Let $v \ne 0$ be a vector in ${\mathbb {R}}^n$ . Consider the Laplacian on ${\mathbb {R}}^n$ with drift $\Delta _{v} = \Delta + 2v\cdot \nabla $ and the measure $d\mu (x) = e^{2 \langle v, x \rangle } dx$ , with respect to which $\Delta _{v}$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1, 1)$ and other sharp endpoint estimates for the Riesz transforms of any order, and also for the vertical and horizontal Littlewood–Paley–Stein functions associated with the heat and the Poisson semigroups.

In this paper we establish the endpoint estimates and Hardy type estimates for the Riesz transform associated with the generalized Schrödinger operator.

Let λ > 0 and let

be the Bessel operator on ℝ+ := (0,∞). We show that the oscillation operator 𝒪(RΔλ,∗) and variation operator 𝒱ρ(RΔλ,∗) of the Riesz transform RΔλ associated with Δλ are both bounded on Lp(ℝ+, dmλ) for p ∈ (1,∞), from L1(ℝ+, dmλ) to L1,∞(ℝ+, dmλ), and from L∞(ℝ+, dmλ) to BMO(ℝ+, dmλ), where ρ ∈ (2,∞) and dmλ(x) := x2λ dx. As an application, we give the corresponding Lp-estimates for β-jump operators and the number of up-crossings.

Let ${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+{\mathcal{V}}$ be a Schrödinger operator on $\mathbb{R}^{n},n\geq 3$, where ${\mathcal{V}}$ is a potential satisfying an appropriate reverse Hölder inequality. In this paper, we prove the boundedness of the Riesz transforms and the Littlewood–Paley square function associated with Schrödinger operators ${\mathcal{L}}$ in some new function spaces, such as new weighted Bounded Mean Oscillation (BMO) and weighted Lipschitz spaces, associated with ${\mathcal{L}}$. Our results extend certain well-known results.

Let L be a one-to-one operator of type ω in L2(ℝn), with ω∈[0, π/2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. Let p(·): ℝn→(0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space $H_L^{p(\cdot )} ({\open R}^n)$ associated with L. By means of variable tent spaces, the authors establish the molecular characterization of $H_L^{p(\cdot )} ({\open R}^n)$. Then the authors show that the dual space of $H_L^{p(\cdot )} ({\open R}^n)$ is the bounded mean oscillation (BMO)-type space ${\rm BM}{\rm O}_{p(\cdot ),{\kern 1pt} L^ * }({\open R}^n)$, where L* denotes the adjoint operator of L. In particular, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of $H_L^{p(\cdot )} ({\open R}^n)$ and show that the fractional integral L−α for α∈(0, (1/2)] is bounded from $H_L^{p(\cdot )} ({\open R}^n)$ to $H_L^{q(\cdot )} ({\open R}^n)$ with (1/p(·))−(1/q(·))=2α/n, and the Riesz transform ∇ L−1/2 is bounded from $H_L^{p(\cdot )} ({\open R}^n)$ to the variable Hardy space Hp(·)(ℝn).

Let $H=-\unicode[STIX]{x1D6E5}+V$ be a Schrödinger operator with some general signed potential $V$. This paper is mainly devoted to establishing the $L^{q}$-boundedness of the Riesz transform $\unicode[STIX]{x1D6FB}H^{-1/2}$ for $q>2$. We mainly prove that under certain conditions on $V$, the Riesz transform $\unicode[STIX]{x1D6FB}H^{-1/2}$ is bounded on $L^{q}$ for all $q\in [2,p_{0})$ with a given $2<p_{0}<n$. As an application, the main result can be applied to the operator $H=-\unicode[STIX]{x1D6E5}+V_{+}-V_{-}$, where $V_{+}$ belongs to the reverse Hölder class $B_{\unicode[STIX]{x1D703}}$ and $V_{-}\in L^{n/2,\infty }$ with a small norm. In particular, if $V_{-}=-\unicode[STIX]{x1D6FE}|x|^{-2}$ for some positive number $\unicode[STIX]{x1D6FE}$, $\unicode[STIX]{x1D6FB}H^{-1/2}$ is bounded on $L^{q}$ for all $q\in [2,n/2)$ and $n>4$.

We raise a question of whether the Riesz transform on $\mathbb{T}^{n}$ or $\mathbb{Z}^{n}$ is characterized by the ‘maximal semigroup symmetry’ that the transform satisfies. We prove that this is the case if and only if the dimension is one, two or a multiple of four. This generalizes a theorem of Edwards and Gaudry for the Hilbert transform on $\mathbb{T}$ and $\mathbb{Z}$ in the one-dimensional case, and extends a theorem of Stein for the Riesz transform on $\mathbb{R}^{n}$. Unlike the $\mathbb{R}^{n}$ case, we show that there exist infinitely many linearly independent multiplier operators that enjoy the same maximal semigroup symmetry as the Riesz transforms on $\mathbb{T}^{n}$ and $\mathbb{Z}^{n}$ if the dimension $n$ is greater than or equal to three and is not a multiple of four.

Let $w$ be either in the Muckenhoupt class of ${{A}_{2}}\left( {{\mathbb{R}}^{n}} \right)$ weights or in the class of $QC\left( {{\mathbb{R}}^{n}} \right)$ weights, and let ${{L}_{w}}\,:=\,-{{w}^{-1}}\,\text{div}\left( A\nabla \right)$ be the degenerate elliptic operator on the Euclidean space ${{\mathbb{R}}^{n}}$, $n\,\ge \,2$. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space $H_{{{L}_{w}}}^{p}\,\left( {{\mathbb{R}}^{n}} \right)$ associated with ${{L}_{w}}$ for $p\,\in \,(0,\,1]$, and when $p\,\in \,(\frac{n}{n+1},\,1]$ and $w\,\in \,{{A}_{{{q}_{0}}}}\left( {{\mathbb{R}}^{n}} \right)$ with ${{q}_{0}}\,\in \,[1,\,\frac{p(n+1)}{n})$, the authors prove that the associated Riesz transform $\nabla L_{w}^{-1/2}$ is bounded from $H_{{{L}_{w}}}^{p}\,\left( {{\mathbb{R}}^{n}} \right)$ to the weighted classical Hardy space $H_{w}^{p}\left( {{\mathbb{R}}^{n}} \right)$.

We prove a uniformcontrol as $z\,\to \,0$ for the resolvent ${{(P-z)}^{-1}}$ of long range perturbations $P$ of the Euclidean Laplacian in divergence form by combining positive commutator estimates and properties of Riesz transforms. These estimates hold in dimension $d\,\ge \,3$ when $P$ is defined on ${{\mathbb{R}}^{d}}$ and in dimension $d\,\ge \,2$ when $P$ is defined outside a compact obstacle with Dirichlet boundary conditions.

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.