Long time decay estimates in real Hardy spaces for evolution equations with structural dissipation

Abstract

In this paper we derive asymptotic-in-time linear estimates in Hardy spaces $H^p(\mathbb {R}^{n})$ for the Cauchy problem for evolution operators with structural dissipation. The obtained estimates are a natural extension of the known $L^p-L^q$ estimates, $1\le p\le q\le \infty$, for these models. Different, standard, tools to work in Hardy spaces, are used to derive optimal estimates.

Appendix: Some results about Hardy spaces

We recall how the Hardy spaces $H^p({\mathbb {R}}^n)$ are presented by Fefferman and Stein in [5]. Fix, once for all, a radial nonnegative function $\varphi \in C_{c}^{\infty }({\mathbb {R}}^n)$ supported in the unit ball with integral equal to 1. For $u\in \mathcal {S}'({\mathbb {R}}^n)$ we define the maximal function $M_{\varphi }u$ by

$$\begin{aligned} M_{\varphi }u(x)=\sup _{0<t<\infty }|(u*\varphi _t)(x)|, \end{aligned}$$

where $\varphi _t(x)=t^{-n}\varphi (x/t)$.

Definition 4

Let $0<p<\infty$. A tempered distribution $u\in \mathcal {S}'({\mathbb {R}}^n)$ belongs to $H^p({\mathbb {R}}^n)$ if and only if $M_{\varphi }u\in L^p({\mathbb {R}}^n)$, i.e.,

$$\begin{aligned} \Vert u\Vert _{H^p}\doteq \Vert M_{\varphi }u\Vert _{L^p}<\infty . \end{aligned}$$

For $p=\infty$, we set $H^\infty ({\mathbb {R}}^n)=L^\infty ({\mathbb {R}}^n)$.

The spaces $H^p({\mathbb {R}}^n)$ are independent of the choice of $\varphi \in C_{c}^{\infty }({\mathbb {R}}^n)$ with $\int _{{\mathbb {R}}^n} \varphi (x)\,dx\ne 0$. For $p=1$, $\Vert u\Vert _{H^1}$ is a norm and $H^1({\mathbb {R}}^n)$ is a normed space densely contained in $L^1({\mathbb {R}}^n)$. For $p>1$, $\Vert u\Vert _{H^p}$ is a norm equivalent to the usual $L^p$ norm and we denote $H^p({\mathbb {R}}^n)=L^p({\mathbb {R}}^n)$, by abusing notation. For $0<p\le 1$, the space $H^p({\mathbb {R}}^n)$ is a complete metric space with the distance

$$\begin{aligned} d(u,v)=\Vert u-v\Vert _{H^p}^p, \quad u,v\in H^p({\mathbb {R}}^{n}). \end{aligned}$$

Although $H^p({\mathbb {R}}^n)$ is not locally convex for $0<p<1$ and $\Vert u\Vert _{H^p}$ is not truly a norm (it is a quasi-norm [25]), we will still refer to $\Vert u\Vert _{H^p}$ as the “norm” of u, as it is customary.

A useful property of Hardy spaces is a pointwise estimate for the Fourier transform of $H^p$ functions, with $p\in (0,1]$ (see Corollary 7.21 in [6]):

$$\begin{aligned} |\hat{f}(\xi )| \lesssim {\left| \xi \right| }^{n\left( \frac{1}{p}-1\right) }\,\Vert f\Vert _{H^p}, \qquad p\in (0,1]. \end{aligned}$$

(77)

Moreover, the following integral estimate holds (see [24], page 128):

$$\begin{aligned} \left( \int _{\mathbb {R}^n} {\left| \xi \right| }^{n\left( p-2\right) }\,|\hat{f}(\xi )|^p\,d\xi \right) ^{\frac{1}{p}}\lesssim \Vert f\Vert _{H^p}, \qquad p\in (0,1]. \end{aligned}$$

(78)

Now let us present some standard tools in Hardy spaces. First we recall the real interpolation due to Fefferman, Riviera and Sagher (see Remark 1, page 66 in [25] for more details).

Theorem 6

(Real interpolation) Let $0<p_{0}<p<p_{1}\le \infty$, and $\theta \in (0,1)$ be such that $p^{-1}\;\doteq \;\theta p^{-1}_{0}+ (1-\theta )p^{-1}_{1}$. Then, there exists $C=C(p_{0},p_{1},\theta ,n)>0$ such that

$$\begin{aligned} \Vert u\Vert _{H^{p}(\mathbb {R}^{n})}\le C\Vert u\Vert ^{\theta }_{H^{p_{0}}(\mathbb {R}^{n})}\Vert u\Vert ^{1-\theta }_{H^{p_{1}}(\mathbb {R}^{n})}. \end{aligned}$$

Let $K \in \mathcal {S}'(\mathbb {R}^{n})$ be a tempered distribution and consider the convolution operator $T:\mathcal {S}(\mathbb {R}^{n}) \rightarrow \mathcal {S}(\mathbb {R}^{n})$ given by $Tf=K *f$. The operator T may also be written as $\widehat{Tf}(\xi )=m(\xi )\widehat{f}(\xi )$ where $m=\widehat{K} \in \mathcal {S}'(\mathbb {R}^{n})$ is the Fourier transform of K. To simplify the notation we rename the operator as $T_{m}$. In this work two important ingredients will be used for the boundedness of operators acting on $H^{p}(\mathbb {R}^{n})$.

The first ingredient is a version of the celebrated Mikhlin-Hörmander multiplier theorem for Hardy spaces (see [14]; see also page 232 in [23]).

Theorem 7

Let $0<p<\infty$ and $k=\max { \left\{ [n(1/p-1/2)]+1,[n/2]+1 \right\} }$. Suppose that $m \in \mathcal C^{k}(\mathbb {R}^{n}\backslash \left\{ 0 \right\} )$ and

$$\begin{aligned} \left| \partial _\xi ^{\beta }m(\xi )\right| \le C\, |\xi |^{-|\beta |}, \quad |\beta |\le k. \end{aligned}$$

Then $T_{m}$ is continuously bounded from $H^{p}(\mathbb {R}^{n})$ into itself.

Variants of this theorem assuming conditions on the support of $m(\xi )$ can be found in [14]. In particular, we recall the following result, which is derived from a special case of Theorem 2 in [14].

Theorem 8

Let $p\in (0,2)$ and $k=\max \left\{ [n(1/p-1/2)]+1,[n/2]+1 \right\}$. Suppose that $m \in \mathcal C^{k}(\mathbb {R}^{n}\backslash \left\{ 0\right\} )$, $m(\xi )=0$ if $|\xi |\ge 1$, and

$$\begin{aligned} \left| \partial _\xi ^{\beta }m(\xi )\right| \le C\,(A|\xi |^{-1})^{|\beta |}, \quad |\beta |\le k, \end{aligned}$$

with some constant $A\ge 1$. Then $T_{m}$ is bounded from $H^{p}(\mathbb {R}^{n})$ into itself and

$$\begin{aligned} \Vert T_{m}f\Vert _{H^{p}(\mathbb {R}^{n})}\le C\, A^{n \left( \frac{1}{p}-\frac{1}{2} \right) }\Vert f\Vert _{H^{p}(\mathbb {R}^{n})}. \end{aligned}$$

We conclude this appendix recalling an important version of $L^{p}-L^{q}$ estimates for fractional integration in the context of Hardy spaces (page 135 in [24]). Let $I_r$ be the Riesz potential with order $r>0$, defined by means of $I_rf(\xi )\doteq \mathfrak {F}^{-1}(|\xi |^{-r}\mathfrak F f(\xi ))$. We notice that $I_r(I_s f)=I_{r+s}f$. If $r\in (0,n)$, the Riesz potential may be represented for sufficiently smooth f by

$$\begin{aligned} I_r f (x) = c_{n,r} \int _{\mathbb {R}^n}\frac{f(y)}{|x-y|^{n-r}}\,dy, \end{aligned}$$

for suitable $c_{n,r}$.

Theorem 9

Consider $r>0$ and $0<p<n/r$. Then, there exists $C=C(r,p)>0$ such that

$$\begin{aligned} \Vert I_{r}f\Vert _{H^{q}(\mathbb {R}^{n})}\le C \Vert f\Vert _{H^{p}(\mathbb {R}^{n})}, \qquad \frac{1}{q}=\frac{1}{p}-\frac{r}{n}\,. \end{aligned}$$