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Calderón reproducing formulas and new Besov spaces associated with operators. (English) Zbl 1241.46020
Let \(\mathcal{X}\) be a space of polynomial upper bound on volume growth, which might not be doubling, and \(L\) be the generator of an analytical semigroup \(\{e^{-tL}\}_{t>0}\) acting on \(L^2(\mathcal{X})\) whose heat kernel satisfies an upper bound of Poisson type. In this paper, for \(-1<\alpha<1\) and \(1\leq p,\,q\leq \infty\), the authors introduce a new class of Besov spaces \(B_{p,q}^{\alpha,L}(\mathcal{X})\) associated with the operator \(L\). This new Besov space \(B_{p,q}^{\alpha,L}(\mathcal{X})\) is defined to be the space of all elements belonging to some “distribution space” with the norm \[ \|f\|_{B_{p,q}^{\alpha,L}(\mathcal{X})}:=\left\{\int_{0}^\infty \left( t^\alpha\left\| tLe^{-tL} (f)\right\|_{L^p(\mathcal{X})} \right)^q\frac{dt}{t}\right\}^{1/q}<\infty, \] which generalizes the notion of the classical Besov space and has wide applications.
One of the main aims of this paper is to find the relationship between the classical and the new Besov spaces. By assuming that \(L\) has some conservation property, the authors show that the classical Besov space \(B_{p,q}^{\alpha}(\mathbb R^n)\) is continuously embedded in \(B_{p,q}^{\alpha,L}(\mathbb R^n)\). Moreover, if the heat kernels \(\{p_t(x,\,y)\}_{t>0}\) of \(\{e^{-tL}\}_{t>0}\) further satisfy some Hölder continuity, the authors prove the equivalence \(B_{p,q}^{\alpha}(\mathbb R^n)=B_{p,q}^{\alpha,L}(\mathbb R^n)\). Also, by establishing four different Calderón reproducing formulas, which are of independent interest, the authors obtain many basic properties of \(B_{p,q}^{\alpha,L}(\mathcal{X})\), such as the embedding theorem, some norm equivalences and the boundedness of the fractional integral \(L^{-\gamma}\) for \(\gamma\in(0,\,1)\).
As an application, when the operator \(L:=-\Delta+V\) is a Schrödinger operator on the Euclidean space \(\mathbb{R}^n\) with the nonnegative potential \(V\) satisfying some reverse Hölder estimates, the authors study the Besov space \(B_{1,1}^{0,L}(\mathbb{R}^n)\) and prove that the classical Besov space \(B_{1,1}^{0}(\mathbb{R}^n)\) is strictly contained in \(B_{1,1}^{0,L}(\mathbb{R}^n)\). Moreover, by using one of the Calderón reproducing formulas, a new molecular characterization of \(B_{1,1}^{0,L}(\mathbb{R}^n)\) is also obtained.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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