[Paper Review] Sharp decay estimates in Lorentz spaces for nonnegative Schrödinger heat semigroups

Norisuke Ioku, Kazuhiro Ishige+1 more|arXiv (Cornell University)|Oct 30, 2013

Spectral Theory in Mathematical Physics8 references3 citations

TL;DR

This paper establishes sharp decay estimates for the heat semigroup $ e^{-tH} $ generated by nonnegative Schrödinger operators $ H = -\Delta + V $ with radially symmetric potentials $ V(r) \sim \omega r^{-2} $ in Lorentz spaces $ L^{p,\sigma} \to L^{q,\theta} $, where $ \omega > -\omega_* = -(N-2)^2/4 $. The key result is a complete classification of decay rates depending on the criticality of $ H $ and the parameters $ p,q,\sigma,\theta $, with precise asymptotic behavior $ \|e^{-tH}\|_{(L^{p,\sigma} \to L^{q,\theta})} \asymp t^{-\frac{N}{2}+A} (\log t)^{\gamma} $ for large $ t $, where $ A $ is determined by the spectral behavior at infinity.

ABSTRACT

Let $H:=-Δ+V$ be a nonnegative Schrödinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is a radially symmetric function decaying quadratically at the space infinity. In this paper we consider the Schrödinger heat semigroup $e^{-tH}$, and make a complete table of the decay rates of the operator norms of $e^{-tH}$ in the Lorentz spaces as $t o\infty$.

Motivation & Objective

• To derive sharp large-time decay estimates for the operator norm $ \|e^{-tH}\|_{(L^{p,\sigma} \to L^{q,\theta})} $ of the Schrödinger heat semigroup in Lorentz spaces.
• To classify the decay rates based on the criticality of the Schrödinger operator $ H = -\Delta + V $, particularly when $ V(r) \sim \omega r^{-2} $ with $ \omega > -(N-2)^2/4 $.
• To extend previous $ L^p \to L^q $ results to the full Lorentz scale $ L^{p,\sigma} \to L^{q,\theta} $, including endpoint and logarithmic corrections.
• To characterize the dependence of decay rates on the parameters $ p,q,\sigma,\theta $, especially in the critical and subcritical regimes.

Proposed method

• The analysis relies on the existence of a positive radial harmonic function $ U(r) \sim r^{-A} $ for $ H $, where $ A $ is determined by the spectral parameter $ \omega $ and dimension $ N $.
• The authors use real interpolation theory and Lorentz space interpolation inequalities, particularly the Riesz-Thorin and Marcinkiewicz interpolation theorems, to control operator norms between Lorentz spaces.
• Key estimates are derived via duality and duality-based bounds, including the use of $ \|e^{-tH}\|_{(L^{\alpha,\sigma} \to L^{\beta,\theta})} \asymp t^{-\frac{N}{2}+A} (\log t)^{\gamma} $, where $ \alpha = N/(N-A) $, $ \beta = N/A $.
• The proof proceeds by case analysis on the indices $ \sigma, \theta $, distinguishing cases based on whether $ \sigma = \theta' $, $ \sigma < \theta' $, or $ \sigma > \theta' $, using interpolation and duality to bound the norms.
• The authors establish sharp lower and upper bounds by combining estimates from $ L^\alpha \to L^2 $, $ L^\alpha \to L^\beta $, and $ L^\alpha \to L^{\beta,\theta} $, using the behavior of the heat kernel and the harmonic function at infinity.
• The logarithmic corrections arise from the interplay between the decay of the potential and the growth of the harmonic function, particularly in the critical case where $ A > 0 $.

Experimental results

Research questions

• RQ1 What are the sharp large-time decay rates of the Schrödinger heat semigroup $ e^{-tH} $ in Lorentz spaces $ L^{p,\sigma} \to L^{q,\theta} $ for nonnegative Schrödinger operators with inverse-square potentials?
• RQ2 How do the decay rates depend on the criticality of the operator $ H $, as determined by the parameter $ \omega $ in $ V(r) \sim \omega r^{-2} $?
• RQ3 What is the role of the exponent $ A $, defined via $ A = \frac{N-2 \pm \sqrt{(N-2)^2 + 4\omega}}{2} $, in determining the decay behavior?
• RQ4 How do logarithmic corrections emerge in the decay estimates, and for which Lorentz space parameters $ (p,\sigma,q,\theta) $ do they appear?
• RQ5 Can the decay rates in Lorentz spaces be fully classified, and how do they compare to the classical $ L^p \to L^q $ decay rates?

Key findings

• The operator norm $ \|e^{-tH}\|_{(L^{p,\sigma} \to L^{q,\theta})} $ decays as $ t^{-\frac{N}{2}+A} (\log t)^{\gamma} $ as $ t \to \infty $, where $ A $ is the exponent from the harmonic function $ U(r) \sim r^{-A} $.
• For $ A = 0 $, the decay is $ \|e^{-tH}\|_{(L^{p,\sigma} \to L^{q,\theta})} \asymp t^{-\frac{N}{2}(\frac{1}{p}-\frac{1}{q})} $, matching the heat kernel decay, indicating no logarithmic correction.
• When $ 0 < A < N/2 $, the decay rate is $ \|e^{-tH}\|_{(L^{p,\sigma} \to L^{q,\theta})} \asymp t^{-\frac{N}{2}+A} (\log t)^{\gamma} $, with $ \gamma $ depending on $ \sigma $ and $ \theta $.
• The logarithmic factor arises only when the target space $ L^{q,\theta} $ is not the endpoint $ L^\infty $ or $ L^1 $, and its exponent depends on the dual indices.
• The sharpness of the estimates is confirmed through duality and interpolation techniques, including the use of $ \|e^{-tH}\|_{(L^{\alpha,\sigma} \to L^{\beta,\theta})} \asymp t^{-\frac{N}{2}+A} (\log t)^{\gamma} $ for $ \alpha = N/(N-A) $, $ \beta = N/A $.
• The complete classification of decay rates is established across all $ (p,q,\sigma,\theta) \in \Lambda $, with distinct behaviors in the subcritical and critical regimes.