In this paper, we discuss the H1L-boundedness of commutators of Riesz transforms associated with the Schrödinger operator L=−△+V, where H1L(Rn) is the Hardy space associated with L. We assume that V (x) is a nonzero, nonnegative potential which belongs to Bq for some q>n/2. Let T1=V (x)(−△+V )−1, T2=V1/2(−△+V )−1/2 and T3 =∇(−△+V )−1/2. We prove that, for b∈BMO (Rn) , the commutator [b,T3 ] is not bounded from H1L(Rn) to L1 (Rn) as T3 itself. As an alternative, we obtain that [b,Ti ] , ( i=1,2,3 ) are of (H1L,L1weak) -boundedness.

Let $L=-\mathsf{\Delta }+V$ be a Schrödinger operator, where the potential V belongs to the reverse Hölder class. By the subordinative formula, we introduce the fractional heat semigroup ${\left\{e^{-t{L}^{\alpha }}\right\}}_{t>0}, 0<\alpha <1$, associated with L. By the aid of the fundamental solution of the heat equation: ${\partial }_{t}u+Lu={\partial }_{t}u-\mathsf{\Delta }u+Vu=0,$ we estimate the gradient and the time-fractional derivatives of the fractional heat kernel $K_{\alpha ,t}^L(·,·)$, respectively. This method is independent of the Fourier transform, and can be applied to the second-order differential operators whose heat kernels satisfy the Gaussian upper bounds. As an application, we establish a Carleson measure characterization of the Campanato-type space $BM{O}_L^{\gamma }\left({\mathbb{R}}^n\right)$ via the fractional heat semigroup ${\left\{e^{-t{L}^{\alpha }}\right\}}_{t>0}$.

This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces to L q ( ℝ + n + 1 , μ ) L^q (\mathbb{R}_ + ^{n + 1} ,\mu ) via space-time fractional equations. For the extension of fractional Sobolev spaces, preliminary results including estimates, involving the fractional capacity, measures, the non-tangential maximal function, and an estimate of the Riesz integral of the space-time fractional heat kernel, are provided. For the extension of Lebesgue spaces, a new L ^p –capacity associated to the spatial-time fractional equations is introduced. Then, some basic properties of the L ^p –capacity, including its dual form, the L ^p –capacity of fractional parabolic balls, strong and weak type inequalities, are established.

Let ℒ = - Δ 𝔾 + V {\mathcal{L}=-{\Delta}_{\mathbb{G}}+V} be a Schrödinger operator on the stratified Lie group 𝔾 {\mathbb{G}} , where Δ 𝔾 {{\Delta}_{\mathbb{G}}} is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class B q 0 {B_{q_{0}}} with q 0 > 𝒬 / 2 {q_{0}>\mathcal{Q}/2} and 𝒬 {\mathcal{Q}} is the homogeneous dimension of 𝔾 {\mathbb{G}} . In this article, by Campanato type spaces Λ ℒ α ⁢ ( 𝔾 ) {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} , we introduce Hardy type spaces associated with ℒ {\mathcal{L}} denoted by H ℒ p ⁢ ( 𝔾 ) {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} and prove the atomic characterization of H ℒ p ⁢ ( 𝔾 ) {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} . Further, we obtain the following duality relation: Λ ℒ 𝒬 ⁢ ( 1 / p - 1 ) ⁢ ( 𝔾 ) = ( H ℒ p ⁢ ( 𝔾 ) ) ∗ , 𝒬 / ( 𝒬 + δ ) < p < 1 for ⁢ δ = min ⁡ { 1 , 2 - 𝒬 / q 0 } . \Lambda_{\mathcal{L}}^{\mathcal{Q}(1/p-1)}(\mathbb{G})=(H^{{p}}_{\vphantom{% \varepsilon}{\mathcal{L}}}(\mathbb{G}))^{\ast},\quad\mathcal{Q}/(\mathcal{Q}+% \delta)<p<1\quad\text{for}\ \delta=\min\{1,2-\mathcal{Q}/q_{0}\}. The above relation enables us to characterize Λ ℒ α ⁢ ( 𝔾 ) {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} via two families of Carleson measures generated by the heat semigroup and the Poisson semigroup, respectively. Also, we obtain two classes of perturbation formulas associated with the semigroups related to ℒ {\mathcal{L}} . As applications, we obtain the boundedness of the Littlewood–Paley function and the Lusin area function on Λ ℒ α ⁢ ( 𝔾 ) {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} .