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BOUNDEDNESS OF DIFFERENTIAL TRANSFORMS FOR FRACTIONAL HEAT SEMIGROUPS GENERATED BY SCHRÖDINGER OPERATORS
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- Journal:
- Journal of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 25 November 2024, pp. 1-35
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Let
$L=-\Delta +V$ be a Schrödinger operator in 
${\mathbb R}^n$ with 
$n\geq 3$, where 
$\Delta $ is the Laplace operator denoted by 
$\Delta =\sum ^{n}_{i=1}({\partial ^{2}}/{\partial x_{i}^{2}})$ and the nonnegative potential V belongs to the reverse Hölder class 
$(RH)_{q}$ with 
$q>n/2$. For 
$\alpha \in (0,1)$, we define the operator 
$$ \begin{align*} T_N^{L^{\alpha}} f(x) =\sum_{j=N_1}^{N_2} v_j(e^{-a_{j+1}L^\alpha} f(x)-e^{-a_{j}L^\alpha} f(x)) \quad \mbox{for all }x\in \mathbb R^n, \end{align*} $$where
$\{e^{-tL^\alpha } \}_{t>0}$ is the fractional heat semigroup of the operator L, 
$\{v_j\}_{j\in \mathbb Z}$ is a bounded real sequence and 
$\{a_j\}_{j\in \mathbb Z}$ is an increasing real sequence.We investigate the boundedness of the operator
$T_N^{L^{\alpha }}$ and the related maximal operator 
$T^*_{L^{\alpha }}f(x):=\sup _N \vert T_N^{L^{\alpha }} f(x)\vert $ on the spaces 
$L^{p}(\mathbb {R}^{n})$ and 
$BMO_{L}(\mathbb {R}^{n})$, respectively. As extensions of 
$L^{p}(\mathbb {R}^{n})$, the boundedness of the operators 
$T_N^{L^{\alpha }}$ and 
$T^*_{L^{\alpha }}$ on the Morrey space 
$L^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})$ and the weak Morrey space 
$WL^{\rho ,\theta }_{1,\kappa }(\mathbb {R}^{n})$ has also been proved.
