Renjin Jiang

Maximal function characterizations of Hardy spaces associated with magnetic Schrödinger operators

Abstract. Let be a magnetic Schrödinger operator on , , where and . In this paper, the authors establish the equivalent characterizations of the Hardy space for , defined by the Lusin area function associated with , in terms of the radial maximal functions and the non-tangential maximal functions associated with and , respectively. This gives an affirmative answer to an open problem of Xuan Thinh Duong et al. [Ark. Mat. 44 (2006), 261–275]. The boundedness of the Riesz transforms , , from to is also presented, where is the closure of in and .

Localized Hardy spaces associated with operators

Let L be a linear operator in L 2(ℝ n ) and generate an analytic semigroup {e −tL } t≥0 with kernels satisfying an upper bound of Poisson type. In this article, the authors introduce the localized Hardy space via molecules and show that , where and are Hardy spaces associated with L and L + I, respectively. Characterizations of via the localized Lusin area function and are established. Then, the authors introduce the localized BMO space bmo L (ℝ n ) and prove that the dual of is bmo L*(ℝ n ), where L* denotes the adjoint operator of L in L 2(ℝ n ). The John–Nirenberg inequality for elements in bmo L (ℝ n ) and a characterization of bmo L (ℝ n ) via BMO L (ℝ n ) are also established, where BMO L (ℝ n ) is the BMO space associated with L. As applications, the authors obtain the characterizations of the localized Hardy space associated to the Schrödinger operator L = −Δ + V, where is a nonnegative potential, in terms of the localized Lusin-area functions and the localized radial maximal functions.

Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants

For n≥1 and α ∈ (−1, 1), let Hα,2 be the space of harmonic functions u on the upper half-space ℝ+n+1 satisfying sup(x0,r)∈ℝ+n+1r−(2α+n)∫B(x0,r)∫0r|∇ x,tu(x,t)|2tdtdx < ∞, and ℒ2,n+2α be the Campanato space on ℝn. We show that Hα,2 coincides with e−t−Δℒ 2,n+2α for all α ∈ (−1, 1), where the case α ∈ [0, 1) was originally discovered by Fabes, Johnson and Neri [E. B. Fabes, R. L. Johnson and U. Neri, Spaces of harmonic functions representable by Poisson integrals of functions in BMO and ℒp,λ, Indiana Univ. Math. J. 25 (1976) 159–170] and yet the case α ∈ (−1, 0) was left open. Moreover, for the scaling invariant version of Hα,2, ℋα,2, which comprises all harmonic functions u on ℝ+n+1 satisfying sup(x0,r)∈ℝ+n+1r−(2α+n)∫B(x0,r)∫0r|∇ x,tu(x,t)|2t1+2αdtdx < ∞, we show that ℋα,2 = e−t−Δ(−Δ)α2ℒ2,n+2α, where (−Δ)α 2ℒ2,n+2α is the collection of all functions f such that (−Δ)− α2f are in ℒ2,n+2α. Analogues for solutions to the heat equation are also established. As an application, we show that the spaces (−Δ)α 2ℒ2,n+...

Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators

Let w be either in the Muckenhoupt class of A2(R ) weights or in the class of QC(R) weights, and Lw := −w −1 div(A∇) the degenerate elliptic operator on the Euclidean space R, n ≥ 2. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space H p Lw (R) associated with Lw for p ∈ (0, 1] and, when p ∈ ( n n+1 , 1] and w ∈ Aq0 (R ) with q0 ∈ [1, p(n+1) n ), the authors prove that the associated Riesz transform ∇L −1/2 w is bounded from H p Lw (R) to the weighted classical Hardy space H p w(R ).

Hamilton’s gradient estimates and a monotonicity formula for heat flows on metric measure spaces

Abstract In this paper, we extend the Hamilton’s gradient estimates (Hamilton 1993) and a monotonicity formula of entropy (Ni 2004) for heat flows from smooth Riemannian manifolds to (non-smooth) metric measure spaces with appropriate Riemannian curvature-dimension condition.

Relative ∞-capacity and its affinization

Abstract In this paper, the so-called relative ∞ {\infty} -capacity is introduced and investigated in a close connection to the viscosity solution of the ∞ {\infty} -Laplace equation. We not only show that the relative ∞ {\infty} -capacity equals the limit of the p-th root of the relative p-capacity as p → ∞ {p\to\infty} and hence has a simple geometric characterization in terms of the Euclidean distance, but also establish several basic properties for the relative ∞ {\infty} -capacity. Consequently, we apply the relative ∞ {\infty} -capacity to the embedding theory of the ∞ {\infty} -Sobolev space. More geometrically, we affinize the relative ∞ {\infty} -capacity and its fundamental features as much as possible.

A note on transport equation in quasiconformally invariant spaces

Abstract In this note, we study the well-posedness of the Cauchy problem for the transport equation in the BMO space and certain Triebel–Lizorkin spaces.

Hardy and Hardy-Sobolev Spaces on Strongly Lipschitz Domains and Some Applications

Abstract Let Ω ⊂ Rn be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, Hpr (Ω)and Hpz (Ω), and Hardy-Sobolev spaces, H1,pr (Ω) and H1,pz,0 (Ω) on , for p ∈ ( n/n+1, 1]. The authors establish grand maximal function characterizations of these spaces. As applications, the authors obtain some div-curl lemmas in these settings and, when is a bounded Lipschitz domain, the authors prove that the divergence equation div u = f for f ∈ Hpz (Ω) is solvable in H1,pz,0 (Ω) with suitable regularity estimates.