Comparison of the classical BMO with the BMO spaces associated with operators and applications

Abstract

Let L be a generator of a semigroup satisfying the Gaussian upper bounds. A new BMO L space associated with L was recently introduced in [15] and [16]. We discuss applications of the new BMO L spaces in the theory of singular integration. For example we obtain BMO L estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space BMO L might coincide with or might be essentially different from the classical BMO space.

Classes of Hardy spaces associated with operators, duality theorem and applications

“…To begin with, let us recall some basic facts about the Neumann Laplacian N on R n , which was studied in [12]. In what follows, R n + denotes the upper-half space in R n , i.e.,…”

“…On the other hand, from Theorem 4.1 of [12] and Proposition 5.3 of [19], this operator N generates the spaces L N (α, 2, 0) with 0 ≤ α < n −1 such that (i) L(α, 2, 0) ⊆ L N (α, 2, 0).…”

“…H(x n y n ), (6.6) where H : R → {0, 1} is the Heaviside function, given by H(t) = 0 if t < 0; 1 if t ≥ 0. See Section 2 of [12].…”

“…For more properties of the space H 1 L (R n ) and the BMO L (R n ) space, we refer the reader to [5], [17], [18], [13] and [12].…”

“…comparison with the classical H p (R n ) spaces, an important feature of the H p L (R n ) spaces is that they tightly connect the operators considered, which may be an effective tool in the study of singular integral operators associated with the operator L; see [5], [12], [15], [17] and [18] for more details.…”