# Contractivity of Neumann Laplacean

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),

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:

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

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):

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

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.

• Consider what happens for $\phi=1$. Aug 19, 2021 at 13:05
• Since $S(t)1=1$ ultracontractivity holds only in bounded intervals in the form written above. Aug 19, 2021 at 13:12
• Now I understand. So we can assume it on intervals of the type $(0,T]$ not just on $(0,1]$. There is nothing special with 1 then. Thanks a lot! Aug 19, 2021 at 13:39
• But when $p=q=+\infty$ then we can assume it on $(0,\infty)$, right? Aug 19, 2021 at 13:42
• Does it happen to know some place where I can find a proof of that fact? Thanks a lot! Aug 19, 2021 at 14:11

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).