[Paper Review] Weighted Besov and Triebel--Lizorkin spaces associated to operators

Huy‐Qui Bui, Anh Tuan Bui+1 more|arXiv (Cornell University)|Sep 8, 2018

Advanced Harmonic Analysis Research13 references3 citations

TL;DR

This paper develops weighted Besov and Triebel-Lizorkin spaces associated with a nonnegative self-adjoint operator $L$ on a space of homogeneous type, under Gaussian heat kernel bounds. It establishes continuous characterizations, atomic decompositions, and identifications with classical function spaces, proving boundedness of fractional powers and spectral multipliers in these new scales for the full range $0 < p,q \leq \infty$, $\alpha \in \mathbb{R}$, and $w \in A_\infty$. The key contribution is a unified theory extending classical function spaces to the operator setting with full generality in indices and weights.

ABSTRACT

Let $X$ be a space of homogeneous type and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ satisfying Gaussian upper bounds on its heat kernels. In this paper we develop the theory of weighted Besov spaces $\dot{B}^{α,L}_{p,q,w}(X)$ and weighted Triebel--Lizorkin spaces $\dot{F}^{α,L}_{p,q,w}(X)$ associated to the operator $L$ for the full range $0

Motivation & Objective

• To extend the theory of Besov and Triebel-Lizorkin spaces to the setting of nonnegative self-adjoint operators on spaces of homogeneous type.
• To develop weighted versions of these function spaces for the full range $0 < p,q \leq \infty$, $\alpha \in \mathbb{R}$, and $w \in A_\infty$.
• To establish continuous characterizations, atomic decompositions, and identifications with classical function spaces such as Hardy, BMO, and Sobolev spaces.
• To prove boundedness of fractional powers and spectral multipliers of Laplace transform type in the new function spaces.

Proposed method

• Use of Calderón reproducing formulas involving functions in $\mathscr{S}_m(\mathbb{R})$ to define and characterize the new function spaces.
• Application of Gaussian upper bounds on the heat kernel $p_t(x,y)$ to derive kernel estimates and maximal function bounds.
• Introduction of a new class of distributions adapted to the operator $L$ to handle the full range of indices.
• Employment of Littlewood-Paley theory and square function characterizations via Lusin functions and spectral projections.
• Use of weighted norm inequalities, including the Fefferman-Stein inequality, for $A_\infty$ weights.
• Establishment of atomic decompositions via dyadic decompositions and molecules adapted to the operator $L$.

Experimental results

Research questions

• RQ1 How can weighted Besov and Triebel-Lizorkin spaces associated with a nonnegative self-adjoint operator $L$ be defined and characterized in the full range $0 < p,q \leq \infty$?
• RQ2 Under what conditions do the new function spaces coincide with classical function spaces such as $L^p_w$, Hardy, BMO, and Sobolev spaces?
• RQ3 What is the boundedness behavior of the fractional power $L^{s/2}$ on these new function spaces?
• RQ4 How do spectral multipliers of Laplace transform type act on the new function spaces?
• RQ5 Can the theory be extended to general operators satisfying only Gaussian heat kernel bounds, without additional regularity assumptions?

Key findings

• The weighted Besov space $\dot{B}^{\alpha,L}_{p,q,w}(X)$ admits a continuous characterization via square functions and is equivalent to the norm defined by the Calderón reproducing formula.
• Atomic decompositions are established for both $\dot{B}^{\alpha,L}_{p,q,w}(X)$ and $\dot{F}^{\alpha,L}_{p,q,w}(X)$, with atoms satisfying cancellation and size conditions adapted to $L$.
• The space $\dot{F}^{\alpha,L}_{p,q,w}(X)$ is identified with the weighted Hardy space $H^p_{L,w}(X)$ when $p \leq 1$, and with the weighted BMO space $\text{BMO}_{L,w}(X)$ when $p = \infty$.
• The fractional power $L^{s/2}$ maps $\dot{B}^{\alpha+s,L}_{p,q,w}(X)$ boundedly into $\dot{B}^{\alpha,L}_{p,q,w}(X)$, and similarly for Triebel-Lizorkin spaces.
• The spectral multiplier $\tilde{m}(L)$ of Laplace transform type is bounded on $\dot{F}^{\alpha,L}_{p,q,w}(X)$ for $0 < p < \infty$, $0 < q \leq \infty$, and on $\dot{B}^{\alpha,L}_{p,q,w}(X)$ for $0 < p,q \leq \infty$.
• The theory applies to a wide class of operators, including sub-Laplacians on Lie groups of polynomial growth and Laplace-Beltrami operators on certain Riemannian manifolds, under only the Gaussian heat kernel bound.