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.
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Acknowledgements
The results for the linear problem in this contribution are a variant of the one contained in the master thesis of the second author, who has been a student at University of Bari.
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Appendix
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
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.,
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
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
Moreover,
Another number fixes the order of the moment conditions which the functions in Hardy spaces shall verify. Indeed,
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
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
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.
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
for some constant a ≥ 0 and A ≥ 1, then \(m\in \mathcal {M}(\mathcal {H}^p(\mathbb {R}^n))\)and
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
for some constant a ≥ 0 and A ≥ 1, then \(m\in \mathcal {M}(\mathcal {H}^p(\mathbb {R}^n))\)and
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
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
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D’Abbicco, M., De Luca, A. (2020). Long Time Decay Estimates in Real Hardy Spaces for the Double Dispersion Equation. In: Boggiatto, P., et al. Advances in Microlocal and Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36138-9_11
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DOI: https://doi.org/10.1007/978-3-030-36138-9_11
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