On Hardy spaces of local and nonlocal operators. (English) Zbl 1312.42026
This paper deals with the characterization of \(L^p\) Hardy spaces on a domain \(D\) of \({\mathbb R}^d\) by means of Hardy-Stein identities, motivated by the study of martingale Hardy and Bergman spaces, in connection with quadratic variation of martingales and the operator carré du champ. The setting of the paper includes Hardy spaces for the fractional Laplacian \(\Delta^{\alpha/2}\) under conditioning by a fixed \(\alpha\)-harmonic function \(h\), for which the authors derive nonlocal and conditional extensions of the classical Hardy-Stein inequalities.
Reviewer: Nicolas Privault (Singapore)
MSC:
42B30 | \(H^p\)-spaces |
42B35 | Function spaces arising in harmonic analysis |
60G51 | Processes with independent increments; Lévy processes |
60G52 | Stable stochastic processes |
60J75 | Jump processes (MSC2010) |
60J50 | Boundary theory for Markov processes |
30H10 | Hardy spaces |
30H20 | Bergman spaces and Fock spaces |
35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
35R11 | Fractional partial differential equations |
31B25 | Boundary behavior of harmonic functions in higher dimensions |