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Morrey estimates for parabolic nondivergence operators of Hörmander type. (English) Zbl 1230.35142

Let \(\Omega\) be a bounded open subset of \({\mathbb{R}}^{n}\). Let \(X_{1}\),…, \(X_{q}\) be a system of real smooth vector fields satisfying the Hörmander rank condition in \(\Omega\). The authors consider the parabolic operator in nondivergence form \[ H_{1}\equiv\partial_{t}-\sum_{i,j=1}^{q}a_{ij}(t,x)X_{i}X_{j}-\sum_{j=1}^{q}b_{j}(t,x)X_{j}-c(t,x)\,, \] where the coefficients \(a_{ij}\) are bounded and measurable in an open subset \(U\) of \({\mathbb{R}}\times\Omega\), have vanishing mean oscillation with respect to the sub-elliptic metric induced by the vector fields \(X_{1}\),…, \(X_{q}\), and satisfy a uniform ellipticity condition. Under appropriate assumptions on the coefficients \(a_{ij}\), \(b_{j}\), \(c\) of \(H_{1}\), the authors estimate the norm of a function \(f\) in a Sobolev space built on a Morrey space in a relatively compact open subset \(U'\) of \(U\) in terms of the norm of \(H_{1}f\) in a lower order Sobolev space built on a Morrey space in \(U\) and of a Morrey norm of \(f\) in \(U\). Here the spatial derivatives associated to the definition of the Sobolev spaces built on a Morrey space are taken with respect to the vector fields \(X_{1}\),…, \(X_{q}\).

MSC:

35R05 PDEs with low regular coefficients and/or low regular data
35B45 A priori estimates in context of PDEs
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
35K10 Second-order parabolic equations
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References:

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