Maximal function characterizations for Hardy spaces associated with nonnegative self-adjoint operators on spaces of homogeneous type

Abstract

Abstract. Let X be a metric measure space with a doubling measure and L be a nonnegative selfadjoint operator acting on L 2 (X). Assume that L generates an analytic semigroup e −tL whose kernels p t (x, y) satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables x and y. In this article we continue a study in [21] to give an atomic decomposition for the Hardy spaces H p L,max (X) in terms of the nontangential maximal function associated with the heat semigroup of L, and hence we establish characterizations of Hardy spaces associated to an operator L, via an atomic decomposition or the nontangential maximal function. We also obtain an equivalence of H p L,max (X) in terms of the radial maximal function.

Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type

“…We then prove a generalization of (2) and (3) to h (4) for 0 < p ≤ 1, which is the content of Theorem 2.4. This can be viewed as a local version of those in [38]. If one further assumes (A3) and (A4) then one can add h p at (X) to picture for n n+δ < p ≤ 1, which is the content of Theorem 2.7.…”

“…If one further assumes (A3) and (A4) then one can add h p at (X) to picture for n n+δ < p ≤ 1, which is the content of Theorem 2.7. We remark that the ideas in the proof of Theorem 2.4 rely on the innovations in [38], although some significant modifications are needed, not least of which the development of an inhomogeneous Calderón reproducing formula (Proposition 3.6).…”

Frame decomposition and radial maximal semigroup characterization of Hardy spaces associated to operators

“…Remark 6.2. We should note that the equivalency of the radial maximal function characterization and the nontangential maximal function characterizations of H 1 L (R n ) have been obtained in [40,36]. Our Theorem 6.1 provides a different proof by using the frame decomposition.…”

“…A maximal function characterization of H 1 L (R n ) In this section, we continue the discussion from Section 4 regarding a characterization of the Hardy space H 1 L (R n ) in terms of radial maximal function under the following conditions: (H1) ′ L is a second order non-negative self-adjoint operator on L 2 (R n ); (H2) ′ The kernel of e −tL , denoted by p t (x, y), is a measurable function on R n × R n and satisfies a Gaussian upper bound, that is p t (x, y) ≤ Ct −n/2 exp − |x − y| 2 ct for all t > 0, and x, y ∈ R n , where C and c are positive constants. The space H 1 L (R n ) involves some different characterizations, see for examples, [2,18,19,22,27,28,29,35,36,40]. If an operator L satisfies conditions (H1) ′ and (H2) ′ , then for any M ≥ 1, 1 < q ≤ ∞, Define the spaces H 1 L,max (R n ) as the completion of L 2 (R n ) in the norms given by the L 1 (R n ) norm of the maximal function, i.e., f H 1 L,max (R n ) = f + h L 1 (R n ) .…”

“…Yang [40] obtained it under the additional assumption that the kernel of the heat semigroup satisfies Hölder's continuity. In [36], L. Song and L.X. Yan got rid of the additional assumption of [40] and proved the radial maximal function characterization of Hardy spaces associated to L.…”