Long Time Decay Estimates in Real Hardy Spaces for the Double Dispersion Equation

Abstract

We study the Cauchy problem for the linear generalized double dispersion equation and derive long time decay estimates for the solution in L ^p spaces and in real Hardy spaces.

Appendix

We recall how the Hardy spaces $\mathcal {H}^p(\mathbb {R}^n)$ are presented by Fefferman and Stein [11]. We use the notation $\mathcal {H}^p$ instead of the classical notation H ^p to avoid possible confusion with the Sobolev space W ^{p, 2}.

Fix, once for all, a radial nonnegative function $\phi \in C^\infty _c(\mathbb {R}^n)$ supported in the unit ball with integral equal to 1. For $u\in \mathcal {S}'(\mathbb {R}^n)$ we define the maximal functionM _ϕu by

$$\displaystyle \begin{aligned} M_\phi u(x)=\sup_{0<t<\infty}|(u\ast\phi_t)(x)|, \end{aligned}$$

where ϕ _t(x) = t ⁻ⁿϕ(x∕t).

Definition 2

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

$$\displaystyle \begin{aligned} \Vert u\Vert_{\mathcal{H}^p}=\Vert M_\phi u\Vert_{L^p}<\infty. \end{aligned}$$

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

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

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

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

The property $f\in \mathcal {H}^p$ can be characterized by appropriate singular integrals in a way that has some analogy with the earlier maximal characterization [14, Theorem C]: a function f ∈ L ² belongs to $\mathcal {H}^p$ when p ∈ (0, 1], if and only if f ∈ L ^p and R _αf ∈ L ^p, for |α|≤ k, where k = 1 + [(n − 1)(1∕p − 1)], and R _αf denotes the Riesz transform of f, defined via the Fourier transform by

$$\displaystyle \begin{aligned} \widehat{R_\alpha f}(\xi)=(i\xi\lvert \xi \rvert^{-1})^\alpha\hat{f}(\xi). \end{aligned}$$

Moreover,

$$\displaystyle \begin{aligned} \|f\|{}_{\mathcal{H}^p} \approx \sum_{|\alpha|\leq k} \|R_\alpha f\|{}_{L^p}. \end{aligned}$$

Another number fixes the order of the moment conditions which the functions in Hardy spaces shall verify. Indeed,

$$\displaystyle \begin{aligned} \int_{\mathbb{R}^n} x^\alpha f(x)\,dx =0, \qquad \forall\,|\alpha|\leq [n(1/p-1)] \end{aligned}$$

for any function $f\in \mathcal {H}^p\cap \mathcal C_c^\infty$.

In Theorem 2 we use a variant of the celebrated Mikhlin-Hörmander multiplier theorem for Hardy spaces (see [13]) to obtain the boundedness of operators acting on $\mathcal {H}^p(\mathbb {R}^n)$.

Definition 3

Let m be a bounded function on $\mathbb {R}^n$ and consider the operator T _m defined by

$$\displaystyle \begin{aligned} T_m f= \mathscr{F}^{-1} \bigl(m(\xi)\hat{f}(\xi)\bigr). \end{aligned} $$

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We say that m is a Fourier multiplier for $\mathcal {H}^p$ if $T_m f\in \mathcal {H}^p$ for all $f\in \mathcal {H}^p$ and

$$\displaystyle \begin{aligned} \Vert T_m f\Vert _{\mathcal{H}^p} \leq C \Vert f\Vert _{\mathcal{H}^p}; \end{aligned} $$

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in other words, if T _m can be extended to a bounded operator from $\mathcal {H}^p$ to $\mathcal {H}^p$.

In this context, $\mathcal {M}(\mathcal {H}^p)$ denotes the set of all the Fourier multipliers for $\mathcal {H}^p$. The norm $\Vert m\Vert _{\mathcal {M}(\mathcal {H}^p)}$ is defined to be the operator norm of T _m in $\mathcal {H}^p$, i.e.

$$\displaystyle \begin{aligned} \Vert m\Vert_{\mathcal{M}(\mathcal{H}^p)}=\sup_{f\in \mathcal{H}^p, f\neq 0}\frac{\Vert T_m f\Vert_{\mathcal{H}^p}}{\Vert f\Vert_{\mathcal{H}^p}}. \end{aligned} $$

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Theorem 3

Let p ∈ (0, 2), and θ = θ(n, p) = n(1∕p − 1∕2). Assume that $m \in C^k(\mathbb {R}^n)$, with m(ξ) = 0 in a neighborhood of the origin, and $k=\max \lbrace [\theta ],[\frac {n}{2}]\rbrace +1$. If

$$\displaystyle \begin{aligned} |\partial_{\xi}^{\gamma} m(\xi)|\leq |\xi|{}^{-a\theta}(A|\xi|{}^{a-1})^{|\gamma|}, \ |\gamma|\leq k, \end{aligned}$$

for some constant a ≥ 0 and A ≥ 1, then $m\in \mathcal {M}(\mathcal {H}^p(\mathbb {R}^n))$and

$$\displaystyle \begin{aligned} \Vert m \Vert _{\mathcal{M}(\mathcal{H}^p(\mathbb{R}^n))}\leq CA^\theta, \end{aligned}$$

where C > 0 is a constant independent of A.

Theorem 4

Let p ∈ (0, 2), and θ = θ(n, p) = n(1∕p − 1∕2). Assume that $m \in C^k(\mathbb {R}^n\setminus \{0\})$, with m(ξ) = 0 for $\lvert \xi \rvert \geq 1$, and $k=\max \lbrace [\theta ],[\frac {n}{2}]\rbrace +1$. If

$$\displaystyle \begin{aligned} |\partial_{\xi}^{\gamma} m(\xi)|\leq |\xi|{}^{a\theta}(A|\xi|{}^{-a-1})^{|\gamma|}, \ |\gamma|\leq k, \end{aligned}$$

for some constant a ≥ 0 and A ≥ 1, then $m\in \mathcal {M}(\mathcal {H}^p(\mathbb {R}^n))$and

$$\displaystyle \begin{aligned} \Vert m \Vert _{\mathcal{M}(\mathcal{H}^p(\mathbb{R}^n))}\leq CA^\theta, \end{aligned}$$

where C is a constant independent of A.

Let I _r be the Riesz potential with order r > 0, defined by means of $I_r f(\xi )=\mathscr {F}^{-1}(|\xi |{ }^{-r}\hat {f}(\xi ))$. If r ∈ (0, n), then there exists c _n,r such that

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

and sufficiently smooth f. Real Hardy spaces have the property that the Hardy-Littlewood-Sobolev theorem for Riesz potential, valid in L ^p spaces, with p > 1, extends to $\mathcal {H}^p$, with p ∈ (0, ∞), see [14, Theorem F].

Theorem 5

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

$$\displaystyle \begin{aligned} \Vert I_r f\Vert_{\mathcal{H}^q(\mathbb{R}^n)}\leq C\Vert f\Vert_{\mathcal{H}^p(\mathbb{R}^n)}, \quad \frac{1}{q}=\frac{1}{p}-\frac{r}{n}. \end{aligned}$$