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Chen, Peng; Duong, Xuan Thinh; Yan, Lixin

\(L^p\)-bounds for Stein’s square functions associated to operators and applications to spectral multipliers. (English) Zbl 1277.42011

J. Math. Soc. Japan 65, No. 2, 389-409 (2013).
The authors prove a Hörmander-type spectral multiplier theorem for non-negative, self-adjoint operators applicable to, for example, sub-Laplacians acting on Lie groups of polynomial growth and Schrödinger operators with rough potentials. They assume that the kernel \(p_t(x, y)\) of the semigroup \(e^{-tL}\) generated by the operator \(-L\) satisfies the Gaussian upper bound \[ |p_t(x,y)| \leq \frac{C}{V(x,t^{1/m})} \exp\left(-\frac{d(x, y)^{m/(m-1)}}{ct^{1/(m-1)}}\right) \] for all \(t > 0\) and \(x, y \in X\), where \((X, d,\mu)\) is an appropriate metric measure space and \(V(x,r)\) is the \(\mu\)-measure of the ball of radius \(r\) centered at \(x\). Here \(C, c > 0\) and \(m \geq 2\). They also assume the kernel \(K_{F(L^{1/m})}(x, y)\) of the operator \(F(L^{1/m})\) satisfies \[ \int_X |K_{F(L^{1/m})}(x, y)|^2 d\mu(x) \leq \frac{C}{V(y,t^{-1})}\|F (t\cdot)\|_{L^q}^2 \] for some \(2 \leq q \leq \infty\), any \(t > 0\) and all Borel functions \(F\) supported on \([0, t]\).
The theorem is proved by first showing \(L^p\) bounds for an associated square function \[ \mathcal{G}_\delta(L)f(x) = c_{m\delta}\left( \int_0^\infty \left|\frac{\partial}{\partial R}S^{\delta + 1}(L)f(x)\right|^2 RdR \right)^{1/2}, \] first introduced by E. M. Stein for \(L = -\Delta\) in his study of Bochner-Riesz means \(S_R^\delta (L) = (I - L/R^m)^\delta_+\) of order \(\delta\).
The multiplier theorem states that if \(F\) is a locally absolutely continuous function on \((0, \infty)\) and \[ B := \|F\|_{L^\infty} + \left( \sup_{R > 0} R\int_R^{2R} |F'(\lambda)|^2 d\lambda \right)^{1/2} < \infty, \] then \(F(L)\) is bounded on \(L^p(X)\) for all \(p \in (0,\infty)\) satisfying \((n + 1 - 2/q)|1/p - 1/2| < 1/2\). In addition, \[ \|F(L)\|_{L^p(X) \to L^p(X)} \leq CB \] with \(C\) independent of \(F\).
Reviewer: David Rule (Linköping)

Cited in 9 Documents

MSC:

42B15 Multipliers for harmonic analysis in several variables
42B25 Maximal functions, Littlewood-Paley theory
47F05 General theory of partial differential operators

Keywords:

Stein’s square function; Bochner-Riesz means; non-negative self-adjoint operator; heat semigroup; space of homogeneous type
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\textit{P. Chen} et al., J. Math. Soc. Japan 65, No. 2, 389--409 (2013; Zbl 1277.42011)
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