Elliptic operators with unbounded diffusion, drift and potential terms

TL;DR

This paper proves that certain elliptic operators with unbounded coefficients generate analytic semigroups in L^p spaces, extending previous results and establishing properties like compactness and ultracontractivity.

Contribution

It generalizes existing results by showing that elliptic operators with unbounded diffusion, drift, and potential terms generate strongly continuous analytic semigroups under specific conditions.

Findings

Generates analytic semigroup in L^p for specified parameters

Establishes the semigroup's consistency and immediate compactness

Proves ultracontractivity of the semigroup

Abstract

We prove that the realization $A_p$ in $L^p(\mathbb{R}^N),\,1<p<\infty$, of the elliptic operator $A=(1+|x|^{\alpha})\Delta+b|x|^{\alpha-1}\frac{x}{|x|}\cdot \nabla-c|x|^{\beta}$ with domain $D(A_p) =\{ u \in W^{2,p}(\mathbb{R}^N)\, |\, Au \in L^p(\mathbb{R}^N)\}$ generates a strongly continuous analytic semigroup $T(\cdot)$ provided that $\alpha >2,\,\beta >\alpha -2$ and any constants $b\in \mathbb{R}$ and $c>0$. This generalizes the recent results in [A.Canale, A. Rhandi, C. Tacelli, Ann. Sc. Norm. Super. Pisa CI. Sci. (5), 2016] and in [G.Metafune, C.Spina, C.Tacelli, Adv. Diff. Equat., 2014]. Moreover we show that $T(\cdot)$ is consistent, immediately compact and ultracontractive.

Equations

$$Au(x)=q(x)\Delta u(x)+b|x{|}^{\alpha -1}\frac{x}{|x|}\cdot \nabla u-V(x)u(x),x\in {\mathbb{R}}^N,$$

$$D_{p,max}(A)=\{u\in L^p({\mathbb{R}}^N)\cap W_{loc}^{2,p}({\mathbb{R}}^N):Au\in L^p({\mathbb{R}}^N)\}.$$

$$D_p(A):=\{u\in W^{2,p}({\mathbb{R}}^N):Vu,(1+|x{|}^{\alpha -1})\nabla u,(1+|x{|}^{\alpha })D^{2}u\in L^p({\mathbb{R}}^N)\}.$$

$$|\nabla f(x){|}^2=\sum _{i=1}^N|D_{i}f(x){|}^2,|D^{2}f(x){|}^2=\sum _{i,j=1}^N|D_{ij}f(x){|}^2.$$

$$D_{max}(A)=\{u\in C_b({\mathbb{R}}^N)\cap W_{loc}^{2,p}({\mathbb{R}}^N)\text{for all}1\le p<\infty :Au\in C_b({\mathbb{R}}^N)\}.$$

$$\left\{\begin{array}[]{ll}u_{t}(t,x)=Au(t,x)&x\in\mathbb{R}^{N},\ t>0,\\ u(0,x)=f(x)&x\in\mathbb{R}^{N}\;,\end{array}\right.$$

$$u\in C([0,+\infty )\times {\mathbb{R}}^N)\cap C_{loc}^{1+\frac{\sigma }{2},2+\sigma }((0,+\infty )\times {\mathbb{R}}^N)$$

$$A\psi \le C\psi .$$

$$A\psi$$

$$\le \{\gamma (N+\gamma -2)+|b|\gamma \}(1+|x{|}^{\alpha })|x{|}^{\gamma -2}-c(1+|x{|}^{\gamma })|x{|}^{\beta }.$$

$$\{\gamma (N+\gamma -2)+|b|\gamma \}(1+|x{|}^{\alpha })|x{|}^{\gamma -2}\le c(1+|x{|}^{\gamma })|x{|}^{\beta }+C(1+|x{|}^{\gamma }).$$

$$\{\begin{array}{ll}& w_t(x,t)=L_{0}w(t,x)+V(x)u(t,x),t>0,\\ & w(0,x)=0x\in {\mathbb{R}}^N.\end{array}$$

$$\lambda u-Au=f.$$

$$\lambda u-Au=\frac{(1+|x{|}^{\alpha })}{\sqrt{\varphi }}\left[-\Delta v+Uv+\frac{V+\lambda }{1+|x{|}^{\alpha }}v\right],$$

$$U=-\frac{1}{4}{\left|\frac{\nabla \varphi }{\varphi }\right|}^2+\frac{1}{2}\frac{\Delta \varphi }{\varphi }.$$

$$-Hv=\frac{\sqrt{\varphi }}{1+|x{|}^{\alpha }}f,$$

$$H=\Delta -U-\frac{V+\lambda }{1+|x{|}^{\alpha }}.$$

$$v(x)={\int }_{{\mathbb{R}}^N}G(x,y)\frac{\sqrt{\varphi (y)}}{1+|y{|}^{\alpha }}f(y)dy,$$

$$u(x)=Lf(x):=\frac{1}{\sqrt{\varphi (x)}}{\int }_{{\mathbb{R}}^N}G(x,y)\frac{\sqrt{\varphi (y)}}{1+|y{|}^{\alpha }}f(y)dy.$$

$$\mathcal{V}$$

$$=\left(\frac{1}{1+|x{|}^{\alpha }}-\frac{1}{(1+|x{|}^{\alpha }{)}^2}\right)|x{|}^{\alpha -2}\left(\frac{b^2}{4}-\frac{b\alpha }{2}\right)+\frac{|x{|}^{\alpha -2}}{1+|x{|}^{\alpha }}\frac{b}{2}(N+\alpha -2)+\frac{c|x{|}^{\beta }+\lambda }{1+|x{|}^{\alpha }}$$

$$=\frac{|x{|}^{\alpha -2}}{1+|x{|}^{\alpha }}\left(\frac{b^2}{4}+b\left(\frac{N-2}{2}\right)\right)+\frac{|x{|}^{\alpha -2}}{(1+|x{|}^{\alpha }{)}^2}\left(-\frac{1}{4}{b}^2+\frac{1}{2}b\alpha \right)+\frac{c|x{|}^{\beta }}{1+|x{|}^{\alpha }}+\frac{\lambda }{1+|x{|}^{\alpha }}.$$

$$\frac{|x{|}^{2\alpha -2}}{(1+|x{|}^{\alpha })}\left(\frac{b^2}{4}-\frac{b\alpha }{2}\right)+|x{|}^{\alpha -2}\frac{b}{2}(N+\alpha -2)+c|x{|}^{\beta }$$

$$C_1(1+|x{|}^{\beta -\alpha })\le \mathcal{V}\le C_2(1+|x{|}^{\beta -\alpha })\text{ if }\beta \ge \alpha ,$$

$$C_3\frac{1}{1+|x{|}^{\alpha -\beta }}\le \mathcal{V}\le C_4\frac{1}{1+|x{|}^{\alpha -\beta }}\text{ if }\alpha -2<\beta <\alpha$$

$${\left(\frac{1}{|B|}{\int }_B{V}^q(x)dx\right)}^{1/q}\le C\left(\frac{1}{|B|}{\int }_{B}V(x)dx\right)$$

$$\Vert V{\Vert }_{L^{\infty }(B)}\le C\left(\frac{1}{|B|}{\int }_{B}V(x)dx\right)$$

$$|G(x,y)|\le \frac{C_k}{{\left(1+m(x)|x-y|\right)}^k}\cdot \frac{1}{|x-y{|}^{N-2}},x,y\in {\mathbb{R}}^N,$$

$$\frac{1}{m(x)}:=\underset{r>0}{\sup }\left\{r:\frac{1}{r^{N-2}}{\int }_{B(x,r)}\mathcal{V}(y)dy\le 1\right\},x\in {\mathbb{R}}^N.$$

$$m(x)\ge C{\left(1+|x|\right)}^{\frac{\beta -\alpha }{2}}.$$

$$G(x,y)\le C_k\frac{1}{1+|x-y{|}^k{\left(1+|x|\right)}^{\frac{\beta -\alpha }{2}k}}\frac{1}{|x-y{|}^{N-2}},x,y\in {\mathbb{R}}^N$$

$$\Vert |x{|}^{\gamma }Lf{\Vert }_p\le C\Vert f{\Vert }_p.$$

Full text

1. Introduction

Starting from the 1950’s, the theory of linear second order elliptic operators with bounded coefficients has widely been studied. In recent years there has been a surge of activity focused on the case of unbounded coefficients. Let us recall some recent results concerning elliptic operators having polynomial coefficients.

In ([13], [14]) it is proved that, under suitable assumptions on $\alpha$, the operator $(s+|x|^{\alpha})\Delta$, $s=0,1$, generates an analytic semigroup in $L^{p}(\mathbb{R}^{N})$. In [9] (resp. [3]) the generation of an analytic semigroup of the $L^{p}$-realization of the Schrödinger-type operators $(1+|x|^{\alpha})\Delta-|x|^{\beta}$ in $L^{p}(\mathbb{R}^{N})$ for $\alpha\in[0,2)$ and $\beta>2$ (resp. $\alpha>2$, $\beta>\alpha-2$) is obtained. In [9] some estimates for the associated heat kernel are provided.

Under suitable growth assumptions on the functions $q$ and $F$, it is proved in [5] that the operator $\mathcal{A}u=q\Delta u+F\cdot\nabla u$ admits realizations generating analytic semigroups in $L^{p}(\mathbb{R}^{N})$ for any $p\in[1,+\infty]$ and in $C_{b}(\mathbb{R}^{N})$.

Concerning the operator $(1+|x|^{\alpha})\Delta+b|x|^{\alpha-2}x\cdot\nabla$, G. Metafune et al. in [16] showed the generation of an analytic semigroup in $L^{p}(\mathbb{R}^{N})$ in the case where $\alpha>2$, $\frac{N}{N-2+b}<p<\infty$, and the semigroup is contractive if and only if $p\geq\frac{N+\alpha-2}{N-2+b}$. Domain characterization and spectral properties as well as kernel estimates have been also proved. More recently in [10] the authors showed that the operator $L=|x|^{\alpha}\Delta+b|x|^{\alpha-2}x\cdot\nabla-c|x|^{\alpha-2}$ generates a strongly continuous semigroup in $L^{p}(\mathbb{R}^{N})$ if and only if $s_{1}+\min\{0,2-\alpha\}<\frac{N}{p}<s_{2}+\max\{0,2-\alpha\}$, where $s_{i}$ are the roots of the equation $c+s(N-2+b-s)=0$. Moreover the domain of the generator is also characterized.

At this point it is important to note that the techniques used in [10] are completely different from our and lead to results which are not comparable with our case ($\beta>\alpha-2$).

In this paper, we are interested in studying quantitative and qualitative properties in $L^{p}(\mathbb{R}^{N}),\,1<p<\infty$, of the elliptic operator

[TABLE]

where $\alpha>2,\,\beta>\alpha-2$, $q(x)=(1+|x|^{\alpha})$ and $V(x)=c|x|^{\beta}$, with $b\in\mathbb{R}$ and $c>0$.

Let $A_{p}$ be the realization of $A$ in $L^{p}(\mathbb{R}^{N})$ endowed with the maximal domain

[TABLE]

After proving a priori estimates, we deduce that the maximal domain $D_{p,max}(A)$ of the operator $A$ coincides with

[TABLE]

So, we show in the main result of this paper that, for any $1<p<\infty$, the realization $A_{p}$ of $A$ in $L^{p}(\mathbb{R}^{N})$, with domain $D_{p}(A)$ generates a positive strongly continuous and analytic semigroup $(T_{p}(t))_{t\geq 0}$ for $p\in(1,\infty)$. This semigroup is also consistent, irreducible, immediately compact and ultracontractive.

The paper is divided as follows. In section 2 we recall the solvability of the elliptic and parabolic problems in spaces of continuous functions. In Section 3 we introduce the definition of the reverse Hölder class and recall some results given in [19] and in [3] to study the solvability of the elliptic problem in $L^{p}(\mathbb{R}^{N})$. In section 4 we prove that the maximal domain of the operator $A$ coincides with the weighted Sobolev space $D_{p}(A)$, and we state and prove the main result of this paper.

Notation. In general we use standard notations for function spaces. We denote by $L^{p}(\mathbb{R}^{N})$ and $W^{2,p}(\mathbb{R}^{N})$ the standard $L^{p}$ and Sobolev spaces, respectively. Following the notation in [3], for any $k\in\mathbb{N}\cup\{\infty\}$ we denote by $C^{k}_{c}(\mathbb{R}^{N})$ the set of all functions $f:\mathbb{R}^{N}\to\mathbb{R}$ that are continuously differentiable in $\mathbb{R}^{N}$ up to $k$-th order and have compact support (say ${\rm supp}(f)$). The space $C_{b}(\mathbb{R}^{N})$ is the set of all bounded and continuous functions $f:\mathbb{R}^{N}\to\mathbb{R}$, and we denote by $\|f\|_{\infty}$ its sup-norm, i.e., $\|f\|_{\infty}=\sup_{x\in\mathbb{R}^{N}}|f(x)|$. We use also the space $C_{0}(\mathbb{R}^{N}):=\{f\in C_{b}(\mathbb{R}^{N}):\lim_{|x|\to\infty}f(x)=0\}.$ If $f$ is smooth enough we set

[TABLE]

For any $x_{0}\in\mathbb{R}^{N}$ and any $r>0$ we denote by $B(x_{0},r)\subset\mathbb{R}^{N}$ the open ball, centered at $x_{0}$ with radius $r$. We simply write $B(r)$ when $x_{0}=0$. The function $\chi_{E}$ denotes the characteristic function of the set $E$, i.e., $\chi_{E}(x)=1$ if $x\in E$, $\chi_{E}(x)=0$ otherwise. Finally, by $x\cdot y$ we denote the Euclidean scalar product of the vectors $x,y\in\mathbb{R}^{N}$.

2. Solvability in $C_{0}(\mathbb{R}^{N})$

In this short section we briefly recall some properties of the elliptic and parabolic problems associated with $A$ in spaces of continuous functions.

Let us first consider the operator $A$ on $C_{b}(\mathbb{R}^{N})$ with its maximal domain

[TABLE]

It is known, cf. [8, Chapter 2, Section 2], that to the associated parabolic problem

[TABLE]

where $f\in C_{b}(\mathbb{R}^{N})$, one can associate a semigroup $(T(t))_{t\geq 0}$ of bounded operator in $C_{b}(\mathbb{R}^{N})$ such that $u(t,x)=T(t)f(x)$ is a solution of (2.1) in the following sense:

[TABLE]

and $u$ solves (2.1) for any $f\in C_{b}(\mathbb{R}^{N})$ and some $\sigma\in(0,1)$. Moreover, in our case the solution is unique. This can be seen by proving the existence of a Lypunov function for $A$, i.e., a positive function $\varphi(x)\in C^{2}(\mathbb{R}^{N})$ such that $\lim_{|x|\to\infty}\varphi(x)=+\infty$ and $A\varphi-\lambda\varphi\leq 0$ for some $\lambda>0$.

Proposition 1**.**

Assume that $\alpha>2$ and $\beta>\alpha-2$. Let $\psi=1+|x|^{\gamma}$ where $\gamma>2$ then there exists a constant $C>0$ such that

[TABLE]

Proof.

An easy computation gives

[TABLE]

Since $\beta>\alpha-2$, it follows that there exists $C>0$ such that

[TABLE]

Thus, $\psi$ is a Lyapunov function for $A$. ∎

As in [3] one can prove the following result.

Proposition 2**.**

Assume that $N>2,\,\alpha>2$ and $\beta>\alpha-2$. Then the semigroup $(T(t))$ is generated by $(A,D_{max}(A))\cap C_{0}(\mathbb{R}^{N})$ and maps $C_{0}(\mathbb{R}^{N})$ into $C_{0}(\mathbb{R}^{N})$.

Proof.

We split the proof into two steps.

Step 1. Here we assume that $b>2-N$.

Let $L_{0}$ be the operator defined by $L_{0}:=q(x)\Delta+b|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla$, and as a result of [16, Proposition 2.2 (ii)], we have that the minimal semigroup $(S(t))$ is generated by $(L_{0},D_{max}(L_{0}))\cap C_{0}(\mathbb{R}^{N})$. Moreover the resolvent and the semigroup map $C_{b}(\mathbb{R}^{N})$ into $C_{0}(\mathbb{R}^{N})$ and are compact.

Set $v(t,x)=S(t)f(x)$ and $u(t,x)=T(t)f(x)$ for $t>0,x\in\mathbb{R}^{N}$ and $0\leq f\in C_{b}(\mathbb{R}^{N})$. Then the function $w(t,x)=v(t,x)-u(t,x)$ solves

[TABLE]

So, applying [8, Theorem 4.1.3], we have $w\geq 0$ and hence $T(t)\leq S(t)$. Thus, $T(t)\mathbf{1}\in C_{0}(\mathbb{R^{N}})$, for any $t>0$ (see [13, Proposition 2.2 (iii)]). Therefore $T(t)$ is compact and $C_{0}(\mathbb{R}^{N})$ is invariant for $T(t)$ (cf. [8, Theorem 5.1.11]).

Step 2. Assume now that $b\leq 2-N$.

Let $f\in C_{0}(\mathbb{R}^{N})$. Since $C_{c}^{\infty}(\mathbb{R}^{N})$ is dense in $C_{0}(\mathbb{R}^{N})$, there is a sequence $(f_{n})\subset C_{c}^{\infty}(\mathbb{R}^{N})$ such that $\lim_{n\to\infty}\|f_{n}-f\|_{\infty}=0$.

On the other hand, it follows from Theorem 3 that the operator $A_{p}$ with domain $D_{p,max}(A)$ generates an analytic semigroup $T_{p}(t)$ in $L^{p}(\mathbb{R}^{N})$, and $D_{p}(A)$ is continuously embedded into $W^{2,p}(\mathbb{R}^{N})$. Hence, by Theorem 2 and Sobolev’s embedding theorem, $T(t)f_{n}=T_{p}(t)f_{n}\in D_{p}(A)\subset W^{2,p}(\mathbb{R}^{N})\hookrightarrow C_{0}(\mathbb{R}^{N})$ for $p>\frac{N}{2}$. Since $f_{n}\rightarrow f$ uniformly, it follows that $T(t)f_{n}\rightarrow T(t)f$ uniformly. Hence $T(t)f\in C_{0}(\mathbb{R}^{N})$. ∎

Remark 1**.**

As one sees from the proof of the above proposition, in the case where $b>2-N$ the semigroup $(T(t))$ generated by $(A,D_{max}(A))\cap C_{0}(\mathbb{R}^{N})$ is compact.

3. Solvability of $\lambda u-Au=f$ in $L^{p}(\mathbb{R}^{N})$

In the previous section we have proved the existence and uniqueness of the elliptic and parabolic problems in $C_{0}(\mathbb{R}^{N})$. In this section we study the solvability of the equation $\lambda u-A_{p}u=f$ for $\lambda>\lambda_{0}$, where $\lambda_{0}$ is a suitable positive constant.

Let $f\in L^{p}(\mathbb{R}^{N})$ and consider the equation

[TABLE]

Let $\phi=(1+|x|^{\alpha})^{b/\alpha}$ and set $u=\frac{v}{\sqrt{\phi}}$. A simple computation gives

[TABLE]

where

[TABLE]

Then solving (3.1) is equivalent to solve

[TABLE]

where $H$ is the Schrödinger operator defined by

[TABLE]

If we denote by $G(x,y)$ the Green function of $H$, a solution of (3.3) is given by

[TABLE]

and hence a solution of (3.1) should be

[TABLE]

First, we have to show that $L$ is a bounded operator in $L^{p}(\mathbb{R}^{N})$. For this purpose we need to estimate $G.$

We focus our attention to the operator $H$. Evaluating the potential $\mathcal{V}=U+\frac{V+\lambda}{1+|x|^{\alpha}}$, it follows that

[TABLE]

We can choose $\lambda_{0}>0$ such that for every $\lambda\geq\lambda_{0}$ the potential $\mathcal{V}$ is positive. Indeed, since $\beta>\alpha-2$ the function

[TABLE]

has a nonpositive minimum $\mu$ in $\mathbb{R}^{N}$. So, one takes $\lambda_{0}>-\mu.$

On the other hand, since $\mathcal{V}(0)=\lambda>0$ and $\mathcal{V}$ behaves like $|x|^{\beta-\alpha}$ as $|x|\to\infty$ we have the following estimates

[TABLE]

for some positive constants $C_{1},C_{2},C_{3},C_{4}$.

At this point we can use bounds of $G$ obtained by [19] in the case of positive potentials belonging to the reverse Hölder class $B_{q}$ for some $q\geq N/2$.

We recall that a nonnegative locally $L^{q}$-integrable function $V$ on $\mathbb{R}^{N}$ is said to be in $B_{q},\,1<q<\infty$, if there exists $C>0$ such that the reverse Hölder inequality

[TABLE]

holds for every ball $B$ in $\mathbb{R}^{N}$. A nonnegative function $V\in L_{\rm loc}^{\infty}(\mathbb{R}^{N})$ is in $B_{\infty}$ if

[TABLE]

for every ball $B$ in $\mathbb{R}^{N}$.

As regards the potential $\mathcal{V}$, by (3.5) and since $\beta-\alpha>-2$, we have that $\mathcal{V}\in B_{N/2}$. So, it follows from [19, Theorem 2.7] that for any $k\in\mathbb{N}$ there is a constant $C_{k}>0$ such that

[TABLE]

where the auxiliary function $m$ is defined by

[TABLE]

In [3] a lower bound for the auxiliary function associated to the potential $\tilde{V}=\frac{|x|^{\beta}}{1+|x|^{\alpha}}$ was obtained. Since $\mathcal{V}\geq C_{1}\tilde{V}$ for some positive constant $C_{1}$, we have $m(x)\geq\widetilde{m}(x)$, where $\frac{1}{\widetilde{m}(x)}:=\sup_{r>0}\left\{r\,:\,\frac{1}{r^{N-2}}\int_{B(x,r)}C_{1}\tilde{V}(y)dy\leq 1\right\}$. Replacing $\tilde{V}$ with $C_{1}\tilde{V}$ in [3, Lemma 3.1, Lemma 3.2] we obtain $\widetilde{m}(x)\geq C_{2}\left(1+|x|\right)^{\frac{\beta-\alpha}{2}}$. So we have

Lemma 1**.**

Let $\alpha-2<\beta$. There exists $C=C(\alpha,\beta,N)$ such that

[TABLE]

Finally by (3.6) and the previous lemma we can estimate the Green function $G$

Lemma 2**.**

Let $G(x,y)$ denotes the Green function of the Schrödinger operator $H$ and assume that $\beta>\alpha-2$. Then

[TABLE]

for any $k>0$ and some constant $C_{k}>0$ depending on $k$.

We can prove now the boundedness in $L^{p}(\mathbb{R}^{N})$ of the operator $L$ given by (3.4)

Lemma 3**.**

Assume that $\alpha>2$, $N>2$ and $\beta>\alpha-2$. Then there exists a positive constant $C=C(\lambda)$ such that for every $0\leq\gamma\leq\beta$ and $f\in L^{p}(\mathbb{R}^{N})$

[TABLE]

Proof.

Recall the function $\phi(x)=(1+|x|^{\alpha})^{b/\alpha}$. Let $\Gamma(x,y)=\sqrt{\frac{\phi(y)}{\phi(x)}}\frac{G(x,y)}{1+|y|^{\alpha}}$, $f\in L^{p}(\mathbb{R}^{N})$ and

[TABLE]

We have to show that

[TABLE]

By setting $\Gamma_{0}=\frac{G(x,y)}{1+|y|^{\alpha}}$, we have $\Gamma(x,y)=\left(\frac{1+|y|^{\alpha}}{1+|x|^{\alpha}}\right)^{b/(2\alpha)}\Gamma_{0}(x,y)$. Moreover if we set $L_{0}f(x):=\int_{\mathbb{R}^{N}}\Gamma_{0}(x,y)f(y)dy,\,x\in\mathbb{R}^{N}$, then [3, Lemma 3.4] gives

[TABLE]

For $x\in\mathbb{R}^{N}$ let us consider the regions $E_{1}:=\{|x-y|\leq\frac{1}{2}(1+|y|)\}$ and $E_{2}:=\{|x-y|>\frac{1}{2}(1+|y|)\}$ and write

[TABLE]

In $E_{1}$ we have $1+|y|\leq 1+|x|+|x-y|\leq 1+|x|+\frac{1}{2}(1+|y|)$ and hence $\frac{1}{2}(1+|y|)\leq 1+|x|$. Thus,

[TABLE]

Therefore there are constants $C,\,\widetilde{C}>0$ such that $\left(\frac{1+|y|^{\alpha}}{1+|x|^{\alpha}}\right)^{b/(2\alpha)}\leq\widetilde{C}\left(\frac{1+|y|}{1+|x|}\right)^{b/2}\leq C2^{|b|/2}$ and $\Gamma(x,y)\leq C\Gamma_{0}(x,y)$ in $E_{1}$. So, we have

[TABLE]

By (3.11) it follows that $\||x|^{\gamma}u_{1}\|_{p}\leq C\|f\|_{p}$.

As regards the region $E_{2}$, we have, by Hölder’s inequality,

[TABLE]

We propose to estimate first $\int_{E_{2}}|x|^{\gamma}\Gamma(x,y)dy$. In $E_{2}$ we have $1+|y|\leq 2|x-y|$ and $1+|x|\leq 1+|y|+|x-y|\leq 3|x-y|$, then

[TABLE]

From (3.9) and by the symmetry of $G$ it follows that

[TABLE]

For every $k>\beta-N+2+|b|/2$, taking into account that $\frac{1}{|x-y|}\leq 2\frac{1}{1+|y|}$, we get

[TABLE]

Since $\beta-\alpha+2>0$ we can choose $k$ such that $\frac{k}{2}(\beta-\alpha+2)+N-2-\beta-|b|/2>N$, then there is a constant $C_{1}>0$ such that

[TABLE]

Moreover by (3.9) as above we have

[TABLE]

Taking into account that $\frac{1}{|x-y|}\leq 3\frac{1}{1+|x|}$, arguing as above we obtain

[TABLE]

for some constant $C_{2}>0$. Hence (3) implies

[TABLE]

Thus, by (3.14) and (3.13), we have

[TABLE]

∎

Here and in Section 4 we will need the following covering result, see [4, Proposition 6.1].

Proposition 3**.**

Given a covering $\mathcal{F}=\{B(x,\rho(x))\}_{x\in\mathbb{R}^{N}}$ of $\mathbb{R}^{N}$, where $\rho:\mathbb{R}^{N}\to\mathbb{R}_{+}$ is a Lipschitz continuous function with Lipschitz constant $k<1/2$, there exists a countable subcovering $\{B(x_{n},\rho(x_{n}))\}_{n\in\mathbb{N}}$ of $\mathbb{R}^{N}$ and $\zeta=\zeta(N,k)\in\mathbb{N}$ such that at most $\zeta$ among the double balls $\{B(x_{n},2\rho(x_{n}))\}_{n\in\mathbb{N}}$ overlap.

We propose now to characterize the domain $D_{p,max}(A)$.

Proposition 4**.**

Assume that $N>2$, $\alpha>2$ and $\beta>\alpha-2$. For $1<p<\infty$ the following holds

[TABLE]

Proof.

It suffices to prove that $D_{p,max}(A)\subset\{u\in W^{2,p}(\mathbb{R}^{N}):Au\in L^{p}(\mathbb{R}^{N})\}$. Let $u\in D_{p,max}(A)$. Then $f:=Au\in L^{p}(\mathbb{R}^{N})$. This implies that

[TABLE]

If $\beta\leq\alpha$ then the potential $\widetilde{V}(x):=\frac{c|x|^{\beta}}{1+|x|^{\alpha}}$ is bounded and by standard regularity results for uniformly elliptic operators with bounded coefficients we deduce that $u\in W^{2,p}(\mathbb{R}^{N})$.

Let us assume now that $\beta>\alpha$. Then $\widetilde{V}\in B_{q}$ for all $q\in(1,\infty)$. So, by [1, Theorem 1.1 and Corollary 1.3], we have that $D_{p,max}(\Delta-\widetilde{V})=W^{2,p}(\mathbb{R}^{N})\cap D_{p,max}(\widetilde{V})$ and the following estimate holds

[TABLE]

for all $f\in D_{p,max}(\Delta-\widetilde{V})$ with a constant $C$ independent of $f$.

Fix now $x_{0}\in\mathbb{R}^{N}$ and $R\geq 1$. We propose to prove the following interior estimate

[TABLE]

with a constant $C$ independent of $u$ and $R$. To this purpose take $\sigma\in(0,1)$ and set $\sigma^{\prime}:=\frac{\sigma+1}{2}$. Consider a cutoff function $\vartheta\in C_{c}^{\infty}(\mathbb{R}^{N})$ such that $0\leq\vartheta\leq 1$, $\vartheta(x)=1$ for $x\in B(x_{0},\sigma R)$, $\vartheta(x)=0$ for $x\in B^{c}(x_{0},\sigma^{\prime}R)$, $\|\nabla\vartheta\|_{\infty}\leq\frac{C}{R(1-\sigma^{\prime})}$ and $\|\Delta\vartheta\|_{\infty}\leq\frac{C}{R^{2}(1-\sigma^{\prime})^{2}}$ with a constant $C$ independent of $R$.

In order to simplify the notation we write $\|\cdot\|_{p,r}$ instead of $\|\cdot\|_{L^{p}\left(B(x_{0},r)\right)}$. The function $v=u\vartheta$ belongs to $D_{p,max}(\Delta-\tilde{V})$ and so by (3.15) we have

[TABLE]

where $F(x):=b\frac{|x|^{\alpha-2}}{1+|x|^{\alpha}}x$ and $C$ a positive constant independent of $u$ and $R$, which may change from line to line. Multiplying the above estimate by $R^{2}(1-\sigma^{\prime})^{2}$ and taking into account that $1-\sigma=2(1-\sigma^{\prime})$ we obtain

[TABLE]

So,

[TABLE]

Thus, by [6, Theorem 7.28], for every $\gamma>0$ there exists $\sigma_{\gamma}\in(0,1)$ such that

[TABLE]

Letting $\gamma\to 0$ we deduce that

[TABLE]

Putting (3.18) into (3.17) with a suitable choice of $\varepsilon$ we obtain

[TABLE]

Hence (3.16) follows since $(1-\frac{1}{2})^{2}R^{2}\|\Delta u\|_{p,\frac{R}{2}}\leq\sup_{\sigma\in(0,1)}\left\{R^{2}(1-\sigma)^{2}\|\Delta u\|_{p,\sigma R}\right\}$.

To prove that $u\in W^{2,p}(\mathbb{R}^{N})$ we consider a covering $\{B(x_{n},R/2):n\in\mathbb{N}\}$ of $\mathbb{R}^{N}$ such that at most $\zeta$ among the doubled balls $\{B(x_{n},R):n\in\mathbb{N}\}$ overlap for some $\zeta(N)\in\mathbb{N}$, by Proposition 3. Applying (3.16) with the ball $B(x_{n},R/2)$ we obtain

[TABLE]

This ends the proof. ∎

We show now the invertibility of $\lambda-A_{p}$ in $D_{p,max}(A)$ for all $\lambda\geq\lambda_{0}$, where $\lambda_{0}>0$ is such that $\mathcal{V}\geq 0$ for all $\lambda\geq\lambda_{0}$.

Theorem 1**.**

Assume that $N>2,\,\alpha>2$ and $\beta>\alpha-2$. Then $[\lambda_{0},\infty)\subset\rho(A_{p})$ and $(\lambda-A_{p})^{-1}=L$ for all $\lambda\geq\lambda_{0}$. Moreover there exists $C=C(\lambda)>0$ such that, for every $0\leq\gamma\leq\beta$ and $\lambda\geq\lambda_{0}$, the following holds

[TABLE]

Proof.

First we prove the injectivity of $\lambda-A_{p}$ for $\lambda\geq\lambda_{0}$. Let $u\in D_{p,max}(A)$ such that $\lambda u-A_{p}u=0$. We have to distinguish two cases. The first one is when $b\leq 0$. In this case, by (3.2) we have $Hv=\Delta v-\mathcal{V}v=0$ with $v=u\sqrt{\phi}\in D_{p,max}(H)=W^{2,p}(\mathbb{R}^{N})\cap D_{p,max}\left(\mathcal{V}\right)$, (see [17] or [1]). Then multiplying $Hv$ by $v|v|^{p-2}$ and integrating by part (see [12]) over $\mathbb{R}^{N}$, we have

[TABLE]

Then we have $v\equiv 0$ and hence $u\equiv 0$.

The second case is when $b>0$. For this we multiply $\frac{1}{1+|x|^{\alpha}}(\lambda u-A_{p}u)$ by $u|u|^{p-2}$ and using the fact that $u\in W^{2,p}$, by Proposition 4, we have

[TABLE]

Hence, $u\equiv 0$.

Let now $f\in L^{p}(\mathbb{R}^{N})$ and $u(x)=Lf(x)$ defined by (3.4). Applying Lemma 3 with $\gamma=0$, we have $u\in L^{p}(\mathbb{R}^{N})$. Moreover $u_{n}:=Lf_{n}$ satisfies $\lambda u_{n}-A_{p}u_{n}=f_{n}$ for any $f_{n}\in C_{c}^{\infty}(\mathbb{R}^{N})$ approximating $f$ in $L^{p}(\mathbb{R}^{N})$. Thus, $\lim_{n\to\infty}\|u_{n}-u\|_{p}=0$. Since, by local elliptic regularity, $A_{p}$ on $D_{p,max}(A)$ is closed, it follows that $u\in D_{p,max}(A)$ and $\lambda u-A_{p}u=f$. Thus $\lambda-A_{p}$ is invertible and $(\lambda-A_{p})^{-1}\in\mathcal{L}(L^{p}(\mathbb{R}^{N}))$ for all $\lambda\geq\lambda_{0}$.

Finally, (3.19) follows from (3.10). ∎

The following result shows that the resolvent in $L^{p}(\mathbb{R}^{N})$ and $C_{0}(\mathbb{R}^{N})$ coincides.

Theorem 2**.**

Assume that $N>2$, $\beta>\alpha-2$ and $\alpha>2$. Then, for all $\lambda\geq\lambda_{0}$, $(\lambda-A_{p})^{-1}$ is a positive operator on $L^{p}(\mathbb{R}^{N})$. Moreover, if $f\in L^{p}(\mathbb{R}^{N})\cap C_{0}(\mathbb{R}^{N})$, then $(\lambda-A_{p})^{-1}f=(\lambda-A)^{-1}f$.

Proof.

The positivity of $(\lambda-A_{p})^{-1}$ follows from Theorem 1 and the positivity of $L$.

For the second assertion take $f\in C_{c}^{\infty}(\mathbb{R}^{N})$ and set $u:=(\lambda-A_{p})^{-1}f$. Since the coefficients of $A$ are Hölder continuous, by local elliptic regularity (cf. [6, Theorem 9.19]), we know $u\in C^{2+\sigma}_{loc}(\mathbb{R}^{N})$ for some $0<\sigma<1$. On the other hand, $u\in W^{2,p}(\mathbb{R}^{N})$ by Proposition 4.

If $p\geq\frac{N}{2}$ then, by Sobolev’s inequality, $u\in L^{q}(\mathbb{R}^{N})$ for all $q\in[p,+\infty)$. In particular, $u\in L^{q}(\mathbb{R}^{N})$ for some $q>\frac{N}{2}$ (cf. [2, Corollary 9.13]) and hence $Au=-f+\lambda u\in L^{q}(\mathbb{R}^{N})$. Moreover, since $u\in C^{2+\sigma}_{loc}(\mathbb{R}^{N})$ it follows that $u\in W^{2,q}_{loc}(\mathbb{R}^{N})$. So, $u\in D_{q,max}(A)\subset W^{2,q}(\mathbb{R}^{N})\subset C_{0}(\mathbb{R}^{N})$ by Proposition 4 and Sobolev’s embedding theorem (cf. [2, Corollary 9.13]).

Let us now suppose that $p<\frac{N}{2}$. Take the sequence $(r_{n})$, defined by $r_{n}=1/p-2n/N$ and set $q_{n}=1/r_{n}$ for $n\in\mathbb{N}$. Let $n_{0}$ be the smallest integer such that $r_{n_{0}}\leq 2/N$ noting that $r_{n_{0}}>0$. Then, $u\in D_{p,max}(A)\subset L^{q_{1}}(\mathbb{R}^{N})\cap L^{p}(\mathbb{R}^{N})$, by the Sobolev embedding theorem. As above we obtain that $u\in D_{q_{1},max}(A)\subset L^{q_{2}}(\mathbb{R}^{N})$. Iterating this argument, we deduce that $u\in D_{q_{n_{0}},max}(A)$. So we can conclude that $u\in C_{0}(\mathbb{R}^{N})$ arguing as in the previous case. Thus, $Au=-f+\lambda u\in C_{b}(\mathbb{R}^{N})$. Again, since $u\in C^{2+\sigma}_{loc}(\mathbb{R}^{N})$, it follows that $u\in W^{2,q}_{loc}(\mathbb{R}^{N})$ for any $q\in(1,+\infty)$. Hence, $u\in D_{max}(A)$. So, by the uniqueness of the solution of the elliptic problem, we have $(\lambda-A_{p})^{-1}f=(\lambda-A)^{-1}f$ for every $f\in C_{c}^{\infty}(\mathbb{R}^{N})$. Thus the assertion follows by density.

∎

4. Characterization of the domain and generation of semigroups

The aim of this section is to prove that the operator $A_{p}$ generates an analytic semigroup on $L^{p}(\mathbb{R}^{N})$, for any $p\in(1,\infty)$, provided that $N>2,\,\alpha>2$ and $\beta>\alpha-2$.

We characterize first the domain of the operator $A_{p}$. More precisely we prove that the maximal domain $D_{p,max}(A)$ coincides with the weighted Sobolev space $D_{p}(A)$ defined by

[TABLE]

In the following lemma we give a complete proof of the weighted gradient and second derivative estimates.

Lemma 4**.**

Suppose that $N>2,\,\alpha>2$ and $\beta>\alpha-2$. Then there exists a constant $C>0$ such that for every $u\in D_{p}(A)$ we have

[TABLE]

[TABLE]

Proof.

Let $u\in D_{p}(A)$. We fix $x_{0}\in\mathbb{R}^{N}$ and choose $\vartheta\in C_{c}^{\infty}(\mathbb{R}^{N})$ such that $0\leq\vartheta\leq 1$, $\vartheta(x)=1$ for $x\in B(1)$ and $\vartheta(x)=0$ for $x\in\mathbb{R}^{N}\setminus B(2)$. Moreover, we set $\vartheta_{\rho}(x)=\vartheta\left(\frac{x-x_{0}}{\rho}\right)$, where $\rho=\frac{1}{4}(1+|x_{0}|)$. We apply the well-known interpolation inequality (cf. [6, Theorem 7.27])

[TABLE]

to the function $\vartheta_{\rho}u$ and obtain for every $\varepsilon>0$,

[TABLE]

where $M=\|\nabla\vartheta\|_{\infty}+\|\Delta\vartheta\|_{\infty}$. Since $2\rho=\frac{1}{2}(1+|x_{0}|)$ we get

[TABLE]

Thus,

[TABLE]

Let $\{B(x_{n},\rho(x_{n}))\}$ be a countable covering of $\mathbb{R}^{N}$ as in Proposition 3 such that at most $\zeta$ among the double balls $\{B(x_{n},2\rho(x_{n}))\}$ overlap.

We write (4) with $x_{0}$ replaced by $x_{n}$ and sum over $n$, we obtain

[TABLE]

Choosing $\varepsilon$ such that $\varepsilon C\zeta<\frac{1}{2(1+|b|)}$ we have

[TABLE]

for some constant $C>0$. Furthermore, by (3.19), we know that $\|(1+|x|^{\beta})u\|_{p}\leq C(\|A_{p}u\|_{p}+\|u\|_{p})$ for every $u\in D_{p}(A)\subset D_{p,max}(A)$ and some $C>0$. Hence,

[TABLE]

As regards the second order derivatives we recall the classical Calderón-Zygmund inequality on $B(1)$

[TABLE]

By rescaling and translating we obtain

[TABLE]

for every $x_{0}\in\mathbb{R}^{N}$, $R>0$ and $v\in W^{2,p}(B(x_{0},R))\cap W^{1,p}_{0}(B(x_{0},R))$. We observe that the constant $C$ does not depend on $R$ and $x_{0}$.

Then we fix $x_{0}\in\mathbb{R}^{N}$ and choose $\rho$ and $\vartheta_{\rho}\in C_{c}^{\infty}(\mathbb{R}^{N})$ as above. Applying (4.5) to the function $\vartheta_{\rho}u$ in $B(x_{0},2\rho)$, we obtain

[TABLE]

Arguing as above we obtain

[TABLE]

The lemma follows from (3.19) and (4.1). ∎

The following result shows that $C_{c}^{\infty}(\mathbb{R}^{N})$ is a core for $A_{p}$, since by Lemma 4 the norm (4.7) is equivalent to the graph norm of $A_{p}$. The proof is based on Theorem 1 and Lemma 4 and it is similar to the one given in [3, Lemma 4.3].

Lemma 5**.**

The space $C_{c}^{\infty}(\mathbb{R}^{N})$ is dense in

[TABLE]

endowed with the norm

[TABLE]

Now, we are ready to show the main result of this section:

Theorem 3**.**

Suppose that $N>2,\,\alpha>2$ and $\beta>\alpha-2$. Then the operator $A_{p}$ with domain $D_{p,max}(A)$ generates an analytic semigroup in $L^{p}(\mathbb{R}^{N})$.

Proof.

Let $f\in L^{p}(\mathbb{R}^{N}),\,\rho>0$. Consider the operator $\widehat{A_{p}}:=A_{p}-\omega$, where $\omega$ is a constant which will be chosen later. It is known that the elliptic problem in $L^{p}(B(\rho))$

[TABLE]

admits a unique solution $u_{\rho}$ in $W^{2,p}(B(\rho))\cap W_{0}^{1,p}(B(\rho))$ for $\lambda>0$, (cf. [6, Theorem 9.15]).

Let us prove that $e^{\pm i\theta}\widehat{A_{p}}$ is dissipative in $B(\rho)$ for $0\leq\theta\leq\theta_{\alpha}$ with suitable $\theta_{\alpha}\in(0,\frac{\pi}{2}]$. To this purpose observe that

[TABLE]

Set $u^{*}=\bar{u}_{\rho}|u_{\rho}|^{p-2}$ and recall that $q(x)=1+|x|^{\alpha}$. Multiplying $\widehat{A_{p}}u_{\rho}$ by $u^{*}$ and integrating over $B(\rho)$, we obtain

[TABLE]

We note here that the integration by part in the singular case $1<p<2$ is allowed thanks to [12]. By taking the real part of the left and the right hand side, we have

[TABLE]

Taking now the imaginary part of the left and the right hand side, we obtain

[TABLE]

We can choose $\omega>0$ such that

[TABLE]

Furthermore,

[TABLE]

where $\tilde{c}=\frac{|\alpha-b|(N-2+\alpha)}{p}$ is a positive constant.

Moreover,

[TABLE]

Setting

[TABLE]

As a result of the above estimates, we conclude

[TABLE]

If $\tan\theta_{\alpha}=l_{\alpha}$, then $e^{\pm i\theta}\widehat{A_{p}}$ is dissipative in $B(\rho)$ for $0\leq\theta\leq\theta_{\alpha}$ . From [18, Theorem I.3.9] it follows that the problem (4.8) has a unique solution $u_{\rho}$ for every $\lambda\in\Sigma_{\theta},\,0\leq\theta<\theta_{\alpha}$ where

[TABLE]

Moreover, there exists a constant $C_{\theta}$ which is independent of $\rho$, such that

[TABLE]

Let us now fix $\lambda\in\Sigma_{\theta}$, with $0<\theta<\theta_{\alpha}$ and a radius $r>0$. We apply the interior $L^{p}$ estimates (cf. [6, Theorem 9.11]) to the functions $u_{\rho}$ with $\rho>r+1$. So, by (4.9), we have

[TABLE]

Using a weak compactness and a diagonal argument, we can construct a sequence $(\rho_{n})\to\infty$ such that the functions $(u_{\rho_{n}})$ converge weakly in $W^{2,p}_{loc}$ to a function $u$ which satisfies $\lambda u-\widehat{A_{p}}u=f$ and

[TABLE]

Moreover, $u\in D_{p,max}(A)$. We have now only to show that $\lambda-\widehat{A_{p}}$ is invertible on $D_{p,max}(A)$ for $\lambda_{0}<\lambda\in\Sigma_{\theta}$. Consider the set

[TABLE]

where $C(r):=\{\lambda\in\mathbb{C}:|\lambda|<r\}$. Since, by Theorem 1, $\lambda_{0}$ is in the resolvent set of $\widehat{A_{p}}$, then $R=\sup E>0$. On the other hand, the norm of the resolvent is bounded by $C_{\theta}/|\lambda|$ in $C(R)\cap\Sigma_{\theta}$. Consequently it cannot explode on the boundary of $C(R)$. Then $R=\infty$ and this ends the proof of the theorem. ∎

Let us show that $D_{p,max}(A)$ and $D_{p}(A)$ coincide.

Proposition 5**.**

Assume that $N>2,\,\alpha>2$ and $\beta>\alpha-2$. Then maximal domain $D_{p,max}(A)$ coincides with $D_{p}(A)$.

Proof.

We have to prove only the inclusion $D_{p,max}(A)\subset D_{p}(A)$.

Let $\tilde{u}\in D_{p,max}(A)$ and set $f=\lambda\,\tilde{u}-A_{p}\tilde{u}$. The operator $A$ in $B(\rho)$, $\rho>0$, is an uniformly elliptic operator with bounded coefficients. Then the Dirichlet problem

[TABLE]

admits a unique solution $u_{\rho}$ in $W^{2,p}(B(\rho))\cap W_{0}^{1,p}(B(\rho))$ (cf. [6, Theorem 9.15]). So, $\widetilde{u_{\rho}}$, the zero extension of $u_{\rho}$ to the complement $B(\rho)^{c}$, belongs to $D_{p}(A)$. Thus, by Lemma 4 and (3.19), we have

[TABLE]

with $C$ independent of $\rho$.

We observe that $u_{\rho}$ is the solution of (4.8) with $\lambda$ replaced with $\lambda-\omega$. Then arguing as in the proof of Theorem 3, by (4.9) and (4.10) for $\lambda>\omega$, we have $\|u_{\rho}\|_{L^{p}(B(\rho))}\leq\frac{C_{1}}{\lambda-\omega}\|f\|_{L^{p}}$ and $\|u_{\rho}\|_{W^{2,p}(B(r))}\leq C_{2}\|f\|_{L^{p}}$ where $r<\rho-1$ and $C_{1},C_{2}$ are positive constants which do not depend on $\rho$.

Using a standard weak compactness argument we can construct a sequence $\widetilde{u_{\rho_{n}}}$ which converges to a function $u$ in $W^{2,p}_{loc}(\mathbb{R}^{N})\cap L^{p}(\mathbb{R}^{N})$ such that $\lambda\,u-Au=f$. Since the estimates above are independent of $\rho$, also $u\in D_{p}(A)$. Then $\lambda\tilde{u}-A\tilde{u}=\lambda u-Au$ and since $D_{p}(A)\subset D_{p,max}(A)$ and $\lambda-A$ is invertible on $D_{p,\max}(A)$ by Theorem 1, we have $\tilde{u}=u$. ∎

Proposition 6**.**

For any $f\in L^{p}(\mathbb{R}^{N})$, $1<p<\infty$, and any $0<\nu<1$ and for all $t>0$, the function $T_{p}(t)f$ belongs to $C_{b}^{1+\nu}(\mathbb{R}^{N})$. In particular, the semigroup $(T_{p}(t))_{t\geq 0}$ is ultracontractive.

Proof.

In Theorem 3 we have proved that $A_{p}$ generates an analytic semigroup $T_{p}(\cdot)$ on $L^{p}(\mathbb{R}^{N})$ and in Theorem 2 we have obtained that for $f\in L^{p}(\mathbb{R}^{N})\cap C_{0}(\mathbb{R}^{N})$, $(\lambda-A_{p})^{-1}f=(\lambda-A)^{-1}f$. Hence this shows the coherence of the resolvents on $L^{p}(\mathbb{R}^{N})\cap L^{q}(\mathbb{R}^{N})$ by using a density argument. This will yield immediately that the semigroups are coherent in different $L^{p}$-spaces. One can deduce the result by using the same arguments as in the proof of [9, Proposition 2.6]. ∎

To end this section we study the spectrum of $A_{p}$.

Proposition 7**.**

Assume $N>2$, $\alpha>2$, $\beta>\alpha-2$. Then, for $p\in(1,\infty)$, the resolvent operator $R(\lambda,A_{p})$ is compact in $L^{p}(\mathbb{R}^{N})$ for all $\omega_{0}<\lambda\in\rho(A_{p})$, where $\omega_{0}$ is a suitable positive constant, and the spectrum of $A_{p}$ consists of a sequence of negative real eigenvalues which accumulates at $-\infty$. Moreover, $\sigma(A_{p})$ is independent of $p$.

Proof.

The proof is similar to the one given in [9]. ∎

[TABLE]

**Acknowledgement: ** S-E. Boutiah wishes to thank the Dept. of Information Eng., Electrical Eng. and Applied Mathematics (D.I.E.M) of the University of Salerno for the warm hospitality and a very fruitful and pleasant stay in Salerno, where this paper has been written.