Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

By Dachun Yang, Sibei Yang. 2012

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Let $\mathcal{X}$ be a metric space with doubling measure and L a nonnegative self-adjoint operator in $L^{2}(\mathcal{X})$ satisfying the Davies–Gaffney estimates. Let $\varphi:\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that φ(x,⋅) is an Orlicz function, $\varphi(\cdot,t)\in\mathbb{A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(φ)∈(0,1], and it satisfies the uniformly reverse Hölder inequality of order 2/[2−I(φ)]. In this paper, the authors introduce a Musielak–Orlicz–Hardy space $H_{\varphi,L}(\mathcal{X})$, by the Lusin area function associated with the heat semigroup generated by L, and a Musielak–Orlicz BMO-type space $\mathrm{BMO}_{\varphi,L}(\mathcal{X})$, which is further proved to be the dual space of $H_{\varphi,L}(\mathcal{X})$ and hence whose φ-Carleson measure characterization is deduced. Characterizations of $H_{\varphi,L}(\mathcal{X})$, including the atom, the molecule, and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize $H_{\varphi,L}(\mathcal{X})$ in terms of the Littlewood–Paley $g^{\ast}_{\lambda}$-function $g^{\ast}_{\lambda,L}$ and establish a Hörmander-type spectral multiplier theorem for L on $H_{\varphi,L}(\mathcal{X})$. Moreover, for the Musielak–Orlicz–Hardy space Hφ,L(ℝn) associated with the Schrödinger operator L:=−Δ+V, where $0\le V\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$, the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom, and the molecule; finally, the authors show that the Riesz transform ∇L−1/2 is bounded from Hφ,L(ℝn) to the Musielak–Orlicz space Lφ(ℝn) when i(φ)∈(0,1], and from Hφ,L(ℝn) to the Musielak–Orlicz–Hardy space Hφ(ℝn) when $i(\varphi)\in(\frac{n}{n+1},1]$, where i(φ) denotes the uniformly critical lower type index of φ.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/381907a250bb0854a4c4f103d17ec7f03f36b291

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In Applied Mathematics

  1. Dachun Yang
  2. Sibei Yang
  3. Ji Li
  4. Luong Dang Ky
  5. Xuan Thinh Duong
  6. The Anh Bui
  7. Wen Yuan
  8. Jun Cao

In Mathematical Physics

  1. Dachun Yang
  2. Sibei Yang
  3. Ji Li
  4. Luong Dang Ky
  5. Xuan Thinh Duong
  6. The Anh Bui
  7. Jun Cao
  8. Wen Yuan

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