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A characterization of commutators of parabolic singular integrals. (English) Zbl 1039.42500

García-Cuerva, José (ed.) et al., Fourier analysis and partial differential equations. Proceedings of the conference held in Miraflores de la Sierra, Madrid, Spain, June 15–20, 1992. Boca Raton, FL: CRC Press (ISBN 0-8493-7877-X). Studies in Advanced Mathematics, 195-210 (1995).
The author considers parabolic bilinear singular integrals of the form \[ T[A] f(x,t)= \text{p.v.} \int_{\mathbb{R}} \int_{\mathbb{R}^{n-1}} K(x- y,t-s) [A(x,t)- A(y,s)] f(y,s)\, dy\, ds, \] where the kernel \(K\) is smooth away from the origin and satisfies an appropriate cancellation condition on the unit sphere, and the parabolic homogeneity condition \(K(\lambda x, \lambda^2 t)= \lambda^{-n-2} K(x,t)\). These operators are the parabolic analogues of the Calderón first commutator. The author characterizes the functions \(A(x,t)\) for which \(T[A]\) is bounded in \(L^2 (\mathbb{R}^n)\).
Let \(| (x,t)|\) be the nonisotropic “norm” associated with the group of parabolic dilations \((\lambda x, \lambda^2t)\), \((x,t)\in \mathbb{R}^{n-1} \times \mathbb{R}\), and \(I_p\) the parabolic fractional integral of order 1, \(\widehat {I_p f} (x,t)= \| (x,t)\|^{-1} \widehat f(x,t)\). Let \(\text{BMO}_p\) denote the space of functions of bounded mean oscillation defined with parabolic cubes, and \(I_p (\text{BMO}_p)\) the parabolic BMO Sobolev space given by \[ I_p (\text{BMO}_p)= \{A(x,t): A= I_p a,\;a\in \text{BMO}_p\}. \] The main results proved in the paper are:
Theorem 1. Let \(A\in I_p (\text{BMO}_p)\) satisfy \(\| \nabla_x A(x,t) \|_{L^\infty (\mathbb{R}^n)}< \infty\). If \(K\) satisfies the conditions described above then \(T[A]\) is bounded in \(L^2 (\mathbb{R}^n)\). Theorem 2. There exists a kernel \(K\) satisfying the conditions above such that if \(T[A]\) is bounded in \(L^2 (\mathbb{R}^n)\) then \(\| \nabla_x A(x,t) \|_{L^\infty (\mathbb{R}^n)}< \infty\) and \(A\in I_p (\text{BMO}_p)\).
To prove Theorem 1, the author uses a generalization of the \(T1\) Theorem of David and Journé to the homogeneous spaces setting. For Theorem 2, he shows that one can take \(K\) such that \(T[A]= [H^{1/2}, A]\), where \(H= \Delta- \partial/\partial t\).
For the entire collection see [Zbl 0847.00037].

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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