A local \(Tb\) theorem for square functions and parabolic layer potentials. (English) Zbl 07914959
Summary: In this paper, we give a locally parabolic version of \(Tb\) theorem for a class of vector-valued operators with off-diagonal decay in \(L^2\) and certain quasi-orthogonality on a subspace of \(L^2\), in which the testing functions themselves are also vector-valued. As an application, we establish the boundedness of layer potentials related to parabolic operators in divergence form, defined in the upper half-space \(\mathbb{R}_+^{n+2} := \{(x, t, \lambda)\in\mathbb{R}^{n+1}\times(0, \infty)\}\), with uniformly complex elliptic, \(L^\infty\), \(t\), \(\lambda\)-independent coefficients, and satisfying the De Giorgi/Nash estimates.
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 35J15 | Second-order elliptic equations |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Keywords:
local \(Tb\) theorems; Calderón-Littlewood-Paley family; parabolic layer potentials; Carleson measure estimatesReferences:
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