frac{N}{2}(\frac{1}{p}-\frac{1}{q}) < 1$ (equivalently, $\frac{2}{N} > \frac{1}{p} - \frac{1}{q}$) is satisfied.


Hence, if $\frac{2}{N} > \frac{1}{p} - \frac{1}{q}$, then the resolvent $R(\lambda,\Delta\_N)$ maps $L^p$ into $L^q$; since the range of the resolvent equals the domain of $\Delta\_N$, this means that the domain of the Neumann Laplace operator on $L^p$ is contained in $L^q$ if $\frac{2}{N} > \frac{1}{p} - \frac{1}{q}$.


If the underlying domain $\Omega$ in $\mathbb{R}^N$ has sufficiently smooth boundary, the domain of $\Delta\_N$ in $L^p$ is the Sobolev space $W^{2,p}$; so in this case, what we have shown before amounts to saying that the Sobolev embedding theorem $W^{2,p} \subseteq L^q$ holds if $\frac{2}{N} > \frac{1}{p} - \frac{1}{q}$. Since all we needed for the proof was the ultracontractivity property $(2)$, we conclude that ultracontractivity gives rise to an abstract version of the Sobolev embedding theorem.


More information in this direction can, for instance, be found in Chapter 2 of Davies' book "*Heat kernels and spectral theory*" (1989).


