Some Remarks on Strichartz Estimates for Homogeneous Wave Equation

TL;DR

This paper analyzes Strichartz estimates for the homogeneous wave equation, focusing on endpoint behavior, $L^ abla_x$ bounds, and the impact of radial versus angular data. The authors study the case $r=\infty$, clarifying Knapp-type obstructions that forbid triples such as $(2,\infty,3)$ and $(\infty,\infty,n)$, and show that away from these endpoints many $r=\infty$ estimates hold by generalized interpolation, with angular/radial improvements extending the validity. They demonstrate that incorporating angular regularity yields stronger Strichartz bounds for $r<\infty$ in $n\ge 3$, giving explicit relations like $1/q + n/r = n/2 - b$ and $s = (1+\epsilon)((n-1)/r + 2/q - (n-1)/2)$ under additional constraints. The paper extends estimates to inhomogeneous Sobolev spaces $H^s$ by decomposing $u = \cos(tD) f + D^{-1} \sin(tD) g$, where the cosine part satisfies $H^s$ estimates for all admissible triples with $s = b+$ (often $s = b$), while the sine part requires $b \ge 1$ with extra $(q,r)$ constraints. It also confirms the endpoint failure of $L^4_t L^\infty_x$ when $n=2$, clarifying the precise limits of endpoint Strichartz at low dimensions.

Abstract

We give several remarks on Strichartz estimates for homogeneous wave equation with special attention to the cases of $L^\infty_x$ estimates, radial solutions and initial data from the inhomogeneous Sobolev spaces. In particular, we give the failure of the endpoint estimate $L^4_t L^\infty_x$ for $n=2$.

Figures (5)

• Figure 1: Classical Strichartz Estimate \ref{['Sobolev']}
• Figure 2: Radial improvement of Strichartz Estimate \ref{['Sobolev']}
• Figure 3: Theorem \ref{['inho']}: Strichartz Estimate in $H^{s}$
• Figure 4: Theorem \ref{['d-inh']}, Strichartz Estimate in $H^{s}$
• Figure 5: Strichartz Estimate \ref{['shi2']}

Theorems & Definitions (19)

• Theorem 1
• Corollary 2
• Theorem 3
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 4
• Theorem 5
• Theorem 6