[Paper Review] Algebra properties for Sobolev spaces- Applications to semilinear PDE's on manifolds

Nadine Badr, Frédéric Bernicot+1 more|HAL (Le Centre pour la Communication Scientifique Directe)|Jul 19, 2011

Nonlinear Partial Differential Equations18 references4 citations

TL;DR

This paper establishes algebra properties for generalized Sobolev spaces $W^{s,p} \cap L^\infty$ on Riemannian manifolds using off-diagonal decay estimates and $L^p$ boundedness of Riesz transforms, without requiring pointwise Gaussian bounds on heat kernels. It introduces two novel approaches—paraproducts and functional calculus—and applies them to prove well-posedness of semilinear PDEs, including Schr"odinger and heat equations, under minimal geometric assumptions on the manifold.

ABSTRACT

In this work, we aim to prove algebra properties for generalized Sobolev spaces $W^{s,p} \cap L^\infty$ on a Riemannian manifold, where $W^{s,p}$ is of Bessel-type $W^{s,p}:=(1+L)^{-s/m}(L^p)$ with an operator $L$ generating a heat semigroup satisfying off-diagonal decays. We don't require any assumption on the gradient of the semigroup. To do that, we propose two different approaches (one by a new kind of paraproducts and another one using functionals). We also give a chain rule and study the action of nonlinearities on these spaces and give applications to semi-linear PDEs. These results are new on Riemannian manifolds (with a non bounded geometry) and even in the Euclidean space for Sobolev spaces associated to second order uniformly elliptic operators in divergence form.

Motivation & Objective

• To establish algebra properties for Sobolev spaces $W^{s,p} \cap L^\infty$ on Riemannian manifolds under weak geometric assumptions.
• To develop new analytical tools—paraproducts and functional calculus—for generalized Sobolev spaces associated with operators generating heat semigroups with off-diagonal decay.
• To extend the chain rule and nonlinear mapping properties to these generalized Sobolev spaces, particularly in sub-Riemannian and non-compact settings.
• To prove local well-posedness of semilinear PDEs (Schr"odinger and heat equations) in these spaces using Duhamel's principle and local Lipschitz control of nonlinearities.

Proposed method

• Introduce a new class of paraproducts adapted to operators with off-diagonal decay, enabling decomposition of pointwise products in Sobolev spaces.
• Use functional calculus to characterize Sobolev norms via square functions $S^\rho_\alpha f$, linking them to spectral multipliers $L^\beta f$.
• Establish boundedness of the Riesz transform on $L^p$ for $p$ near 2 as a key assumption, replacing pointwise kernel estimates.
• Apply off-diagonal decay estimates for the semigroup $e^{-tL}$ to control the regularity of nonlinearities without gradient bounds.
• Use Duhamel’s iteration argument to construct solutions to semilinear PDEs in $C^0_I(W^{\alpha,p}_L \cap L^\infty)$, relying on local Lipschitz continuity of the nonlinearity.
• Prove that the nonlinearity $F$ maps $W^{\alpha,p}_L \cap L^\infty$ into itself under smoothness and local Lipschitz conditions, with bounds depending on the $L^\infty$-norm of the input.

Experimental results

Research questions

• RQ1 Can the algebra property for $W^{s,p} \cap L^\infty$ be extended to Riemannian manifolds without assuming pointwise Gaussian bounds on the heat kernel?
• RQ2 Can paraproduct techniques be adapted to generalized Sobolev spaces defined via spectral multipliers of operators with off-diagonal decay?
• RQ3 What conditions on the operator $L$ and the manifold ensure that $F(f) \in W^{\alpha,p}_L \cap L^\infty$ for smooth, locally Lipschitz $F$ and $f \in W^{\alpha,p}_L \cap L^\infty$?
• RQ4 Under what geometric assumptions can well-posedness of semilinear Schr"odinger and heat equations be established in $W^{\alpha,p}_L \cap L^\infty$?

Key findings

• The space $W^{\alpha,p}_L \cap L^\infty$ is an algebra under pointwise multiplication for $\alpha > 0$, $p \in (s_-, s_+)$, under off-diagonal decay and $L^p$-bounded Riesz transform assumptions.
• The paraproduct approach allows decomposition of $fg$ into regular components whose $W^{\alpha,p}_L$ norms are controlled by $\|f\|_{W^{\alpha,p}_L \cap L^\infty}$ and $\|g\|_{W^{\alpha,p}_L \cap L^\infty}$.
• The functional calculus approach characterizes $\|L^\beta f\|_{L^p}$ via square functions $\|S^\rho_\alpha f\|_{L^p}$, providing a new norm equivalence in the generalized Sobolev setting.
• For $\alpha \in (0,1)$, the nonlinearity $F$ is locally Lipschitz on $W^{\alpha,2}_L \cap L^\infty$, with $\|F(u) - F(v)\|_{W^{\alpha,2}_L} \lesssim_R \|u - v\|_{W^{\alpha,2}_L \cap L^\infty}$.
• Well-posedness of the Schr"odinger equation $iu_t + Lu = F(u)$ holds in $C^0_I(W^{\alpha,2}_L \cap L^\infty)$ for small time intervals $I$, provided $\alpha > 0$ and $\alpha > d/2$.
• In sub-Riemannian structures, the restriction $\alpha \in (0,1)$ can be removed, allowing $\alpha > 0$ for the same well-posedness results.