Let L be the divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on (0, ∞) of strictly critical lower type p ω ∈ (0, 1] and ρ(t) = t −1 /ω −1 (t −1 ) for t ∈ (0, ∞). In this paper, the authors study the Orlicz-Hardy space H ω,L (R n ) and its dual space BMO ρ,L * (R n ), where L * denotes the adjoint operator of L in L 2 (R n ). Several characterizations of H ω,L (R n ), including the molecular characterization, the Lusin-area function characterization and the maximal function characterization, are established. The ρ-Carleson measure characterization and the JohnNirenberg inequality for the space BMO ρ,L (R n ) are also given. As applications, the authors show that the Riesz transform ∇L −1/2 and the Littlewood-Paley g-function g L map H ω,L (R n ) continuously into L(ω). The authors further show that the Riesz transform ∇L −1/2 maps H ω,L (R n ) into the classical Orlicz-Hardy space H ω (R n ) for p ω ∈ ( n n+1 , 1] and the corresponding fractional integral L −γ for certain γ > 0 maps H ω,L (R n ) continuously into H ω,L (R n ), where ω is determined by ω and γ , and satisfies the same property as ω. All these results are new even when ω(t) = t p for all t ∈ (0, ∞) and p ∈ (0, 1).

Abstract. Let X be a metric space with doubling measure, L a nonnegative self-adjoint operator in L 2 (X ) satisfying the Davies-Gaffney estimate, ω a concave function on (0, ∞) of strictly lower type p ω ∈ (0, 1] and ρ(t) = t −1 /ω −1 (t −1 ) for all t ∈ (0, ∞). The authors introduce the Orlicz-Hardy space H ω,L (X ) via the Lusin area function associated to the heat semigroup, and the BMO-type space BMO ρ,L (X ). The authors then establish the duality between H ω,L (X ) and BMO ρ,L (X ); as a corollary, the authors obtain the ρ-Carleson measure characterization of the space BMO ρ,L (X ). Characterizations of H ω,L (X ), including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let X = R n and L = −∆ + V be a Schrödinger operator, where V ∈ L 1 loc (R n ) is a nonnegative potential. As applications, the authors show that the Riesz transform ∇L; moreover, if there exist q 1 , q 2 ∈ (0, ∞) such that q 1 < 1 < q 2 and [ω(t q 2 )] q 1 is a convex function on (0, ∞), then several characterizations of the Orlicz-Hardy space H ω,L (R n ), in terms of the Lusin-area functions, the nontangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when ω(t) = t p for all t ∈ (0, ∞) and p ∈ (0, 1).

Let L be a linear operator in L 2 (R n ) and generate an analytic semigroup {e −tL } t 0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞).Let ω on (0, ∞) be of upper type 1 and of critical lower type p0(ω) ∈ (n/(n + θ(L)), 1] and ρ(tWe introduce the Orlicz-Hardy space Hω, L(R n ) and the BMO-type space BMOρ, L(R n ) and establish the John-Nirenberg inequality for BMOρ, L(R n ) functions and the duality relation between Hω, L(R n ) and BMOρ, L * (R n ), where L * denotes the adjoint operator of L in L 2 (R n ).Using this duality relation, we further obtain the ρ-Carleson measure characterization of BMOρ, L * (R n ) and the molecular characterization of Hω, L(R n ); the latter is used to establish the boundedness of theis the Hardy space introduced by Auscher, Duong and McIntosh. These results generalize the existing results by taking ω(t) = t p for t ∈ (0, ∞) and p ∈ (n/(n + θ(L)), 1].

Let φ:Rn×[0,∞)→[0,∞) satisfy that φ(x,·), for any given x∈Rn, is an Orlicz function and φ(·,t) is a Muckenhoupt A∞(double-struckRn) weight uniformly in t∈(0,∞). In this article, the authors introduce the weak Musielak–Orlicz Hardy space WHφ(double-struckRn) via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of WHφ(Rn), respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or gλ*‐function. All these characterizations for weighted weak Hardy spaces WHwp(double-struckRn) (namely, φ(x,t):=w(x)tp for 0.33em all t∈[0,∞) and x∈Rn with p∈(0,1] and w∈A∞(Rn)) are new and part of these characterizations even for weak Hardy spaces WHp(double-struckRn) (namely, φ(x,t):=tp for 0.33em all t∈[0,∞) and x∈Rn with p∈(0,1]) are also new. As an application, the boundedness of Calderón–Zygmund operators from Hφ(Rn) to WHφ(double-struckRn) in the critical case is presented.

Let (X, d, µ) be a RCD * (K, N) space with K ∈ R and N ∈ [1, ∞). We derive the upper and lower bounds of the heat kernel on (X, d, µ) by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the L p boundedness of (local) Riesz transforms.

Please cite this article in press as: R. Jiang, The Li-Yau inequality and heat kernels on metric measure spaces, J. Math. Pures Appl. (2014), http://dx.Suppose that (X, d) is connected, complete and separable, and supp μ = X. We prove that the Li-Yau inequality for the heat flow holds true on (X, d, μ) when K ≥ 0. A Baudoin-Garofalo inequality and Harnack inequalities for the heat flow are established on (X, d, μ) for general K ∈ R. Large time behaviors of heat kernels are also studied.On suppose que (X, d) est connexe, complet et séparable et supp μ = X. Nous démontrons que l'inégalité de Li-Yau pour le flot de la chaleur sur (X, d, μ) est satisfaite lorsque K ≥ 0. De plus, nousétablissons une inégalité de type Baudoin-Garofalo ainsi que des inégalités de Harnack pour ce flot dans la situation plus générale où K ∈ R. Le comportement en temps long des noyaux de la chaleur est aussiétudié.

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Let (X, d, µ) be a complete metric measure space, with µ a locally doubling measure, that supports a local weak L 2 -Poincaré inequality. By assuming a heat semigroup type curvature condition, we prove that Cheegerharmonic functions are Lipschitz continuous on (X, d, µ). Gradient estimates for Cheeger-harmonic functions and solutions to a class of non-linear Poisson type equations are presented.