On the $H^1$--$L^1$ boundedness of operators

Abstract

Abstract. We prove that if q is in (1, ∞), Y is a Banach space, and T is a linear operator defined on the space of finite linear combinations of (1, q)-atoms in R n with the property that sup{ T a Y : a is a (1, q)-atom} < ∞, then T admits a (unique) continuous extension to a bounded linear operator from H 1 (R n ) to Y . We show that the same is true if we replace (1, q)-atoms by continuous (1, ∞)-atoms. This is known to be false for (1, ∞)-atoms.

Anisotropic Hardy-Lorentz spaces and their applications

“…In addition, as an application of the finite atomic decomposition characterizations for H p,q A (R n ) (see Theorem 5.7 below) obtained in Section 5, we establish a criterion on the boundedness of sublinear operators from H p,q A (R n ) into a quasi-Banach space (see Theorem 6.13 below), which is of independent interest; by this criterion, we further conclude that, if T is a sublinear operator and maps all (p, r, s)-atoms with r ∈ (1, ∞) (or all continuous (p, ∞, s)-atoms) into uniformly bounded elements of some quasi-Banach space B, then T has a unique bounded sublinear extension from H p,q A (R n ) into B (see Corollary 6.14 below). This extends the corresponding results of Meda et al [50], Yang-Zhou [74] and Grafakos et al [36] to the present setting. Finally, via the criterion established in Theorem 6.13, we also obtain the boundedness of δ-type Calderón-Zygmund operators from…”

Bilinear decompositions and commutators of singular integral operators

“…From this result, we see that Inequality (1.2) does not suffice to conclude that [b, T ] is bounded from H 1 b (R n ) into L 1 (R n ). In the setting of H 1 (R n ), it is well-known (see [34] or [44] for details) that a linear operator U can be extended to a bounded operator from…”

“…Remark that the interest of dealing with finite atomic decompositions has been underlined recently, for instance in [34,23]. Now, we denote by H 1 fin (R n ) the vector space of all finite linear combinations of ψ-atoms, that is,…”

Boundedness of sublinear operators in Triebel–Lizorkin spaces via atoms

“…From this, they further deduced that a linear operator defined on H 1, q fin (R n ) which maps (1, q, 0)-atoms of H 1 (R n ) or continuous (1, ∞, 0)-atoms of H 1 (R n ) into uniformly bounded elements of some Banach space B uniquely extends to a bounded operator from H 1 (R n ) to B. In [13], the full results of [19] are generalized to H p (X ) and quasi-Banach-valued sublinear operators, where X is an RD-space having "dimension n" in some sense and p ∈ (n/(n + 1), 1].…”

“…Recently, Meda, Sjögren and Vallarino [19] independently obtained a remarkable result by a different method from [25]. For q ∈ (1, ∞], denote by H 1,q fin (R n ) the vector space of all finite linear combinations of (1, q, 0)-atoms of H 1 (R n ) endowed with the norm f H …”