Maximal characterisation of local Hardy spaces on locally doubling manifolds

We prove a radial maximal function characterisation of the local atomic Hardy space $${{\mathfrak {h}}}^1(M)$$ h 1 ( M ) on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to $${{\mathfrak {h}}}^1(M)$$ h 1 ( M ) if and only if either its local heat maximal function or its local Poisson maximal function is integrable. A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. Uchiyama.

Some Properties of Quasinearly Subharmonic Functions and Maximal Theorem for Bergman Type Spaces

Let denote the class of quasinearly subharmonic functions in unit ball . We provide, following result: if and if , then , where is the radial maximal function and , and . Also, we prove a maximal theorem for Bergman type spaces.

REAL-VARIABLE CHARACTERIZATIONS OF HARDY SPACES ASSOCIATED WITH BESSEL OPERATORS

Let λ > 0, p ∈ ((2λ + 1)/(2λ + 2), 1], and [Formula: see text] be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces Hp((0,∞),dmλ) associated with △λ in terms of the radial maximal function, the nontangential maximal function, the grand maximal function, the Littlewood–Paley g-function and the Lusin-area function, where dmλ(x) ≡ x2λ dx. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.

On the Smirnov Class Defined by the Maximal Function

H. O. Kim has shown that contrary to the case of Hp-space, the Smirnov class M defined by the radial maximal function is essentially smaller than the classical Smirnov class of the disk. In the paper we show that these two classes have the same corresponding locally convex structure, i.e. they have the same dual spaces and the same Fréchet envelopes. We describe a general form of a continuous linear functional on M and multiplier from M into Hp, 0 < p ≤ ∞.