Abstract
Consider the metric cone \(X=C(Y)=(0,\infty )_r\times Y\) with metric \(g=dr^2+r^2h\) where the cross section Y is a compact \((n-1)\)-dimensional Riemannian manifold (Y, h). Let \(\Delta _g\) be the positive Friedrichs extension Laplacian on X and let \(\Delta _h\) be the positive Laplacian on Y, and consider the operator \({\mathcal {L}}_V=\Delta _g+V_0 r^{-2}\) where \(V_0\in {\mathcal {C}}^\infty (Y)\) such that \(\Delta _h+V_0+(n-2)^2/4\) is a strictly positive operator on \(L^2(Y)\). In this paper, we prove global-in-time Strichartz estimates without loss regularity for the wave equation associated with the operator \({\mathcal {L}}_V\). It verifies a conjecture in Wang (Remark 2.4 in Ann Inst Fourier 56:1903–1945, 2006) for wave equation. The range of the admissible pair is sharp and the range is influenced by the smallest eigenvalue of \(\Delta _h+V_0+(n-2)^2/4\). To prove the result, we show a Sobolev inequality and a boundedness of a generalized Riesz transform in this setting. In addition, as an application, we study the well-posed theory and scattering theory for energy-critical wave equation with small data on this setting of dimension \(n\ge 3\).
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1 Introduction and main result
1.1 Background: the setting and Strichartz estimate
Suppose that (Y, h) is a compact \((n-1)\)-dimensional Riemannian manifold, the metric cone \(X=C(Y)\) on Y is the space \((0,\infty )_r\times Y\) with \(g=dr^2+r^2h\). The metric cone X has a simplest geometry singularity and it has incomplete metric. One can complete it to \(C^*(Y)=C(Y)\cup P\) where P is its cone tip. Denote \(\Delta _g\) the Friedrichs extension of Laplace-Beltrami from the domain \({\mathcal {C}}_c^\infty (X^\circ )\), compactly supported smooth functions on the interior of the metric cone. There is a number of works to extend the theory of the Laplace operator \(\Delta _g\) on smooth manifolds to certain Riemannian spaces with such conical singularities; for example the spectral theory, see Cheeger [13, 14].
Solutions to the wave equation on cones and related spaces were studied from the perspective of wave diffraction from the cone point, see [20, 21, 57]. In the setting of exact cones, Cheeger and Taylor [15, 16] studied the Laplacian from points of the functional calculus. Melrose and Wunsch [44] proved a propagation of singularities property for solutions to the wave equation on the more general setting of conic manifolds. In addition, the other aspects of Schrödinger operator on the metric cone, even with potentials that are homogeneous of degree \(-2\), also have been extensively studied. For instance, the asymptotical behavior of Schrödinger propagator was considered in [62]. The heat kernel was studied in [43, 47] and Riesz transform kernel was investigated in [30, 42]. The \(L^p\)-estimates were studied in [41] and the restriction estimate for Schrödinger solution was studied by the first author [64]. The Strichartz estimates for Schrödinger were proved on a flat cone by Ford [19], on polygonal domains by Blair–Ford–Herr–Marzuola [6], on exterior polygonal domains by Baskin–Marzuola–Wunsch [4], and on the metric cone by the authors [66]. Regarding the Strichartz estimate for wave equation on cones, Blair–Ford–Marzuola [7] have established the Strichartz inequalities on a flat cone of dimension two, that is, \(Y=\mathbb {S}^1_\rho \). However, one needs the explicit form of wave propagator when \(Y=\mathbb {S}^1_\rho \) in [7], hence the methods of [7] can not be applied to our general setting.
In this paper, we prove the Strichartz estimates for the solution to wave equations on metric cone and, as an application, we study the well-posed theory and scattering theory for the energy-critical nonlinear wave equation. Here we recall the Schrödinger operator \(\mathcal {L}_V=\Delta _g+V\) where \(V=V_0(y) r^{-2}\) and \(V_0(y)\) is a smooth function on the section Y. Our motivation to study this Schrödinger operator is to understand the regularity or singularity of wave propagates in a singular black hole. For example, we refer to [18, 52] for the connection with Schwarzchild black hole or [48, 63] for the Reissner–Nordstöm black hole. With respect to the potential, since the decay of the inverse-square potential is closely related to the angular momentum as \(r\rightarrow \infty \), we are known that inverse square decay of the potential is in some sense critical for the spectral and scattering theory. In context of this paper, we remark here that the inverse-square type potential is homogeneous of degree \(-2\) and is at the boardline of decay in order to guarantee validity of Strichartz estimate; see Goldberg–Vega–Visciglia [25]. The property of the inverse-square type potential near the cone tip, or near infinity-end, or both, brings the singular behavior, however, it is a natural potential. For example [11], the Dirac equation with a Coulomb potential can be recast in the form of a Klein–Gordon equation with an inverse-square type potential.
Consider the solution \(u: I\times X\rightarrow \mathbb {R}\) to the initial value problem (IVP) for the wave equation on metric cone X,
It is well-known that the Strichartz estimate implies the decay and regularity of the solutions to the wave equations, and plays an important role in the studying of nonlinear wave equations. More precisely, let u be the solution to (1.1) and the time interval \(I\subseteq \mathbb {R}\), the Strichartz estimate states an inequality in the form of
where \(\dot{H}^s={\mathcal {L}}_V^{-\frac{s}{2}} L^2(X)\) denotes the homogeneous \(L^2\)-Sobolev space over X and the pairs \((q,\mathrm {r}), (\tilde{q},\tilde{\mathrm {r}})\in [2,\infty ]\times [2,\infty )\) satisfy the wave-admissible condition
and the scaling condition
For \(s\in \mathbb {R}\), we say the pair \((q,\mathrm {r})\in \Lambda _s\) if \((q,\mathrm {r})\in [2,\infty ]\times [2,\infty )\) satisfies (1.3) and (1.4).
Due to the importance of the Strichartz inequalities, there is a lot of work studying the Strichartz inequalities on Euclidean space or manifolds; for example, see [29, 38, 53, 54] and references therein. In the following, we in particular focus on recalling the most relevant work about the Strichartz estimate on a metric cone, or on a slightly different setting of asymptotically conic manifold, or with a perturbation of inverse-square type potentials. Our setting metric cone is close to the asymptotically conic manifold M which, outside some compact set, is isometric to the conical space X away the cone tip. On the non-trapping asymptotically conic manifold M, for Schrödinger equation, Hassell, Tao and Wunsch [31, 32] and Mizutani [46] showed the local-in-time Strichartz estimates; the global-in-time Strichartz inequality including endpoint case was proved by Hassell and the first author in [35] for Schrödinger and in [65] for wave equation; and very recently Bouclet and Mizutani [8] and the authors [67] showed the global-in-time Strichartz estimates on asymptotically conic manifold even with a hyperbolic trapped geodesic. As remarked above, the perturbation of the inverse-square potential is nontrivial. In [9, 10], the additional perturbation of the inverse-square potential was taken into account when they studied the Euclidean standard Strichartz estimate for Schrödinger and wave, which is a tough task. On a flat cone of dimension 2, Blair–Ford–Marzuola [7] have established the Strichartz inequalities for wave by developing a representation of fundamental solution to the wave equation on the flat cone \(C(\mathbb {S}^1_\rho )\) which is also applied to the Schrödinger case in [19].
1.2 Main result and the sketch of proof
In our present general setting, we need to consider the propagator of the dispersive equation associated with the operator \({\mathcal {L}}_V\) which is influenced by the geometry and the inverse-square type potential. The authors [66] proved the full range of global-in-time Strichartz estimates for Schrödinger equation associated with the operator \(\mathcal {L}_V\) which proved Wang’s conjecture [62, Remark 2.4] for Schrödinger.
In this vein (as in [66]), we intend to prove the global-in-time Strichartz estimate for wave equation associated with the operator \(\mathcal {L}_V\), but with innovative aspects to combat difficulties arising from wave equation. More precisely, we prove the following results.
Theorem 1.1
(Global-in-time Strichartz estimate). Assume that (X, g) is a metric cone of dimension \(n\ge 3\). Let \({\mathcal {L}}_V=\Delta _g+V\) where \(r^2V=:V_0(y)\in {\mathcal {C}}^\infty (Y)\) such that \(\Delta _h+V_0(y)+(n-2)^2/4\) is a strictly positive operator on \(L^2(Y)\) and its smallest eigenvalue is \(\nu _0^2\) with \(\nu _0>0\). Suppose that u is the solution of the Cauchy problem (1.1) with the initial data \(u_0\in \dot{H}^{s}, u_1\in \dot{H}^{s-1}\) for \(s\in \mathbb {R}\).
- (i)
If \(V\equiv 0\), then the Strichartz estimate (1.2) holds for all \((q,\mathrm {r})\), \((\tilde{q}, \tilde{\mathrm {r}})\in \Lambda _s\).
- (ii)
If \(V\not \equiv 0\), then the Strichartz estimate (1.2) holds for all \((q,\mathrm {r})\), \((\tilde{q}, \tilde{\mathrm {r}})\in \Lambda _{s,\nu _0}\) where
$$\begin{aligned} \Lambda _{s,\nu _0}=\{(q,\mathrm {r})\in \Lambda _s: 1/\mathrm {r}>1/2-(1+\nu _0)/n \}. \end{aligned}$$(1.5)
Remark 1.1
From the first result \(\mathrm {(i)}\), the geometry of metric cone, possibly having conjugated points, does not influence the Strichartz estimate even though the conjugated points cause the failure of dispersive estimate. From the restriction (1.5), the Strichartz estimate is affected by the positive square root of the smallest eigenvalue of \(\Delta _h+V_0(y)+(n-2)^2/4\). The requirement (1.5) is sharp, see Sect. 6.3.
Remark 1.2
The set \(\Lambda _{s,\nu _0}\) makes sense when \(s\in [0,1+\nu _0)\) otherwise it is empty. Compared with \(\Lambda _s\), one can check that \(\Lambda _{s,\nu _0}=\Lambda _s\) for \(s\in [0,1/2+\nu _0)\), and while \(\Lambda _{s,\nu _0}\subset \Lambda _s\) for \(s\in [1/2+\nu _0, 1+\nu _0)\). In particular, \(V\equiv 0\), hence \(\nu _0> (n-2)/2\) large enough so that \(\Lambda _{s,\nu _0}=\Lambda _s\) for \(s\in \mathbb {R}\), thus the second conclusion is consistent with the first one.
Remark 1.3
If \(\nu _0>\frac{1}{n-1}\), the Strichartz estimates hold for \((q,\mathrm {r})\) such that \((\frac{1}{q},\frac{1}{\mathrm {r}})\) belongs to the region ABCEF when \(n\ge 4\) and ABO when \(n=3\). Compared with the Euclidean case, the Strichartz estimate fails in the region CDOE of Fig. 1. If \(0<\nu _0<\frac{1}{n-1}\), then the line EC is above the line FB, we do not have the Strichartz estimate with \(q=2\). The result illustrates that the smallest eigenvalue of \(\Delta _h+V_0(y)+(n-2)^2/4\) plays an important role in the Strichartz estimate.
Diagrammatic picture of the range of (q, r), when \(\nu _0 > 1/(n - 1)\)
Remark 1.4
The restriction \(1/\mathrm {r}>1/2-(1+\nu _0)/n\) is also necessary for Schrödinger by a similar counterexample constructed in Sect. 6.3. The reason for disappearance of this restriction in the Strichartz estimate of Schrödinger established by the authors [66, Theorem 1.1] is that we only consider the estimate at regularity level \(s=0\). This is same to the case here for wave \(\Lambda _{s,\nu _0}=\Lambda _s\) for \(s\in [0,1/2+\nu _0)\) in which the restriction \(1/\mathrm {r}>1/2-(1+\nu _0)/n\) disappears. The argument for wave needs more techniques on the Sobolev inequality and Riesz transform.
Remark 1.5
Compared with the result involving the derivatives [9, Theorem 9], the result in Theorem 1.1 needs to consider the influences of conical singular geometry and the potential \(V_0(y)r^{-2}\) (rather than \(V_0(y)\equiv c\)).
We sketch the idea and argument of the proof here. The usual method to derive the Strichartz estimate is Keel–Tao’s [38] abstract method in which we need dispersive estimate and \(L^2\)-estimate. In our setting, however, there are two difficulties to prevent us from obtaining the dispersive estimate. The first one arises from the conjugated points from the geometry, and the second one from the inverse-square potential. First, the degeneration of projection between the conjugated points will slow down the dispersive decay estimate of the Schrödinger or wave propagator, which was illustrated in [34, 35]. Second, as discussed in [35, 66] for Schrödinger, it is not possible to obtain a dispersive estimate for half wave operator \(e^{i(t-s)\sqrt{{\mathcal {L}}_V}}\) with norm \(O(|t-s|^{-\frac{n-1}{2}})\) as \(|t-s|\rightarrow \infty \) due to the influence of the negative inverse-square potential; see [9, 10] for the perturbation of inverse-square potential on Euclidean space.
There are two key points, which have been established and used in [67] for Schrödinger equation, to treat the two issues. The first one is to microlocalize the propagator which separates the conjugated points. We achieve this through studying the property of the micro-localized spectral measure associated with the operator \({\mathcal {L}}_0\), i.e. without potential. The second key one is to establish the global-in-time local smoothing estimate which is proved via a variable separating argument.
More precisely, we first show (i) in Theorem 1.1 in which we do not need to consider the potential. To obtain the Strichartz estimate for \({\mathcal {L}}_0\), as in [35, 65, 67], our strategy is Keel–Tao’s abstract method where we need the property of the micro-localized spectral measure to prove dispersive estimate and \(L^2\)-estimate. However, we should modify the argument to adapt to wave equation and sharpen the Strichartz estimate in a Lorentz space. Compared with the Schrödinger, the wave propagator multiplier \(e^{it\lambda }\) is less oscillation than the Schrödinger’s \(e^{it\lambda ^2}\), thus we need a Littlewood-Paley theory in our setting, in particular the Littlewood–Paley square function inequality on Lorentz space. The key is to show a Mikhlin–Hörmander multiplier theorem. We notice that our setting is a measure space in which the wave operator has finite propagation speed and one has doubling condition. Thus, from Chen–Ouhabaz–Sikora–Yan [12], the multiplier estimate on \(L^p\) is a consequence of a spectral measure estimate which can be obtained from the property of micro-localized spectral measure and \(TT^*\)-method. The Littlewood–Paley (LP) square function inequality on Lorentz space is finally obtained from the interpolation characteristic of Lorentz space. Once the LP square function estimate has been established, we may assume that the initial data is frequency localized in \(\{\lambda \sim 2^k\}\). The argument [65] can be modified to prove the Strichartz estimate. We remark that the property of microlocalized spectral measure capturing the figures of the decay and oscillation behavior which plays an important role in this part.
Next we show (ii) in Theorem 1.1. We use a perturbation method [36, 51] to derive (ii) from a local smoothing and the results of (i). The usual way to show a local smoothing estimate is through establishing the resolvent estimate for \({\mathcal {L}}_V\) at low and high frequency. Unlike the usual way, we avoid the resolvent estimate to show the global-in-time local smoothing estimate by using the explicit formulas with separating variables expression. In addition, in particular for obtaining Strichartz estimate at \(q=2\), we need a double end-points inhomogeneous Strichartz estimate for \({\mathcal {L}}_0\) which is not proved in (i). To this end, we modify an argument in [35] to adapt to wave equation. Another difference between wave and Schrödinger should not be ignored, that is, wave’s double end-points inhomogeneous Strichartz estimate involves some negative derivative. This requires us to study the \(L^p\)-boundedness theory of a generalized Riesz transform operator \(\Delta _g^{s/2}{\mathcal {L}}_V^{-s/2}\). It is worth mentioning that the method of [9] is based on the fact the potential \(V_0(y)r^{-2}=cr^{-2}\) independent of y, and the method can not be directly used for the potential in our setting. To obtain our result, we have to resort to the harmonic analysis tools, such as the Sobolev inequality, associated with \({\mathcal {L}}_V\) which are established in our preliminary sections.
1.3 Application: energy-critical wave equation
As an application of the global-in-time Strichartz estimates, we study the nonlinear wave equation on X of dimension \(n\ge 3\)
where \(\dot{H}^1(X)={\mathcal {L}}_V^{-\frac{1}{2}} L^2(X)\) is the homogeneous Sobolev space over X and \(\gamma =\pm 1\) which corresponds to the defocusing and focusing case respectively. Notice that our metric cone X is invariant under the dilation variable change, hence our equation model has symmetries of time translation and scaling dilation but not translation invariant in space. The class of solutions to (1.6) is invariant by the scaling
One can check that the only homogeneous \(L^2\)-based Sobolev space \(\dot{H}^{1}(X)\times L^2(X)\) is invariant under (1.7). The rescaling also remains invariant for the energy of solutions defined by
which is a conserved quantity for (1.6) and where \(d\mu =\sqrt{|g|}dz=r^{n-1}dr dh\). Hence the Cauchy problem (1.6) falls in the class of energy-critical problem. Because of the conserved quantities at the critical regularity, the energy-critical equations have been the most extensively studied instances of NLW. In the Euclidean space, that is \(X=\mathbb {R}^n\) and \(V=0\), for the defocusing energy-critical NLW, it has been known now that the solutions that are global and scatter when the initial data is in \(\dot{H}^1\times L^2\) which could be arbitrarily large, see Grillakis [24], Kapitanski [37], Shatah and Struwe[55], Bahouri and Gérard [3], Tao [59] and the references therein. For the focusing energy-critical NLW in dimensions \( n\in \{3,4, 5\}\), Kenig and Merle [39] obtained the dichotomy-type result under the assumption that \(E (u_0, u_1) < E (W, 0)\), where W denotes the ground state of an nonlinear elliptic equation. From this, it is not an easy thing to study the global existence and scattering theory of the initial value problem with large data in \(\dot{H}^1\times L^2\) even though in the Euclidean space.
In this paper, as an application of Strichartz estimate, we study the global existence and scattering for the Cauchy problem (1.6) with initial data in \(\dot{H}^1\times L^2\) but small enough. Our result for the energy-critical wave equation is the following.
Theorem 1.2
Let X be metric cone of dimension \(n\ge 3\) and \({\mathcal {L}}_V=\Delta _g+V\) as in Theorem 1.1. Let \(\gamma =\pm 1\) and suppose that the initial data \((u_0, u_1)\in \dot{H}^1(X)\times L^2(X)\). Assume the above \(\nu _0>1/2\). Then there exists \(T=T(\Vert (u_0, u_1)\Vert _{H^1(X)\times L^2(X)})>0\) such that the energy-critical Eq. (1.6) is local wellposed in \(I=[0,T)\) and the unique solution u obeys
where
In addition, if \(\Vert (u_0, u_1)\Vert _{\dot{H}^1(X)\times L^2(X)}\le \delta \) for a small enough constant \(\delta \), there exists a global solution u and the solution u scatters in the sense that there are \((u_0^{\pm }, u_1^\pm )\in \dot{H}^1(X)\times L^2(X)\) such that
where
Remark 1.6
This result is similar to the well known result for energy-critical wave equation in Euclidean space and the global existence and scattering theory for small data still holds on the metric cone manifold. Like the Euclidean result, this small initial result is also a cornerstone result for future work with large data on this setting. The assumption on \(\nu _0>1/2\) guaranteeing that the Strichartz estimate holds for all \((q,\mathrm {r})\in \Lambda _s\) with \(s=1\) can be improved, we do not pursue this here.
We prove this result by using Picard iteration argument which was used in Euclidean space, see Tao’s book [60]. The key ingredient is the global-in-time Strichartz estimate in Theorem 1.1.
Finally, we introduce some notation. We use \(A\lesssim B\) to denote \(A\le CB\) for some large constant C which may vary from line to line and depend on various parameters, and similarly we use \(A\ll B\) to denote \(A\le C^{-1} B\). We employ \(A\sim B\) when \(A\lesssim B\lesssim A\). If the constant C depends on a special parameter other than the above, we shall denote it explicitly by subscripts. For instance, \(C_\epsilon \) should be understood as a positive constant not only depending on p, q, n, and M, but also on \(\epsilon \). Throughout this paper, pairs of conjugate indices are written as \(p, p'\), where \(\frac{1}{p}+\frac{1}{p'}=1\) with \(1\le p\le \infty \). We denote \(a_\pm \) to be any quantity of the form \(a\pm \epsilon \) for any small \(\epsilon >0\).
This paper is organized as follows: In Sect. 2, we recall and prove some analysis results such as the spectral measure and the Littlewood–Paley theory in our setting. Section 3 is devoted to the Sobolev inequality and a generalized Riesz transform. In Sect. 4, we prove our main Theorem 1.1 on Strichartz esimate for wave with \({\mathcal {L}}_0\). We prove a double endpoint inhomogeneous Strichartz estimate in Sect. 5. In the section 6, we show a local smoothing estimate and prove Theorem 1.1 for wave with \({\mathcal {L}}_V\). We construct a counterexample to show the sharpness. In the final section, we utilize the Strichartz estimates to show Theorem 1.2.
2 Some analysis results associated with the operator \({\mathcal {L}}_V\)
This paper is devoted to the wave equation associated with the operator \({\mathcal {L}}_V\), hence we need extra harmonic analysis tools which are influenced by the geometry of the cone X and the potential V, even though some ones have been established in previous work [64, 66]. The purpose of this section is to show and recall the analysis tools for usage in the following sections.
2.1 Basic harmonic analysis tools and notation on the metric cone
Recall that the metric cone \(X=C(Y)=(0,\infty )_r\times Y\) is equipped with the metric \(g=dr^2+r^2h\) and the cross section Y is a compact \((n-1)\)-dimensional Riemannian manifold (Y, h). Let \(z=(r, y)\in \mathbb {R}_+\times Y\), then the measure on C(Y) is
For \(1\le p<\infty \), define the \(L^p(X)\) space by the complement of \(\mathcal {C}_0^\infty (X)\) under the norm
Let d (resp. \(d_Y\)) be the distance function on \(X=C(Y)\) (resp. Y) then, for instance see [15], the distance on a metric cone is
with \(z=(r,y)\) and \(z'=(r',y').\) Furthermore, about the distance function, we refer the reader to Li [43, Proposition 1.3, Lemma 3.1] for the following results.
Lemma 2.1
There exist constants c and C such that the following property of the distance function holds
and
Let \(y'\in Y\) and define the ball \(B_Y(y',\delta )=\{y\in Y: d_Y(y',y)\le \delta \}\) and \(z'\in X\) and the ball \(B(z',r)=\{z\in X: d(z',z)\le r\}\). Then there exists C such that
As a consequence, we first have
Lemma 2.2
For \(0<\alpha <1\) and let \(z=(r,y)\), there exists C such that
Proof
By a direct computation and Lemma 2.1, we have
\(\square \)
Next we recall the Hardy–Littlewood–Sobolev inequality in [43, Corollary 1.4], and we provide an alternative argument by using the above lemma.
Proposition 2.1
(Hardy–Littlewood–Sobolev). Let \(0<\alpha <n\), for any function \(f(z)\in L^p(X)\), let
Then, for any \(1<p<q<+\infty \) satisfying
there exists a constant \(A_{p,q}>0\) such that
Proof
From the classical Marcinkiewicz interpolation theorem, we only need to show that there is a constat \(C>0\) such that for any \(\lambda >0\)
For any \(\gamma >0\), define
and
Thus, for any \(\tau >0\),
Without loss of generality, assume that \(\Vert f\Vert _{L^p}=1\). By Hölder’s inequality, we get
where we use that \(\mu \big (B\big ((s_*,m_*),r\big )\big )\sim r^n \) in Lemma 2.1. Choose \(\gamma \) so that \(C_1\gamma ^{-\frac{n}{q}}=\tau \), then
On the other hand, we will show that
Then, we have by Chebyshev’s inequality [27]
Thus, (2.7) follows if let \(\lambda =2\tau \). Now we prove (2.9). Recalling \(z=(r,y),\;z'=(r',y')\), and using the compactness of Y and the Hölder inequality, we obtain
\(\square \)
2.2 Lorentz spaces
In this subsection, we recall the well-known Lorentz space and some properties of this space for our purpose. Let \((X,\mu )\) be a \(\sigma \)-finite measure space and \(f: X\rightarrow \mathbb {R}\) be a measurable function. Define the distribution function of f as
and its rearrangement function as
For \(1\le p<\infty \) and \(1\le r\le \infty \), define the Lorentz quasi-norm
The Lorentz space \(L^{p,r}(X,\mu )\) denotes the space of complex-valued measurable functions f on X such that its quasi-norm \(\Vert f\Vert _{L^{p,r}(X)}\) is finite. From this characterization, \(L^{p,\infty }(X)\) is the usual weak \(L^p\) space, \(L^{p,p}(X)=L^p(X)\) and \(L^{p,r}(X)\subset L^{p,\tilde{r}}(X)\) with \(r<\tilde{r}\).
We refer to [27] for the following properties of Lorentz space. The first one is the Hölder inequality due to O’Neil [49].
Proposition 2.2
(Hölder’s inequality in Lorentz space). Let \(1\le p, p_0, p_1<\infty \) and \(1\le r, r_0, r_1\le \infty \), then
The second one is the duality of the Lorentz space.
Proposition 2.3
(The dual of Lorentz space). The dual of the Lorentz space \((L^{p,r}(X))^*=L^{p', r'}(X)\).
It is more convenient to use their characterization as real interpolates of Lebesgue spaces. We refer to [5]. Suppose that \(B_0\) and \(B_1\) are two Banach spaces which are continuously embedded into a common topological vector space \(\mathcal {V}\), for \(\theta \in (0,1)\) and \(r\in [1,\infty ]\), the real interpolation space \([B_0, B_1]_{\theta ,r}\) consists of the elements \(f\in \mathcal {V}\) which can be written as \(f=\sum \limits _{j\in {\mathbb {Z}}}f_j\) such that \(f_j\in B_0\cap B_1\), \(\{2^{-j\theta }\Vert f_j\Vert _{B_0}\}_j\in \ell ^{r}({\mathbb {Z}})\) and \(\{2^{j(1-\theta )}\Vert f_j\Vert _{B_1}\}_j\in \ell ^{r}({\mathbb {Z}})\). Actually the space is equipped with the norm
We have the following from [5, Theorem 5.3.1]
Proposition 2.4
Let \(1\le p, p_0, p_1<\infty \) and \(1\le r, r_0, r_1\le \infty \), then
- (1)
if \(p_0\ne p_1\), we have
$$\begin{aligned}{}[L^{p_0}, L^{p_1}]_{\theta ,r}=[L^{p_0,r_0}, L^{p_1,r_1}]_{\theta ,r}=L^{p,r}, \quad \frac{1}{p}=\frac{1-\theta }{p_0}+\frac{\theta }{p_1}, ~~1\le r\le \infty ; \end{aligned}$$(2.11) - (2)
if \(p_0=p_1=p\), we have
$$\begin{aligned}{}[L^{p,r_0}, L^{p,r_1}]_{\theta ,r}=L^{p,r}, \quad \frac{1}{r}=\frac{1-\theta }{r_0}+\frac{\theta }{r_1}. \end{aligned}$$(2.12)
2.3 The spectral measure
We first use the separation of variable method to analyze the spectral measure \(dE_{\sqrt{{\mathcal {L}}_V}}\). In this part, we obtain an explicit expression of half wave operator in terms of series of eigenfunctions which allows us to study a local smoothing estimate but not the dispersive estimate. Next, in the case \(V\equiv 0\), we recall an integral expression for a microlocalized spectral measure based on our previous result [66]. This allows us to obtain the dispersive estimate for a microlocalized half-wave operator.
Our operator is
Let \(\Delta _h\) be the positive Laplace–Beltrami operator on (Y, h), we suppose that \(V_0\) is a smooth function on Y such that
is strictly positive on \(L^2(Y)\) in sense that for any \(f\in L^2(Y)\backslash \{0\}\)
Define the set \(\chi _\infty \) to be
For \(\nu \in \chi _\infty \), let \(d(\nu )\) be the multiplicity of \(\lambda _\nu =\nu ^2-\frac{1}{4}(n-2)^2\) as eigenvalue of \(\widetilde{\Delta }_h:=\Delta _h+V_0(y)\). Let \(\{\varphi _{\nu ,\ell }(y)\}_{1\le \ell \le d(\nu )}\) be the eigenfunctions of \(\widetilde{\Delta }_h\), that is
We can decompose \(L^2(Y)\) into
where \(\mathcal {H}^{\nu }=\text {span}\{\varphi _{\nu ,1},\ldots , \varphi _{\nu ,d(\nu )}\}\). Define the orthogonal projection \(\pi _{\nu }\) on \(f\in L^2(X)\)
For any \(f\in L^2(X)\), we can write f in the form of separation of variable
and furthermore
Note that the Riemannian metric h on Y is independent of r , we can use the separation of variable method [15] to write \(\mathcal {L}_V\) in the coordinate (r, y) as
Let \(\nu >-\frac{1}{2}\) and \(r>0\) and define the Bessel function of order \(\nu \) by
Lemma 2.3
Let \(J_\nu (r)\) be the Bessel function defined in (2.20) and \(R\gg 1\), then there exists a constant C independent of \(\nu \) and R such that
and
Proof
The first one is obtained by a direct computation. The inequality (2.22) is a direct consequence of the asymptotically behavior of Bessel function; see [45, Lemma 2.2]. \(\square \)
Let \(f\in L^2(X)\), define the Hankel transform of order \(\nu \) by
On the space \(\mathcal {H}^{\nu }\), we see
Briefly recalling functional calculus on cones [58], for well-behaved functions F, we have by (8.45) in [58]
where \(b_{\nu ,\ell }(\rho )=(\mathcal {H}_{\nu }a_{\nu ,\ell })(\rho )\) with \(g(r,y)=\sum \limits _{\nu \in \chi _\infty }\sum \limits _{\ell =1}^{d(\nu )}a_{\nu ,\ell }(r)~\varphi _{\nu ,\ell }(y)\).
For \(u_0\in L^2(X)\), we write it in the form of separation of variables by (2.17)
therefore we can write the half-wave operator by using (2.25) with \(F(\rho )=e^{it\rho }\)
where \(b_{\nu ,\ell }(\rho )=(\mathcal {H}_{\nu }a_{\nu ,\ell })(\rho )\).
Although we have the expression of the half-wave operator, it is not easy to obtain a dispersive estimate due to the complicated asymptotic behavior of the Bessel function. In our previous paper [66], we derived a microlocalized dispersive estimate for Schrödinger from a micro-localized spectral measure expression associated with \({\mathcal {L}}_0\). The result about the micro-localized spectral measure is an analogue of [35, Proposition 1.5] on the asymptotically conic setting. We record the result on the spectral measure below for convenience.
Proposition 2.5
(Proposition 3.1 [66]). Let (X, g) be metric cone manifold and \({\mathcal {L}}_0=\Delta _g\). Then there exists a \(\lambda \)-dependent operator partition of unity on \(L^2(X)\)
with N independent of \(\lambda \), such that for each \(1 \le j \le N\) we can write
and \(0\le j'\le N\)
with estimates
and
Here \(d(\cdot , \cdot )\) is the distance on X.
2.4 The Littlewood–Paley square function inequality
As a usual reduction to prove Strichartz estimate for wave equation, we may assume the initial data \(u_0\) and \(u_1\) are frequency localized in an annulus \(\{\lambda \sim 2^k\}\) by using a Littlewood–Paley square function inequality. To this end, we prove the Littlewood–Paley square function inequality associated with the positive Laplacian \({\mathcal {L}}_0=\Delta _g\) on metric cone. In [43], Li has proved the Gaussian boundedness of heat kernel of \({\mathcal {L}}_0\). One can follow the argument in [6, 65] to obtain an appropriate Mikhlin–Hörmander multiplier theorem from a spectral multiplier theorem of Alexopolous [1] and the heat kernel estimate and then to prove the Littlewood–Paley inequality. Here we provide an alternative method to show the Littlewood–Paley inequality. The method is based on an estimate on the spectral measure rather than the heat kernel.
Now we state the Littlewood–Paley square function estimate. Let \(\varphi \in C_0^\infty (\mathbb {R}{\setminus }\{0\})\) take values in [0, 1] and be supported in [1 / 2, 2] such that
Proposition 2.6
Let (X, g) be a metric cone of dimensional \(n\ge 3\) as above, and suppose that \({\mathcal {L}}_0=\Delta _g\) is the Laplace–Beltrami operator on (X, g). Then for \(1<p<\infty \), there exist constants \(c_p\) and \(C_p\) depending on p such that
and
Remark 2.1
In this result, we do not consider the influence of the inverse-square potential \(V=V_0(y)r^{-2}\). We remark that the inverse-square type potential plays an important role in the range of p when the potential is negative, for example [40, Theorem 5.3].
Proof
To prove the Littlewood–Paley square function inequality (2.33), one can follow Stein’s [56] classical argument (in \(\mathbb {R}^n\)) involving Rademacher functions and an appropriate Mikhlin–Hörmander multiplier theorem in the following Lemma 2.4. For more details, we refer the reader to [6, 56].
Lemma 2.4
Let \(m\in C^N(\mathbb {R})\) satisfy the Mikhlin-type condition for \(N\ge \frac{n}{2}+1\)
Then for all \(1<p<\infty \)
is a bounded operator where
Then the inequality (2.34) follows from the general Marcinkiewicz interpolation theorem [5, Theorem 5.3.2] and dual argument. Indeed, define the quadratic functional operator for \(f\in L^p(X)\)
then the operator \(\mathcal {G}_{{\mathcal {L}}_0}\) is sublinear and is bounded on \(L^{1+}\) and \(L^{\infty -}\) respectively. Therefore, using the general Marcinkiewicz interpolation theorem [5, Theorem 5.3.2], the operator \(\mathcal {G}_{{\mathcal {L}}_0}\) is bounded on \(L^{p,r}(X)\) for all \(1<p<\infty \) and \(0<r\le \infty \), hence the case \(r=2\) shows the second inequality in (2.34). The other side can be obtained by dual argument. \(\square \)
Now our main task here is to show Lemma 2.4.
Proof
We adopt the argument which are in spirit of [28] and [12]. We first prove the spectral measure estimate by using the \(TT^*\) argument as given in [28]
By Proposition 2.5, it is easy to see that
Let \(P(\lambda )\) be the Poisson operator associated with \({\mathcal {L}}_0\), then \(dE_{\sqrt{{\mathcal {L}}_0}}(\lambda )=(2\pi )^{-1}P(\lambda )\)\(P(\lambda )^*\) as shown in [33]. By using \(TT^*\) argument again, it follows that
Note that the partition of unity \(Id=\sum _{j=0}^N Q_j\) in Proposition 2.5, therefore we have
By \(TT^*\) argument again, we show (2.37).
From [15, formula (0.13)], it follows the finite propagation speed of solutions to \((\partial _t^2+{\mathcal {L}}_0)u=0\). Hence the operator \({\mathcal {L}}_0\) satisfies the finite speed propagation property. By (2.37) and using [12, Propositions 2.4, 9.1 and Theorems 4.1, 5.1], we have that \(m(\sqrt{{\mathcal {L}}_0})\) is bounded on \(L^p(X)\) for all \(1<p<\infty \). \(\square \)
3 Sobolev inequality and a generalized Riesz transform
For our purpose, we consider the Sobolev space, Sobolev inequality and a boundedness of generalized Riesz transform associated with \({\mathcal {L}}_V\) in this section. Recall the notation \(z=(r,y)\) and \(z'=(r',y')\).
For \(s\in \mathbb {R}\), the operator \({\mathcal {L}}_V^{\frac{s}{2}}\) is defined by
where \(dE_{\sqrt{{\mathcal {L}}_V}}\) is the spectral measure of the operator \(\sqrt{{\mathcal {L}}_V}\).
Definition 3.1
(Sobolev space). For \(1\le p<\infty \) and \(s\in \mathbb {R}\), we define the homogeneous Sobolev space \(\dot{H}^s_p(X):={\mathcal {L}}_V^{-\frac{s}{2}} L^p(X)\) over \(L^p(X)\) which consists of the functions f such that \({\mathcal {L}}_V^{s/2}f\in L^p(X)\). In particular \(p=2\), define \(\dot{H}^s(X):=\dot{H}^s_2(X)={\mathcal {L}}_V^{-\frac{s}{2}} L^2(X)\).
Remark 3.1
For all general \(1\le p<\infty \), due to the influence of the inverse-square potential, the Sobolev norm defined here is not equivalent to the analogue one defined by the operator \({\mathcal {L}}_0\) without the potential. For example, we refer the reader to [40] for the Euclidean Laplacian with the inverse-square potential. But for \(p=2\), the two norms are equivalent.
The equivalent of the two Sobolev spaces is closely related to a topics about the boundedness of the generalized Riesz transform operator
and its reverse operator \({\mathcal {L}}_V^{\frac{s}{2}}\Delta _g^{-\frac{s}{2}}\). In [40], the authors studied the equivalent norms in which we replace \({\mathcal {L}}_V\) by \(\Delta +a r^{-2}\) in Euclidean space by starting from its heat kernel estimate. However, as far as we know, there is no result about heat kernel estimate of \({\mathcal {L}}_V\), even though Li [43] proved the heat kernel estimate for \(\Delta _g\) on metric cone. Rather than from heat kernel, we study the problem from the asymptotical behavior of the resolvent \(({\mathcal {L}}_V+1)^{-1}(z,z') \); see [30, Theorem 4.11, Lemma 5.4].
When \(0<s<n\), we can define the operator \(\mathcal L_V^{-\frac{s}{2}}\) by the Riesz potential kernel
Before stating the main results of this section, we show the estimates on the kernels
Lemma 3.1
Let \(Q(z,z')\) and \(G(z,z')\) be the kernels of the operators \(\mathcal {L}_V^{-\frac{s}{2}}\) and \(\nabla _g\Delta _g^{\frac{s-1}{2}}\) respectively. Assume \(0<s<2\), then \(Q(z,z')\) satisfies
and if \(0<s\le 1\) then \(G(z,z')\) satisfies
where \(\nu _0\) (resp. \(\nu _0'\)) is the square root of the smallest eigenvalue of the operator \(\Delta _h+V_0(y)+(n-2)^2/4\) (resp. \(\Delta _h+(n-2)^2/4\)).
Remark 3.2
Note that \(\nu _0'\ge (n-2)/2\), as mentioned in [30, Remark 4.13], one can improve (3.4) through replacing \(\nu _0'\) by \(\nu _1'\), the square root of the second smallest eigenvalue of \(\Delta _h+(n-2)^2/4\).
Proof
We first estimate \(Q(z,z')\). Let \(\chi \,[0,\infty )\rightarrow [0,1]\) be a smooth cutoff function such that \(\chi ([0,1/2])=1\) and \(\chi ([1,\infty ))=0\). Define
and
Since \(\mathcal {L}_V\) is homogeneous of degree \(-2\), then by scaling we have
Now we consider the boundedness of \(Q_1\). By [30, Theorem 4.11], for any \(N>0\), we have
Therefore, for any \(N>1-s\) and \(s<2\), we have by (3.9)
Similarly we consider the boundedness of \(Q_2\). By [30, Theorem 4.11] again, we have for any \(N>0\)
Therefore similarly as estimating (3.9), for \(s<2\), we have
Finally we estimate \(Q_0\). Recall [30, Lemma 5.4], for any \(N>0\), we have
Therefore, we compute that by using \(d(\lambda z, \lambda z')=\lambda d(z,z')\)
We estimate the kernel \(Q_0(z,z')\) for \(s<2\) and \(N>n-s\)
We need a modification to prove (3.3) due to the support of \(\chi \). For instance, from \(Q_1\), we directly see that \(Q(z,z')\lesssim r'^{-n}r^{s}(r/r')^{1-\frac{n}{2}+\nu _0-s}\) when \(r<r'/8\). On the region \(r'/8\le r\le r'/2\), since \(r'/2\le |r-r'|\le d(z,z')\le r+r'\lesssim r' \) thus \(d(z,z')\sim r\sim r'\). Therefore we prove the boundedness of Q on \( r\le r'/2\). We also can prove the boundedness on \( r\ge 2r'\) through the same modification argument. Hence we prove (3.3).
We next estimate G. Notice that the derivative \(\nabla _g\) is of the form that \(r^{-1}\) times a smooth b-derivative for small r, and is a smooth scattering vector field for r large; we refer the reader to [30] for the b-derivative and scattering vector field. Since \(0<s<1\), we can replace the s (resp. \(\nu _0\)) by \(1-s\) (resp. \(\nu _0'\)) to obtain the estimate of the kernel \(\Delta _g^{(s-1)/2}\). Therefore, we finally obtain the estimate of G by multiplying \(r^{-1}\), thus we prove (3.4). \(\square \)
Lemma 3.2
Let \(0\le s<n,\; 1\le p,q\le +\infty \). Let \(K(r,r',y,y')\) be a kernel on the cone X. Define the operator
If
and \(\alpha +\beta =n-s, \beta >0\), then
with
Similarly, if
and \(\alpha +\beta =n-s, \alpha >0\), then (3.13) holds for
Remark 3.3
In particular \(s=0\), then \(q=p\). This special result has been proved in [30, Corollary 5.9]. Here, we extend such result to \(q\ge p.\)
Proof
We use the argument of [30, Corollary 5.9]. Noting that \(d\mu =r^{n-1}dr\;dh\) and the section Y is a compact set, we get
where
and
Perform a substitution \(\tilde{r}=\ln r,\; \tilde{r}'=\ln r'\), then
with
Then, it is easy to see that
with \(\big (\tfrac{n}{q}-\alpha \big )\sigma >0\) guaranteed by (3.14). Especially, taking \(\tfrac{1}{\sigma }=1+\tfrac{1}{q}-\tfrac{1}{p}\ge 1,\) and using generalised Young’s inequality, we obtain
Similarly, we obtain the other case. Hence, Lemma 3.2 follows. \(\square \)
We prove the following Sobolev inequality which is well-known in the Euclidean space.
Proposition 3.1
(Sobolev inequality for \({\mathcal {L}}_V\)). Let \(n\ge 3\) and \(\nu _0\) be as above. Suppose \(0<s<2\), and \(1<p,q<\infty \). Then
holds for \(s=\tfrac{n}{p}-\tfrac{n}{q}\) and
Proof
The proof follows from Lemma 3.1. The estimate (3.20) is equivalent to
where the operator \(T=\mathcal L_V^{-\frac{s}{2}}\) is defined by the Riesz potential kernel
By using Lemma 3.1, we have for \(0<s<2\)
Then by using(3.21), we obtain Proposition 3.1 from Lemma 3.2 when \(r<r'/2\) and \(r>2r'\) and from the Hardy–Littlewood–Sobolev inequality in Proposition 2.1 when \(r\sim r'\). \(\square \)
Corollary 3.1
(Sobolev inequality for \({\mathcal {L}}_V\)). If \(q\ge 2\) and \(p\ge 2\) satisfying (3.21), the above result holds for \(s>0\).
Remark 3.4
The restriction on s is \(0<s<1+\nu _0\). Indeed, from the facts \(p\ge 2\) and \(2\le q<n/\max \{\frac{n}{2}-1-\nu _0, 0\}\), it follows \(s=\frac{n}{p}-\frac{n}{q}<1+\nu _0\).
Proof
Choose \(\{s_j\}_{j=0}^k\) with \(s_0=0, s_k=s\) such that \(0<s_{j+1}-s_{j}<1\) and and \(\{q_j\}_{j=0}^k\) with \(q_0=q, q_k=p\) such that \(2\le q_j<n/\max \{\frac{n}{2}-1-\nu _0,0\}\) for \(j=0,\cdots k-1\). Thus, we apply Proposition 2.1 to obtain
Therefore, we show
\(\square \)
In the rest of this subsection, we consider the boundedness of the operator
When \(s=1\), the boundedness of this operator has been established by Lin–Hassell [30]. For the following purpose of the establishment of Strichartz estimate, we need the following result
Proposition 3.2
Let \(n\ge 4\) and suppose that \(s=\frac{2}{n-1}\) and \(\nu _0\) is in above such that \(\nu _0>\frac{1}{n-1}\). Then the operator
is bounded.
Before proving this proposition, we show
Lemma 3.3
The following inequality holds for \(q\in (1,\infty )\)
Proof
Indeed partial result is a dual consequence of Riesz transform boundedness. More precisely, Lin–Hassell [30, Theorem 1.1], [42] has shown
is bounded for \(p\in (1, n/\max \{\frac{n}{2}-\nu '_1,0\})\) where \(\nu '_1>0\) is the square root of the second smallest eigenvalue of the operator \(\Delta _h+(n-2)^2/4\). If \(Y=\mathbb {S}^{n-1}\), then \(\nu _1'>\frac{n}{2}\) since the kth eigenvalue of \(\Delta _{\mathbb {S}^{n-1}}\) is \(k(k+n-2)\). However, for the general Y, \(\nu '_1>(n-2)/2\), one has the boundedness for \(p\in (1,n)\) at least. By the dual argument [2], we have
for all \(q\in (n/(n-1),\infty )\). On the other hand, one can use the method in [23] to show the following Poincaré inequalities for \(p=1\)
where \(B=B(z,r)\) and \(f_B=\frac{1}{\mu (B)}\int _B f d\mu \). A result in [2, Theorem 0.7] claimed that the doubling condition and Poincaré inequality implies the reverse Riesz transform boundedness. Hence (3.25) holds for \(q\in (1,\infty )\) which also was stated in [2, Page 535] for our setting. \(\square \)
Now we prove Proposition 3.2. Write \(\Delta _g^{\frac{s}{2}}=\Delta _g^{\frac{1}{2}}\Delta _g^{\frac{s-1}{2}}\), by using (3.25), it suffices to establish
Let \(G(z,z')\) denote the kernel of the operator \(\nabla _g\Delta _g^{\frac{s-1}{2}}\) and \(Q(z',z'')\) denote the kernel of the operator \({\mathcal {L}}_V^{-\frac{s}{2}}\). Recall \(z=(r,y)\), therefore by using Lemma 3.1, the kernel \(G(z,z')\) and \(Q(z',z'')\) satisfy
and
Define the operator
where the kernel \(K(z,z'')\) is given by
To prove Proposition 3.2, it suffices to show
Proposition 3.3
For \(0<s<1\), there exists a constant C
provided
We postpone the proof for a moment. Note \(\nu _0>1/(n-1)\), \(\nu _0'=(n-2)/2\) and \(s=2/(n-1)\), the \(p=2(n-1)/(n+1)\) satisfies the condition (3.32), hence it proves Proposition 3.2 once we have shown this proposition.
The proof of Proposition 3.3
We divide the kernel \(K(z,z'')\) into several cases.
Case1:\(2r\le \frac{r''}{2}\). A simple computation shows
The estimate of\(K_{1,1}(z,z'')\): In this case, since \(r'<\frac{r}{2}<\frac{r''}{2}\), we have
and
Hence, we get
Thus, an application of Lemma 3.2 yields the \(L^p\)-boundedness for \(K_{1,1}(z,z'')\).
The estimate of\(K_{1,2}(z,z'')\): In this case, since \(\frac{r}{2}<r'<2r<\frac{r''}{2}\), we have
and
Hence, by Lemma 2.2, we obtain
Hence, by using Lemma 3.2 again, we obtain the \(L^p\)-boundedness for \(K_{1,2}(z,z'')\).
The estimate of\(K_{1,3}(z, z'')\): We can further decompose
We first consider \(K_{1,31}(z,z'')\). In this case, we have \(2r<r'<\frac{r''}{2}.\) Thus,
and
This implies
Thus, an application of Lemma 3.2 yields the \(L^p\)-boundedness for \(K_{1,31}(z,z'')\).
Next consider \(K_{1,32}(z,z'')\). In this term, we have \(2r<\frac{r''}{2}<r'<2r''.\) Thus,
and
This implies
Finally, we consider \(K_{1,33}(z,z'')\). In this case, we have \(2r<2r''<r'.\) Thus,
and
Since \(\nu _0'>(n-2)/2\), this implies
Therefore, by using Lemma 3.2, we obtain the boundedness of \(K_{1,3}\). In sum, in the case \(2r<\frac{r''}{2}\), we prove \(K(z,z'')\) is bounded as an operator on \(L^p(X)\) provided
Case 2\(\frac{r}{2}>2r''\). We decompose
The estimate of\(K_{2,1}\): In this region, we have \(r'<\frac{r''}{2}<\frac{r}{2}.\) And so
and
Hence, we get
The estimate of\(K_{2,2}\): In this region, we have \(\frac{r''}{2}<r'<2r''<\frac{r}{2}.\) And so
and
Hence, we get
The estimate of\(K_{2,3}\): We further decompose
The contribution of\(K_{2,31}\): In this region, we have \(2r''<r'<\frac{r}{2}.\) And so
and
Hence, we get
The contribution of\(K_{2,32}\): In this region, we have \(2r''<\frac{r}{2}<r'<2r.\) And so
and
Hence, we get
The contribution of\(K_{2,33}\): In this region, we have \(2r''<2r<r'.\) And so
and
Hence, we get
Overall, in the case \(\frac{r}{2}>2r\), by using Lemma 3.2, we show \(K(z,z'')\) is bounded as an operator on \(L^p(X)\) provided
Case 3\(\frac{r''}{4}\le r\le 4r''\). We decompose
The estimate of\(K_{3,1}\): In this region, we have \(r'<\frac{r}{2}\le 2r''.\) If \(r'\ge \frac{r''}{2}\), then one has \(r\sim r'\sim r''\) which can be done as treating \(K_{3,2}\). Hence we only consider \(r'<\frac{r''}{2}\), and so
and
Hence, we get
When \(r\sim r''\), it is easy to prove that
The estimate of\(K_{3,2}\): In this region, we have \(r'\sim r\sim r''.\) And so
and
Therefore, we prove
Similarly, we can prove \(\int _{r\sim r''}K_{3,2}(z,z'') d\mu (z)\lesssim 1\).
The estimate of\(K_{3,3}\): In this region, we have \(r'>2r\ge \frac{r''}{2}.\) Similarly, we only consider \(r'>2r''\). And so
and
Hence, we get
Note that \(r\sim r''\), it is easy to prove that
To conclude, in the case that \(r\sim r''\), by using Schur test lemma, we prove \(K(z,z'')\) is bounded on \(L^p(X)\) for all \(1<p<\infty \). Collecting all the cases, therefore we finish the proof of Proposition 3.3. \(\square \)
4 Strichartz estimates for wave equation with \({\mathcal {L}}_0\)
In this section, we prove the Strichartz estimates for wave equation associated with \({\mathcal {L}}_0\), i.e. without potential, that is, the result (i) of Theorem 1.1 when \(V=0\). The argument here is close to [35, 65] but with necessary modifications. For the sake of being self-contained and convenient, we sketch the main steps.
4.1 Microlocalized propagator
We begin to decompose the half-wave propagator by using the partition of unity \(1=\sum \nolimits _{k\in {\mathbb {Z}}}\varphi (2^{-k}\lambda )\) as in (2.32). Define
We further microlocalize (in phase space) the half-wave propagators adapting to the partition of unity operator
where \(Q_j(\lambda )\) is as in Proposition 2.5. Then the operator \(U_{j,k}(t) U_{j,k}(s)^*\) is given
4.2 \(L^2\)-estimate and dispersive estimate
In this subsection, we prove the two key estimates, i.e. the energy estimate and dispersive estimate. Before stating our result, we recall two results in [35]. The results can be directly applied to our setting if we consider the problems on the region away from the cone tip, in which as mentioned in the introduction they almost are the same. Recall that \(Q_j\) with \(j\ge 1\) are micro-localized away from the cone tip.
By using [35, Lemma 8.2] (see also [26, Lemmas 5.3 and 5.4]), we can divide \((j,j')\), \(1 \le j,j' \le N\) into three classes
so that
if \((j,j') \in J_{near}\), then \(Q_j(\lambda ) dE_{\sqrt{{\mathcal {L}}_\mathbf {0}}}(\lambda )Q_{j'}(\lambda )^*\) satisfies the conclusions of Proposition 2.5;
if \((j,j') \in J_{non-inc}\), then \(Q_j(\lambda )\) is not incoming-related to \(Q_{j'}(\lambda )\) in the sense that no point in the operator wavefront set (microlocal support) of \(Q_j(\lambda )\) is related to a point in the operator wavefront set of \(Q_{j'}(\lambda )\) by backward bicharacteristic flow;
if \((j,j') \in J_{non-out}\), then \(Q_j(\lambda )\) is not outgoing-related to \(Q_{j'}(\lambda )\) in the sense that no point in the operator wavefront set of \(Q_j(\lambda )\) is related to a point in the operator wavefront set of \(Q_{j'}(\lambda )\) by forward bicharacteristic flow.
And we further exploit the not-incoming or not-outgoing property of \(Q_j(\lambda )\) with respect to \(Q_{j'}(\lambda )\) to obtain the Schwartz kernel of \(Q_j(\lambda ) dE_{\sqrt{{\mathcal {L}}_\mathbf {0}}}(\lambda )Q_{j'}(\lambda )^*\)
Lemma 4.1
For \(\lambda > 0\) and \((j,j') \in J_{non-out}\). Then, we can write the Schwartz kernel of \(Q_j(\lambda ) dE_{\sqrt{{\mathcal {L}}_\mathbf {0}}}(\lambda )Q_{j'}(\lambda )^*\) as a multiple of \(|dg dg'|^{1/2} |d\lambda |\) as the sum of a finite number of terms of the form
in the region \(\sigma = r'/r \le 2\), \(r \ge \delta \), or
in the region \(r \le \delta , r' \le \delta \), where in each case, \(\Phi< - \epsilon < 0\) and a is a smooth function compactly supported in the v and s variables (where present), such that \(|(\lambda \partial _\lambda )^N a|\le C_N\). In each case, we may assume that \(k \le n-1\); if \(k=0\) in (4.4) or (4.6), or \(k=1\) in (4.5) then there is no variable v, and no v-integral. Again, the key point is that in each expression, the phase function is strictly negative.
If, instead, \(Q_j\) is not incoming-related to \(Q_{j'}\), then the same conclusion holds with the reversed sign: the Schwartz kernel can be written as a finite sum of terms with a strictly positive phase function.
Remark 4.1
For \(\sigma \ge 1/2\), the Schwartz kernel has a similar description, as follows immediately from the symmetry of the kernel under interchanging the left and right variables.
Proof
Note \((j,j')\in J_{non-out}\), thus \(j,j'\ge 1\). Since \(j,j'\ge 1\) away from cone tip, this result is essentially proved in [35, Lemma 8.3, Lemma 8.5]. Since our setting has scaling symmetry, we do not need to state the result in high and low frequency respectively. The key point is that the sign of the phase function can be determined. \(\square \)
The main results of this subsection are the \(L^2\)-estimate and dispersive estimates.
Proposition 4.1
Let \(U_{j,k}(t)\) be defined in (4.2). Then there exists a constant C independent of \(t, z, z'\) for all \(j,j'\ge 0, k\in {\mathbb {Z}}\) such that
and the following dispersive estimates on \(U_{j,k}(t) U_{j',k}(s)^*\) hold:
If \((j,j') \in J_{near}\) or \((j,j')=(0,j'), (j,0)\), then for all \(t \ne s\) we have
$$\begin{aligned} \big \Vert U_{j,k}(t)U^*_{j',k}(s)\big \Vert _{L^1\rightarrow L^\infty }\le C 2^{k(n+1)/2}(2^{-k}+|t-s|)^{-(n-1)/2}, \end{aligned}$$(4.8)If \((j,j')\) such that \(Q_j\) is not outgoing related to \(Q_{j'}\), and \(t<s\), then
$$\begin{aligned} \big \Vert U_{j,k}(t)U^*_{j',k}(s)\big \Vert _{L^1\rightarrow L^\infty }\le C 2^{k(n+1)/2}(2^{-k}+|t-s|)^{-(n-1)/2}, \end{aligned}$$(4.9)Similarly, if \((j,j')\) such that \(Q_j\) is not incoming related to \(Q_{j'}\), and \(s<t\), then
$$\begin{aligned} \big \Vert U_{j,k}(t)U^*_{j',k}(s)\big \Vert _{L^1\rightarrow L^\infty }\le C 2^{k(n+1)/2}(2^{-k}+|t-s|)^{-(n-1)/2}. \end{aligned}$$(4.10)
Remark 4.2
The dispersive inequalities (4.9) and (4.10) are used to prove endpoint to endpoint inhomogeneous Strichartz estimate; see Sect. 6.
Proof
The inequalities (4.7) and (4.8) are essentially proved [65, Section 3]. Indeed, note that the operators \(\varphi (2^{-k}\lambda )\) and \(Q_j(\lambda )\) are bounded on \(L^2\), thus the microlocalized propagator \(U_{j,k}(t)\) is bounded from \(L^2(X)\) to itself due to the spectral theory on Hilbert space. From above result, if \((j,j') \in J_{near}\) or \((j,j')=(0,j'), (j,0)\), we have the expression of microlocalized spectral mearsue in Proposition 2.5 which is same as the one used in [65]. Then by the stationary phase argument, we have
where \(\varphi \in C_c^\infty ([1/2, 2])\) and takes value in [0, 1]. We refer the reader to [65, Section 3] for details.
We only prove (4.10) since the argument to prove (4.9) is analogous. Assume that \(Q_j\) is not incoming-related to \(Q_{j'}\), and then consider (4.10). By [35, Lemma 5.3], \(U_{j,k}(t) U_{j',k}(s)^*\) is given by
Then we need to show that for \(s<t\) and \(k\in {\mathbb {Z}}\)
By scaling, it suffices to show \(k=0\), that is
If \(t-s<1\), since \(\tilde{\varphi }\) is compactly supported, the estimate follows from the uniform boundedness of (4.4)–(4.6). Now we consider \(t-s \ge 1\). Let \(\phi \in C_c^\infty ([\frac{1}{2},2])\) be such that \(\sum _m\phi (2^{-m}(t-s)\lambda )=1\), define
Plug the decomposition
into the integral (4.13). In addition, we substitute for \(Q_j(\lambda ) dE_{\sqrt{{\mathcal {L}}_0}}(\lambda )Q^*_{j'}(\lambda )\) one of the expressions in Lemma 4.1 to obtain
Hence it implies (4.13) since \(|t-s|>1\).
For \(m \ge 1\), we substitute again one of the expressions in Lemma 4.1. Since the other cases follow from the similar argument, we only consider the expression (4.6). Define \(\bar{\lambda }= (t-s)\lambda \), we obtain by scaling
We observe that the overall exponential factor is invariant under the differential operator
Note that its adjoint is \(L^t = -L\), we apply \(L^N\) to the exponential factors, and integrate by parts N times. Since \(\Phi \ge 0\) according to Lemma 4.1, and since we have an estimate \(|(\bar{\lambda }\partial _{\bar{\lambda }} )^N (\tilde{\varphi } a)| \le C_N\), we gain a factor \(\bar{\lambda }^{-1} \sim 2^{-m}\) each time, thus we estimate for \(t-s>1\)
Hence we prove (4.13) by summing over \(m\ge 0\), thus (4.10) follows. \(\square \)
4.3 Abstract Stirchartz estimate on Lorentz space
To prove the Strichartz estimate, we sharpen the semiclassical version of Strichartz estimates [65, Proposition 4.1] to Lorentz space \(L^{r,2}\) by following abstract Keel–Tao’s Strichartz estimates theorem.
Proposition 4.2
Let \((X,\mathcal {M},\mu )\) be a \(\sigma \)-finite measured space and \(U: \mathbb {R}\rightarrow B(L^2(X,\mathcal {M},\mu ))\) be a weakly measurable map satisfying, for some constants C, \(\alpha \ge 0\), \(\sigma , h>0\),
Then for every pair \(q,\mathrm {r}\in [2,\infty ]\) such that \((q,\mathrm {r},\sigma )\ne (2,\infty ,1)\) and
there exists a constant \(\tilde{C}\) only depending on C, \(\sigma \), q and r such that
where \(\Lambda (h)=h^{-(\alpha +\sigma )(\frac{1}{2}-\frac{1}{\mathrm {r}})+\frac{1}{q}}\).
Proof
For convenience, we write down the proof by repeating the argument in [65] but with minor modification of the interpolation. If \((q,\mathrm {r},\sigma )\ne (2,\infty ,1)\) is on the line \(\frac{1}{q}+\frac{\sigma }{\mathrm {r}}=\frac{\sigma }{2}\), we replace \((|t-s|+h)^{-\sigma }\) by \(|t-s|^{-\sigma }\) and then we closely follow Keel–Tao’s argument [38, Sections 3–7] to show (4.16). We remark here that the alternative interpolation argument in [38, Section 6] shows the inequalities sharpened to Lorentz space. So we only consider \(\frac{1}{q}+\frac{\sigma }{\mathrm {r}}<\frac{\sigma }{2}\). By the \(TT^*\) argument, it suffices to show
Using the bilinear interpolation of (4.15) in [38, Lemma 6.1], we have
Therefore, we see by Hölder’s and Young’s inequalities for \(\frac{1}{q}+\frac{\sigma }{\mathrm {r}}<\frac{\sigma }{2}\)
This proves (4.16). \(\square \)
4.4 Homogeneous Strichartz estimates
Now we show the homogeneous Strichartz estimate. Let u solve
then for \(q,\mathrm {r}\ge 2\), the square function estimates (2.34) and Minkowski’s inequality show that
where \(u_k\) is defined by
where \(\varphi \) is as in (2.32). Applying the operator \(\varphi (2^{-k}\sqrt{{\mathcal {L}}_0})\) to the wave equation, we obtain
where \(f_k=\varphi (2^{-k}\sqrt{{\mathcal {L}}_0})u_0\) and \(g_k=\varphi (2^{-k}\sqrt{{\mathcal {L}}_0})u_1\). Let \(U(t)=e^{it\sqrt{{\mathcal {L}}_0}}\), then we write
For our purpose, we need the following
Proposition 4.3
Let \(f=\varphi (2^{-k}\sqrt{{\mathcal {L}}_0})f\) for \(k\in {\mathbb {Z}}\) and \(U(t)=e^{it\sqrt{{\mathcal {L}}_0}}\), we have
where the admissible pair \((q,\mathrm {r})\in [2,\infty ]^2\) satisfies (1.3) and \(s=n(\frac{1}{2}-\frac{1}{\mathrm {r}})-\frac{1}{q}\).
Proof
Let \(\alpha =(n+1)/2\), \(\sigma =(n-1)/2\) and \(h=2^{-k}\), by Proposition 4.1, we have the estimates (4.15) for \(U_{j,k}(t)\). Then it follows from Proposition 4.2 that
On the other hand, we have
Let \(\widetilde{\varphi } \in C_0^\infty (\mathbb {R}{\setminus }\{0\})\) take values in [0, 1] such that \(\widetilde{\varphi }\varphi =\varphi \), hence we can write
Notice \(f=\varphi (2^{-k}\sqrt{{\mathcal {L}}_0})f\), then \(\widetilde{\varphi }(2^{-k'}\sqrt{{\mathcal {L}}_0})f\) vanishes if \(|k-k'|\gg 1\). Hence we obtain
Therefore, we prove this proposition. \(\square \)
By (4.18) and (6.2) and (4.22), we have that
By Littlewood-Paley theory again (2.34), we prove
4.5 Inhomogeneous Strichartz estimates
In this subsection, we derive the inhomogeneous Strichartz estimate from the homogeneous Strichartz estimate by using Christ–Kiselev lemma [17]. Recall the half-wave operator \(U(t)=e^{it\sqrt{{\mathcal {L}}_0}}: L^2\rightarrow L^2\) and in last subsection we have just proved that
holds for all \((q,\mathrm {r},s)\) satisfying (1.3) and (1.4). Given \(s\in \mathbb {R}\) and \((q,r)\in \Lambda _s\), define the operator \(\mathbf{T}_s\) by
By the dual of Lorentz space in Proposition 2.3, we have
where \(1-s=n(\frac{1}{2}-\frac{1}{\tilde{\mathrm {r}}})-\frac{1}{\tilde{q}}\). It shows that
Note that \(s=n(\frac{1}{2}-\frac{1}{\mathrm {r}})-\frac{1}{q}\) and \(1-s=n(\frac{1}{2}-\frac{1}{\tilde{\mathrm {r}}})-\frac{1}{\tilde{q}}\), thus \((q,\mathrm {r}), (\tilde{q},\tilde{\mathrm {r}})\) satisfy (1.4). By the Christ-Kiselev lemma [17], we thus obtain for \(q>\tilde{q}'\),
Notice that for all \((q,\mathrm {r}), (\tilde{q},\tilde{\mathrm {r}})\in \Lambda _s\), one must have \(q>\tilde{q}'\).
Therefore, we conclude that:
Proposition 4.4
For any \(s\in \mathbb {R}\), let \((q,\mathrm {r}), (\tilde{q},\tilde{\mathrm {r}})\in \Lambda _s\) and let u be the solution to
the following Strichartz estimates hold:
Remark 4.3
This result concludes the full range set of global-in-time Strichartz estimates both in homogenous and inhomogeneous inequalities when \(V=0\). Hence, by embedding inequality of Lorentz space, we prove Theorem 1.1 when \(V=0\).
5 Inhomogeneous Strichartz estimates with \(q=\tilde{q}=2\)
In the next section, we need the following result on the double endpoint inhomogeneous Strichartz estimate.
Proposition 5.1
Let \(\mathrm {r}={2(n-1)}/{(n-3)}\) and \(F=\varphi (2^{-k}\sqrt{{\mathcal {L}}_0})F\), we have the following inequality
As a consequence, we have
Corollary 5.1
Let \(\mathrm {r}={2(n-1)}/{(n-3)}\), the following inequality holds
Proof
This is a consequence of the Littlewood–Paley theory in Lemma 2.6. \(\square \)
Remark 5.1
This inhomogeneous inequality is not included in the above estimate(4.28) since if \(q=\tilde{q}=2\), then at least, one of \((q,\mathrm {r}), (\tilde{q},\tilde{\mathrm {r}})\) is not in \(\Lambda _s\). However, if we only consider the inhomogeneous Strichartz estimate, we can obtain this endpoint estimate (5.2) by following the argument of [38] and [35], although at this moment we only have the microlocalized dispersive estimates (4.8)–(4.10). For more inhomogeneous estimates, we refer the reader to [22, 61] where the propagator satisfies the classical dispersive estimate.
Proof
Recall \(U(t)=e^{it\sqrt{{\mathcal {L}}_0}}\), then
Hence to show (5.2), it suffices to show the bilinear form estimate
where \(\mathrm {r}=2(n-1)/(n-3)\) and \(T_k(F,G)\) is the bilinear form
where \(U_k=\sum _{0\le j\le N} U_{j,k}\) defined in (4.1).
On the other hand, we have proved that for all \((q,\mathrm {r})\in \Lambda _s\) with \(s=n(\frac{1}{2}-\frac{1}{\mathrm {r}})-\frac{1}{q}\)
By duality, we have
In particular \(q=2, \mathrm {r}=\frac{2(n-1)}{n-3}\), it follows that for all \(0\le j,j'\le N\),
We need the following bilinear estimates
Lemma 5.1
Let \(U_{j,k}(t)\) be defined as in (4.2), then for each pair \((j,j')\in \{ 0, 1, \dots , N \}^2 \) there exists a constant C such that, for each k, either
or
We postpone the proof for a moment. Hence for every pair \((j,j')\), we have by (5.6) or subtracting (5.7) from (5.5)
Finally by summing over all j and \(j'\), we obtain (5.3). Once we prove Lemma 5.1, we complete the proof of Proposition 5.1. \(\square \)
Proof of Lemma 5.1
Without loss of generality, by scaling argument, we may assume \(k=0\). In the case that \((j,j') \in J_{near}\) or \((j,j')=(j,0)\) or \((j,j')=(0,j')\), we have the dispersive estimate (4.8). We apply the argument of [38, Sections 4–7] to obtain (5.6). If \((j,j') \in J_{non-out}\), we obtain (5.6) adapting the argument in [38] due to the dispersive estimate (4.10) when \(\tau < t\). Finally, in the case that \((j,j') \in J_{non-inc}\), we obtain (5.7) since we have the dispersive estimate (4.9) for \(\tau > t\). We mention here that we have sharpened the inequality to the Lorentz norm by the interpolation as remarked in [38, Sections 6 and 10]. \(\square \)
6 Strichartz estimates for wave equation with \({\mathcal {L}}_V\)
In this section, we prove the Strichartz estimate for \({\mathcal {L}}_V\) by using Proposition 4.4 and establishing a local smoothing estimate.
6.1 A local-smoothing estimate
In this subsection, we prove a global-in-time local-smoothing estimate. It worths pointing out that we directly prove the local smoothing estimate avoiding the usual method via resolvent estimate of \({\mathcal {L}}_V\).
Proposition 6.1
Let u be the solution of (1.1), then there exists a constant C independent of \((u_0,u_1)\) such that
where \(z=(r,y)\in X\), \(1/2<\beta <1+\nu _0\) with \(\nu _0>0\) such that \(\nu _0^2\) is the smallest eigenvalue of \(\Delta _h+V_0(y)+(n-2)^2/4\).
Remark 6.1
In [10], Burq et al. established the resolvent estimate and thus proved a same estimate, called Morawetz estimate, in Euclidean space with \(\beta =1\).
Proof
We modify the proof of the argument in our previous paper [66] for Schrödinger. A key observation is that the norms in the both sides of the local smoothing are based on \(L^2\)-space which allows us to use orthogonality of eigenfunctions. Without loss of generality, we assume \(u_1=0\). Since
we only consider the estimate of \(e^{it\sqrt{{\mathcal {L}}_V}}u_0\). Recall
By (2.25) with \(F(\rho )=e^{it\rho }\), we will estimate
By the Plancherel theorem with respect to time t, it suffices to estimate
Using the orthogonality, one has
then we see that the above is equal to
To estimate it, we make a dyadic decomposition into the integral. Let \(\chi \) be a smoothing function supported in [1, 2], we see that the above is less than
Let
Then we have the following inequality
We postpone the proof for a moment. By (6.6) we turn to estimate
Note that if \(\frac{1}{2}<\beta <1+\nu _0 \) the summations in R converges and further converges to \(\Vert u_0\Vert ^2_{\dot{H}^{\beta -\frac{1}{2}}(X)}\). Hence we prove (6.1). Now we are left to prove (6.6). To this end, we break it into two cases.
Case 1: \(R\lesssim 1\). Since \(\rho \sim 1\), thus \(r\rho \lesssim 1\). By (2.21), we obtain
$$\begin{aligned} Q_{\nu ,\ell }(R,M)&\lesssim \int _{R}^{2R}\int _{0}^\infty \Big | \frac{ (r\rho )^{\nu }(r\rho )^{-\frac{n-2}{2}}}{2^{\nu }\Gamma \left( \nu +\frac{1}{2}\right) \Gamma \left( \frac{1}{2}\right) }b_{\nu ,\ell }(M\rho )\chi (\rho )\Big |^2 d\rho dr\\&\lesssim R^{2\nu -n+3}M^{-n}\Vert b_{\nu ,\ell }(\rho )\chi \left( \frac{\rho }{M}\right) \rho ^{\frac{n-1}{2}}\Vert ^2_{L^2}. \end{aligned}$$Case 2: \(R\gg 1\). Since \(\rho \sim 1\), thus \(r\rho \gg 1\). We estimate by (2.22) in Lemma 2.3
$$\begin{aligned} Q_{\nu ,\ell }(R,M)&\lesssim R^{-(n-2)}\int _{0}^\infty \big |b_{\nu ,\ell }(M\rho )\chi (\rho )\big |^2\int _{R}^{2R}\big |J_{\nu }( r\rho )\big |^2 dr d\rho \\&\lesssim R^{-(n-2)}\int _{0}^\infty \big |b_{\nu ,\ell }(M\rho )\chi (\rho )\big |^2 d\rho \\&\lesssim R^{-(n-2)}M^{-n}\Vert b_{\nu ,\ell }(\rho )\chi \left( \frac{\rho }{M}\right) \rho ^{\frac{n-1}{2}}\Vert ^2_{L^2}. \end{aligned}$$Thus we prove (6.6). Therefore, we prove the local smoothing estimate.
\(\square \)
Remark 6.2
By constructing the similar counterexample as in Sect. 6.3, we can see the restriction \(\beta <1+\nu _0\) is necessary for (6.1). However,
- (i)
if \(V=0\), then \(\beta <\frac{n}{2}\) is required since the positive square root of the smallest eigenvalue of \(\Delta _h+(n-2)^2/4\) is greater than \((n-2)/2\);
- (ii)
if the initial data, say \(u_0\), belongs to \(\bigoplus _{\nu \in \chi _\infty ,\nu >k} \mathcal {H}^{\nu }\cap \dot{H}^{\beta -\frac{1}{2}}(X)\) where \(k>\nu _0\), then one can relax the restriction on \(\beta \) to \(\beta <1+k\).
6.2 The proof of Strichartz estimates
Let v be as in Proposition 4.4 with \(F=0\) and suppose that u solves the equation
we have by the Duhamel formula
From the spectral theory on \(L^2\), we have the Strichartz estimate for \((q,\mathrm {r})=(\infty , 2)\). By using the Sobolev inequality in Proposition 3.1, we obtain
where \(s=n(1/2-1/\mathrm {r})<2\) and \(2\le \mathrm {r}<n/\max \{\frac{n}{2}-1-\nu _0, 0\}\). Note that the restriction \(s<2\) implies \(\mathrm {r}<2n/(n-4)\) which is a artificial restriction, thus we can get rid of this restriction by using an iterating argument as in Corollary 3.1.
If \(\nu _0>1/(n-1)\), then we have
which is corresponding to \(0<s<1+\nu _0\). On the other hand, by Proposition 4.4 with \(s\in \mathbb {R}\) and Hölder’s inequality in Proposition 2.2, we show that
Now our main task is to estimate
Note that if the set \(\Lambda _s\) is not empty, we must have \(s\ge 0\). Indeed, if \((q,\mathrm {r})\in \Lambda _s\), then
Therefore, without loss of generality, we may assume \(s>0\).
Now we argue Theorem 1.1 by considering the following four cases.
Case I\( 0<s<\frac{1}{2}+\nu _0, q>2\). Let \(\frac{1}{2}<\beta <n/2\), by using Proposition 6.1 and Remark 6.2, we define the operator
Thus from the proof of the local smoothing estimate, it follows that T is a bounded operator. By the duality, we obtain that for its adjoint \(T^*\)
which is also bounded. Define the operator
Hence by the Strichartz estimate with \(s=\frac{3}{2}-\beta \), one has
Now we estimate (6.9). Note that
thus by (6.11), we have a minor modification of (6.9)
where we use the local smoothing estimate in Proposition 6.1 again in the last inequality and we need \(1-\nu _0<\beta <3/2\) such that \(1/2<2-\beta <1+\nu _0\). Therefore the above statement holds for all \(\max \{1/2,1-\nu _0\}<\beta <3/2\). By the Christ-Kiselev lemma [17], thus we have shown that for \(q>2\) and \((q,\mathrm {r})\in \Lambda _{s,\nu _0}=\Lambda _s\) with \(s=\frac{3}{2}-\beta \)
We remark that we have proved all \((q,\mathrm {r})\in \Lambda _s\) with \(q>2\) and s such that \(0<s<\min \{1,\frac{1}{2}+\nu _0\}\). Now we relax the restriction to \(s<\frac{1}{2}+\nu _0\) when \(\nu _0\ge 1/2\). For \(1\le s<\frac{1}{2}+\nu _0\) and any \((q,\mathrm {r})\in \Lambda _s\), then there exists a pair \((q,\tilde{\mathrm {r}})\in \Lambda _{\tilde{s}}\) with \(\tilde{s}=1_-\) such that
Indeed, the Sobolev inequality of Corollary 3.1 shows the first inequality and the above result implies the second one.
Therefore we have proved all \((q,\mathrm {r})\in \Lambda _s\) and s such that \(0<s<\frac{1}{2}+\nu _0\) except the endpoint admissible pair with \(q=2\) when \(s\ge s_0:=(n+1)/2(n-1)\) and \(n\ge 4\).
Case II\(0<\nu _0\le \frac{1}{n-1}\). In this case, if \((q,\mathrm {r})\in \Lambda _{s,\nu _0}\), then \(q>2\). Hence it suffices to fix the gap \(\frac{1}{2}+\nu _0\le s<1+\nu _0\). To this end, we split the initial data into two parts: one is projected to \(\mathcal {H}^\nu \) with \(\nu \le 1+\nu _0\) and the other is the remaining terms. Without loss of generality, we assume \(u_1=0\) and divide \(u_0=u_{0,l}+u_{0,h}\) where \(u_{0,h}=u_0-u_{0,l}\) and
For the part involving \(u_{0,h}\), we can repeat the argument of Case I. In this case, as in Remark 6.2, we can use Proposition 6.1 with \(1/2<2-\beta <2+\nu _0\). Thus we obtain the Strichartz estimate on \(e^{it\sqrt{{\mathcal {L}}_V}}u_{0,h}\) for \(\Lambda _{s,\nu _0}\) with \(s\in [\frac{1}{2}+\nu _0,1+\nu _0)\). We remark here that the set \(\Lambda _{s,\nu _0}\) is empty when \(s\ge 1+\nu _0\).
Next we consider the Strichartz estimate for \(e^{it\sqrt{{\mathcal {L}}_V}}u_{0,l}\). We follow the argument of [50] which treated a radial case. Recall
Since \(\nu \in A\), therefore there exists a constant \(C_{\nu _0}\) depending on \(\nu _0\) such that
Let \(\mu =(n-2)/2\) and recall \(\mathcal {H}_\mu \mathcal {H}_\mu =Id\), then it suffices to estimate
For our purpose, we recall [50, Theorem 3.1] which claimed that the operator \(\mathcal {K}^0_{\mu ,\nu }:=\mathcal {H}_\mu \mathcal {H}_\nu \) is continuous on \(L^p_{r^{n-1}dr}([0,\infty ))\) if
Notice \(\mu =\frac{n-2}{2}\), on one hand, we have that both \(\mathcal {K}^0_{\mu ,\nu }\) and \(\mathcal {K}^0_{\nu ,\mu }\) are bounded in \(L^p_{r^{n-1}dr}([0,\infty ))\) provided \(\frac{1}{p}>\frac{1}{2}-\frac{1+\nu }{n}\). One can check that \(\frac{1}{\mathrm {r}}>\frac{1}{2}-\frac{1+\nu _0}{n}\) satisfies the condition since \(\nu \ge \nu _0\). On the other hand, \(\mathcal {H}_\mu [e^{it\rho }\mathcal {H}_\mu ]\) is a classical half-wave propagator in the radial case which has Strichartz estimate with \((q,\mathrm {r})\in \Lambda _s\). In sum, for \((q,\mathrm {r})\in \Lambda _{s,\nu _0}\), we have
In the second inequality, we use [50, Theorem 3.8].
Case III\(\nu _0> \frac{1}{n-1}\), \(q=2\) and \(n\ge 4\). In this case, we aim to prove
Before proving this, we first prove
Indeed, it follows from Corollary 5.1 and Proposition 6.1 with \(\beta =(n-2)/(n-1)\)\((n\ge 4)\) that
Hence this shows (6.19). On the other hand, from Proposition 3.2, we have shown that the operator
is bounded. Note that the operators \({\mathcal {L}}_V\) and \({\mathcal {L}}_0\) are self-adjoint, by dual argument, we see the boundedness of the operator
Therefore we obtain
This gives (6.18).
Let \(s_0=(n+1)/2(n-1)\) and apply the operator \({\mathcal {L}}_V^{(s-s_0)/2}\) with \(s_0\le s<\frac{1}{2}+\nu _0\) to the wave equation, thus by using the above Strichartz estimate, we obtain
Consider \((2,\frac{2n}{n-2s-1})\in \Lambda _s\) with \(s_0\le s<\frac{1}{2}+\nu _0\). One can verify that \(0\le s-s_0<\min \{2, 1+\nu _0\}\) and \(\frac{2n}{n-2s-1}\) satisfies that (3.21). By the Sobolev inequality in Corollary 3.1, we show
In sum, under the condition \(\nu _0>1/(n-1)\), we have proved the Strichartz estimate (1.2) with \(F=0\) for all \((q,\mathrm {r})\in \Lambda _s\) with \(s\in [0, 1/2+\nu _0)\).
Case VI:\(\nu _0> \frac{1}{n-1}\) and \((q,\mathrm {r})\in \Lambda _{s,\nu _0}\) with \(s\in [1/2+\nu _0, 1+\nu _0)\). For \(s\in [1/2+\nu _0, 1+\nu _0)\) and \((q,\mathrm {r})\in \Lambda _s\cap \{(q,\mathrm {r}):\frac{1}{\mathrm {r}}>\frac{1}{2}-\frac{1+\nu _0}{n}\}\), as using the Sobolev inequality of Corollary 3.1 as before, we have that there exists a pair \((q,\tilde{\mathrm {r}})\in \Lambda _{\tilde{s}}\) with \(\tilde{s}=(1/2+\nu _0)_-\) such that
Thus we prove the homogeneous Strichartz estimate stated in Theorem 1.1. We show the inhomogeneous Strichartz estimate by using \(TT^*\)-method as in Sect. 4.5. Therefore we complete the proof of the second conclusion in Theorem 1.1.
6.3 The sharpness of the restriction (1.5)
In this subsection, we construct a counterexample to claim the restriction (1.5) is necessary for Theorem 1.1.
Proposition 6.2
(Counterexample). If \((q,\mathrm {r})\in \Lambda _s\) but \((q,\mathrm {r})\notin \{(q,\mathrm {r}):\frac{1}{\mathrm {r}}>\frac{1}{2}-\frac{1+\nu _0}{n}\}\). Then the Strichartz estimate possibly fails.
Proof
Assume \(u_0=(\mathcal {H}_{\nu _0}\chi )(s)\) is independent of y, where \(\chi \in {\mathcal {C}}_c^\infty ([1,2])\) is valued in [0, 1]. Due to the compact support of \(\chi \) and the unitarity of the Hankel transform \(\mathcal {H}_{\nu _0}\) on \(L^2\), we obtain \(\Vert u_0\Vert _{\dot{H}^s}\le C\). Now we conclude that
when \(\frac{1}{\mathrm {r}}\le \frac{1}{2}-\frac{1+\nu _0}{n}\). We write that
We recall the behavior of \(J_\nu (r)\) as \(r\rightarrow 0+\). For the complex number \(\nu \) with \(\mathrm {Re}(\nu )>-1/2\), see [27, Section B.6], then we have that
where
satisfies
Now we compute for any \(0<\epsilon \ll 1\)
We first observe that by (6.24)
Next we estimate the lower boundedness
where we have used the fact that \(\cos (\rho t)\ge 1/2\) for \(t\in [0, 1/4]\) and \(\rho \in [1,2]\), and
Hence, we obtain if \(\frac{1}{\mathrm {r}}<\frac{1}{2}-\frac{\nu _0+1}{n}\)
And when \(\frac{1}{\mathrm {r}}=\frac{1}{2}-\frac{\nu _0+1}{n}\), we get
\(\square \)
7 Applications: well-posedness and scattering theory
In this section, we prove Theorem 1.2 by using the Strichartz estimates established in Theorem 1.1. We follow the standard Banach fixed point argument to prove this result. For any small constant \(\epsilon >0\), let \(I=[0,T)\), there exists \(T>0\) such that
To this end, we consider the map
on the complete metric space \(B_I\) with the metric \(d(u,v)=\big \Vert u-v\big \Vert _{L_t^{q}L_z^{\mathrm {r}}(I\times X)}\) and where the pair \((q,\mathrm {r})\) is given by (1.10). We can check that \((q,\mathrm {r})\in \Lambda _1\). On the other hand, we observe that if the initial data has small enough size \(\delta \), then by Strichartz estimate
holds for \(T=\infty \); if not, the inequality holds for some small \(T>0\) by the dominated convergence theorem. We need to prove that the operator \(\Phi \) defined by (7.1) is well-defined on \(B_I\) and is a contraction map under the metric d for I.
Let \(u\in B_I\) with \(0<\epsilon \ll 1\). We first consider \(3\le n\le 6\). Then, we have by Strichartz estimate
and
Next we consider the case \(n\ge 7\). By using the Strichartz estimate again, we show
and
Hence for \(n\ge 3\) we have \(\Phi (u)\in B_I\). On the other hand, for \(\omega _1, \omega _2\in B_I\), by Strichartz estimate and choosing \(\epsilon \) sufficiently small, we obtain for \(3\le n\le 6\)
and for \(n\ge 7\)
The standard fixed point argument gives a unique solution u of (1.6) on \(I\times X\) which satisfies the bound (1.9). Therefore if \(\delta \) is small enough, we obtain the global solution; otherwise, we have the local existence.
Next, we turn to show the scattering result. We just prove that u scatters at \(+\infty \), the proof for the scattering at \(-\infty \) is similar. Using Duhaml’s formula, the solution with initial data \((u(0),\dot{u}(0))=(u_0,u_1)\in \dot{H}^1\times L^2\) of (1.6) can be written as
where \(V_0\) is defined by (1.12). Denote the scattering data \((u_0^+,u_1^+)\) by
Then, by Strichartz estimate, we can obtain for \(3\le n\le 6\)
and for \(n\ge 7\)
Thus we prove that u scatters.
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Acknowledgements
The authors would like to thank Andras Vasy and Andrew Hassell for their helpful discussions and encouragement. The first author is grateful for the hospitality of Stanford University where the project was initiated. J. Zhang was supported by National Natural Science Foundation of China (11771041, 11831004) and H2020-MSCA-IF-2017 (790623), and J. Zheng was partially supported by the European Research Council, ERC-2014-CoG, project number 646650 Singwave and ANR-16-TERC-0006-01:ANADEL.
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Zhang, J., Zheng, J. Strichartz estimates and wave equation in a conic singular space. Math. Ann. 376, 525–581 (2020). https://doi.org/10.1007/s00208-019-01892-7
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DOI: https://doi.org/10.1007/s00208-019-01892-7