Strichartz Estimates in Wiener Amalgam Spaces

• Strichartz estimates in Wiener amalgam spaces are defined by replacing classical Lebesgue or Sobolev norms with separate measures of local integrability and global decay.
• This approach refines spatial and temporal control, enabling precise analysis for dispersive equations such as the wave, Schrödinger, and Dirac flows.
• Methodological advances include kernel decomposition, oscillatory integral analysis, and direct retarded bounds to address low-regularity nonlinear wave equations.

Strichartz estimates in Wiener amalgam spaces address dispersive PDE theory by replacing classical Lebesgue or Sobolev norms with norms that separately quantify local regularity and global decay. The formulation of such estimates in amalgam spaces offers a framework capable of refined spatial and temporal control, particularly for linear and nonlinear evolution equations. This approach has yielded new results for the wave equation, the Schrödinger equation (with and without potential), and, more recently, for relativistic flows such as Dirac. The following exposition presents the main definitions, results, methodological advances, and applications concerning Strichartz estimates in Wiener amalgam spaces, with particular attention to the wave equation.

1. Definition and Structure of Wiener Amalgam Spaces

Wiener amalgam spaces $W(L^p,L^q)$ consist of measurable functions $f$ for which the norm

$$\|f\|_{W(p,q)} := \Bigl\|x \mapsto \|f(\cdot)\,\tau_x \varphi(\cdot)\|_{L^p}\Bigr\|_{L^q_x}$$

is finite, where $\varphi \in C_0^\infty(\mathbb{R}^n)$ is any nonzero window, and $\tau_x\varphi(y) = \varphi(y-x)$. This construction can be generalized to $W(A,B)$ for Banach spaces $A,B$ by the analogous composition. Key properties include:

• Inclusion: $p_0 \ge p_1$, $q_0 \le q_1$ $\implies$ $W(p_0,q_0) \subset W(p_1,q_1)$;
• Convolution: $1/p+1=1/p_0+1/p_1$, $1/q+1=1/q_0+1/q_1$ $\implies$ $W(p_0,q_0)*W(p_1,q_1)\subset W(p,q)$;
• Complex interpolation: $$(W(p_0,q_0), W(p_1,q_1))_{[\theta]} = W(p_\theta, q_\theta)$$ for $1/p_\theta = \theta/p_0 + (1-\theta)/p_1$ and similarly for $q_\theta$;
• Duality: If $p,q<\infty$, then $W(p,q)' = W(p',q')$.

The essential distinction between $W(L^p,L^q)$ and $L^p$ is the decoupling of local integrability (from $L^p$) and global decay (from $L^q$). For example, $W(L^p,L^p)=L^p$, $W(L^\infty,L^q)$ controls global decay, and $W(L^p,L^\infty)$ imposes strong local control but is insensitive to decay at infinity. This framework is particularly suited to capturing dispersive effects and localized smoothing.

2. Strichartz Estimates for the Wave Propagator in Wiener Amalgam Spaces

Let $n\geq3$, $\sigma\in (n/4, (n-1)/2)$, and define the wave propagator $U(t):= e^{it\sqrt{-\Delta}}$. The core results, as established by Kim, Koh, and Seo (Kim et al., 2020), are as follows.

2.1 Homogeneous Strichartz Estimates

For $2\leq \widetilde q < q < \infty$, with $r,\widetilde r$ subject to

$$\frac{2n}{n - 2\sigma} < r \leq \widetilde r < \begin{cases} \frac{4}{n-4\sigma+1} & \sigma < n/4+1/[2(n-1)] \ \frac{2n}{n-2\sigma-1} & \sigma \geq n/4+1/[2(n-1)] \end{cases}$$

and

$$\frac{1}{\widetilde q} + \frac{n-1}{\widetilde r} > \frac{n}{2}-\sigma, \qquad \frac{1}{q} + \frac{n}{r} = \frac{n}{2} - \sigma,$$

the estimate

$$\|U(t)f\|_{W(\widetilde q, q)_t W(\widetilde r, r)_x} \lesssim \|f\|_{\dot H^\sigma}$$

holds for all $f \in \dot H^\sigma(\mathbb{R}^n)$. The mixed time-space amalgam norm grants finer separations of local versus global effects in both $x$ and $t$.

2.2 Inhomogeneous (Retarded) Strichartz Estimates

Given $\gamma \in (n/2,(n+1)/2) \cup ((n+1)/2, n-1)$ and exponents satisfying

$$0<\tfrac1q+\tfrac1{q_1}<\tfrac1{\widetilde q}+\tfrac1{\widetilde q_1}\leq1, \quad 2\leq \widetilde q,\widetilde q_1 < \infty,$$

$$\tfrac1{\widetilde r}+\tfrac1{\widetilde r_1}>\max\{\tfrac{n-2\gamma+1}{2},\tfrac{n-\gamma-1}{n}\},$$

$$\tfrac1q+\tfrac1{q_1}+\tfrac n r+\tfrac n{r_1}=n-\gamma,$$

$$\tfrac1{\widetilde q}+\tfrac1{\widetilde q_1}+\tfrac{n-1}{\widetilde r}+\tfrac{n-1}{\widetilde r_1}>n-\gamma,$$

then for all $F(x,s)$,

$$\Bigl\|\int_0^t U(t-s)|\nabla|^{-\gamma} F(\cdot,s)\,ds\Bigr\|_{W(\widetilde q, q)_t W(\widetilde r, r)_x} \lesssim \|F\|_{W(\widetilde q_1', q_1')_t W(\widetilde r_1', r_1')_x}.$$

The absence of a Christ–Kiselev lemma in $W(L^p,L^q)$ necessitates a direct approach for retarded bounds.

3. Analytical Methodology: Kernel Decomposition and Asymptotics

Unlike the Schrödinger case, no explicit integral kernel formula is available for the wave propagator. Kim-Koh-Seo (Kim et al., 2020) circumvent this by expressing the kernel as an oscillatory integral involving Bessel functions, specifically,

$$K_\gamma(x,t) = C_n r^{-(n-2)/2} \int_0^\infty e^{it\omega} \omega^{n/2 - \gamma} J_{(n-2)/2}(r\omega)\,d\omega,$$

with $J_\nu$ the Bessel function of order $\nu$, $r = |x|$.

A pivotal technical component is the asymptotic expansion of $J_\nu(m)$ for large arguments, allowing the extraction of strong oscillatory cancellation between $e^{i(r\omega\pm t\omega)}$ terms. When integrated by parts in $\omega$ and regionally analyzed for $r\omega<1$ (small argument) and $r\omega>1$ (large argument), one obtains precise pointwise kernel bounds: $$|K_\gamma(x,t)| \lesssim \begin{cases} |t|^{-1}\,|x|^{-n+\gamma+1}, & |x|\leq |t|/2 \ |x|^{-n+\gamma}, & |x| \geq |t|/2 \end{cases}$$ with further logarithmic corrections for critical $\gamma$. This analysis is propagated to the amalgam context by establishing $x \mapsto K_\gamma(\cdot, t) * f$ lies in $W(\widetilde r/2, r/2)_x$ with norm $\lesssim |t|^{-\omega}$ for suitable exponents.

Afterwards, the temporal Hardy-Littlewood-Sobolev inequality and the $TT^*$ argument in $W(L^p, L^q)$ close the estimates.

4. Comparison with Classical Strichartz Theory

Classical (Keel–Tao) Strichartz estimates for the wave equation are phrased as

$$\|U(t)f\|_{L^q_t L^r_x} \lesssim \|f\|_{\dot H^\sigma}$$

for wave-admissible $(q,r)$. The Wiener amalgam variant replaces $L^r_x$ by the finer $W(\widetilde r, r)_x$ (with $r<\widetilde r$) and $L^q_t$ by $W(\widetilde q, q)_t$ (with $\widetilde q < q$), resulting in norms that strictly strengthen $L^r$ to encode finer decay at infinity and stronger localization. When $r<\widetilde r$, $W(\widetilde r, r)_x \subset L^r_x$ globally, but $W(\widetilde q,q)_t \subset L^q$ locally only for $\widetilde q < q$. This framework allows explicit trade-offs between local-in-time regularity and global-in-time decay, as well as spatial localization and decay.

5. Applications to Nonlinear Wave Equations

Wiener amalgam Strichartz estimates have been used to prove low-regularity local well-posedness for semilinear wave equations of the form

$$\partial_t^2 u - \Delta u = F_k(u), \quad u(0)=f \in \dot H^\sigma, \quad u_t(0)=g \in \dot H^{\sigma-1},$$

with nonlinearities $|F_k(u)| \lesssim |u|^k$, $|u| |F_k'(u)| \sim |F_k(u)|$. Under $0<\sigma\leq 1/2$ and $1<k<k(\sigma)$ (with $k(\sigma)$ as specified in (Kim et al., 2020)), there exists $T>0$ and a unique solution

$$u \in W(\widetilde q, q)_t([0,T]; W(\widetilde r, r)_x)$$

for any exponents satisfying explicit constraints in the original result.

The strategy utilizes:

• Homogeneous Strichartz in Wiener amalgam for linear terms;
• Retarded (nonhomogeneous) amalgam Strichartz for the Duhamel integral;
• Algebra and Hölder properties of $W(L^p, L^q)$ to prove contraction in the ball $$\|u\|_{W(\widetilde q, q)_t W(\widetilde r, r)_x} \leq M$$ for sufficiently small $T$.

This approach leads to a finer understanding of both local and global properties of solutions in the presence of low-regularity initial data and weak nonlinearities.

6. Connections, Extensions, and Open Problems

• By complex interpolation, the homogeneous estimate range extends to $\sigma \in [0, n/2)$.
• The methodology applies to the Schrödinger propagator $U(t)=e^{it\Delta}$, recovering and improving prior amalgam Strichartz estimates for the Schrödinger equation (Kim et al., 2019, Takizawa, 17 Dec 2025, Cordero et al., 2018).
• The absence of a general Christ-Kiselev lemma in $W(L^p,L^q)$ compels direct treatment of retarded bounds.
• Possible developments include: generalization to other dispersive PDEs, variable coefficient problems, endpoint exponent cases, and nonlinear profile decompositions.
• Open questions include the sharpness of admissible exponent ranges, long-time dynamics under smallness or defocusing structure, and extensions to the relativistic context (Kim et al., 2022).

The amalgam-based Strichartz estimates have initiated applicable refinements for a variety of dispersive equations, offering analytic machinery that distinguishes between local regularity and global decay—features inaccessible in the classical Lebesgue framework.

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Key References:

• "Strichartz estimates in Wiener amalgam spaces and applications to nonlinear wave equations" (Kim et al., 2020)
• "Strichartz estimates for the Schrödinger propagator in Wiener amalgam spaces" (Kim et al., 2019)
• "Strichartz estimates in Wiener amalgam spaces for Schrödinger equations with at most quadratic potentials" (Takizawa, 17 Dec 2025)
• "Strichartz Estimates for the Schrödinger Equation" (Cordero et al., 2018)
• "Strichartz estimates for the Dirac flow in Wiener amalgam spaces" (Kim et al., 2022)