In the Euclidean space, it is known that a function f in L^2 of a ball B, with vanishing average, is the divergence of a vector field F in L^ 2 and the L^2-norm of F in the ball B is controlled by the L^2-norm of f in the same ball B (times a constant ). In this note, we prove a similar result in any Carnot group G for a vanishing average f in L^p, when 1\le p < Q, where Q is the so-called homogeneous dimension of G.
Baldi, A., Franchi, B., Pansu, P. (2024). Primitives of volume forms in Carnot groups. ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI LINCEI. MATEMATICA E APPLICAZIONI, 35(4), 597-617 [10.4171/RLM/1052].
Primitives of volume forms in Carnot groups
Baldi, Annalisa;Franchi, Bruno
;Pansu, Pierre
2024
Abstract
In the Euclidean space, it is known that a function f in L^2 of a ball B, with vanishing average, is the divergence of a vector field F in L^ 2 and the L^2-norm of F in the ball B is controlled by the L^2-norm of f in the same ball B (times a constant ). In this note, we prove a similar result in any Carnot group G for a vanishing average f in L^p, when 1\le p < Q, where Q is the so-called homogeneous dimension of G.| File | Dimensione | Formato | |
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