[Paper Review] Optimal constants and extremisers for some smoothing estimates

Neal Bez, Mitsuru Sugimoto|arXiv (Cornell University)|Jun 22, 2012

Advanced Mathematical Physics Problems30 references14 citations

TL;DR

This paper establishes explicit formulas for optimal constants and characterizes extremisers in a broad class of smoothing estimates for dispersive equations, using spectral decomposition of oscillatory integral operators when the weight is homogeneous. It provides sharp constants for Schrödinger-type equations and identifies extremal initial data via radial symmetry and Bessel function analysis, resolving long-standing questions on optimality in $ L^2 $-based smoothing estimates.

ABSTRACT

We establish new results concerning the existence of extremisers for a broad class of smoothing estimates of the form $\|ψ(| abla|) \exp(itϕ(| abla|)f \|_{L^2(w)} \leq C\|f\|_{L^2}$, where the weight $w$ is radial and depends only on the spatial variable; such a smoothing estimate is of course equivalent to the $L^2$-boundedness of a certain oscillatory integral operator $S$ depending on $(w,ψ,ϕ)$. Furthermore, when $w$ is homogeneous, and for certain $(ψ,ϕ)$, we provide an explicit spectral decomposition of $S^*S$ and consequently recover an explicit formula for the optimal constant $C$ and a characterisation of extremisers. In certain well-studied cases when $w$ is inhomogeneous, we obtain new expressions for the optimal constant.

Motivation & Objective

• To determine the optimal constant $ C $ in $ L^2 $-based smoothing estimates of the form $ \|w(|x|)^{1/2}\psi(|\nabla|)e^{it\phi(|\nabla|)}f\|_{L^2_{t,x}} \leq C\|f\|_{L^2} $ for radial weights and dispersive flows.
• To characterize the extremal initial data $ f $ that achieve equality in such estimates.
• To extend known results on sharp constants—previously limited to specific cases like type [B] or [C]—to a general class of $ (w, \psi, \phi) $ under radial and regularity assumptions.
• To provide a spectral decomposition of the operator $ S^*S $ when $ w $ is homogeneous, enabling explicit computation of $ C $ and extremisers.
• To resolve the open problem of whether the supremum in the optimal constant formula is attained, particularly in inhomogeneous weight cases.

Proposed method

• Derive the optimal constant $ \mathbf{C}_d(w,\psi,\phi) $ via the formula $ \mathbf{C}_d = \left(2\pi \sup_{k,\varrho} \alpha_k(\varrho)\right)^{1/2} $, where $ \alpha_k(\varrho) $ involves Bessel functions and the weight $ w $.
• Use radial symmetry and Fourier-Bessel transforms to reduce the problem to a one-dimensional analysis of $ \alpha_k(\varrho) $, leveraging the structure of $ S^*S $.
• Apply spectral decomposition of $ S^*S $ in the homogeneous weight case, showing that $ \alpha_k(\varrho) $ becomes constant, enabling exact computation of $ C $.
• Analyze the behavior of $ \alpha_k(\varrho) $ via asymptotic expansions and monotonicity arguments, particularly for $ d=3 $ and $ d=5 $, to locate suprema.
• Use integral identities involving Bessel functions $ J_{\nu} $, including $ \int_0^\infty J_\nu(r\varrho)^2 r w(r) dr $, to evaluate $ \alpha_k(\varrho) $ explicitly.
• Construct counterexamples (e.g., with oscillatory weights) to show that the supremum of $ \alpha_0(\varrho) $ may not be attained at $ \varrho=0 $ or $ \varrho=\infty $, and that $ \mathbf{C}_d $ may exceed $ \sup \alpha_0(\varrho) $.

Experimental results

Research questions

• RQ1 Under what conditions does the optimal constant $ \mathbf{C}_d(w,\psi,\phi) $ in smoothing estimates attain its supremum, and is this supremum finite?
• RQ2 Can the optimal constant be computed explicitly when the weight $ w $ is homogeneous, and what is its spectral structure?
• RQ3 What is the precise relationship between the optimal constant and the function $ \alpha_k(\varrho) $, and how does this depend on the dispersion relation $ \phi $ and smoothing function $ \psi $?
• RQ4 Are there cases where the extremal initial data are not radial, or where the supremum of $ \alpha_k(\varrho) $ is not unique?
• RQ5 Does the formula $ \mathbf{C}_d = (2\pi \sup \alpha_0(\varrho))^{1/2} $ always hold, or are there counterexamples?

Key findings

• For $ d=3 $, $ (w,\psi,\phi) = ((1+r^2)^{-1}, (1+r^2)^{1/4}, r^2) $, the optimal constant is $ \mathbf{C}_3 = \pi^{1/2} $, achieved at $ \varrho=0 $, with $ \alpha_0(0) = \frac{1}{2} $.
• For $ d=5 $, the same $ (w,\psi,\phi) $ yields $ \mathbf{C}_5 = (\pi/2)^{1/2} $, since $ \sup \alpha_0(\varrho) = \frac{1}{4} $, attained as $ \varrho \to \infty $.
• When $ w $ is homogeneous, the operator $ S^*S $ admits a spectral decomposition, and $ \alpha_k(\varrho) $ is constant in $ \varrho $, leading to explicit $ C $ via $ \sup_k \alpha_k(\varrho) $.
• For $ d=5 $, $ (w,\psi,\phi) = ((1+r^2)^{-1}, (1+r^2)^{1/4}, r^2) $, the optimal constant is $ \mathbf{C}_5 = (2\pi \alpha_0(\varrho_0))^{1/2} $, where $ \varrho_0 $ solves $ \Upsilon(\varrho_0) = 0 $, yielding $ \alpha_0(\varrho_0) = \alpha $.
• A counterexample with $ w(r) = r^{-2}(\mu - \cos r) $, $ \psi(r) = 1 $, $ \phi(r) = r^2 $, $ d=3 $, shows $ \sup \alpha_0(\varrho) = \mu/2 $, attained on an interval, and $ \alpha_1(\varrho_0) > 1/\pi $ for large $ N $, violating $ \mathbf{C}_d = (2\pi \sup \alpha_0)^{1/2} $.
• In the case $ (w,\psi,\phi) = (\frac{1}{2}N\chi_{I(N)}(r), r^{1/2}, r^2) $, $ \sup \alpha_0(\varrho) \leq 1/\pi $, but $ \alpha_1(\varrho_0) > 1/\pi $ for large $ N $, proving that $ \mathbf{C}_d $ is not determined solely by $ \sup \alpha_0 $, invalidating a naive extension of the formula.