Fefferman inequality for Carnot-Carathéodory vector fields and quasilinear subelliptic equations

G. Di Fazio and P. Zamboni

In his celebrated paper, C.Fefferman ([4]) proved that

(∗)  ∫_{B} |u(x) − u_{B}|^{p}|V (x)| dx ≤ c ∫_{B} |∇u(x)|^{p} dx  ∀u ∈ C^{∞}

in the case p = 2 assuming the potential V to belong in the class L^{r,n−2r}, 1 < r ≤ n/2. In (*) u_{B} is the integral average |B|^{−1} ∫_{B} u(y) dy, and |B| is the Lebesgue measure of B.

Later, Chiarenza and Frasca [1] extended Fefferman’s result - with different proof - assuming V in L^{r,n−pr}, 1 < r ≤ n/p, 1 < p < n. Danielli, Garofalo and Nhieu in [3] introduced a suitable version of Morrey spaces adapted to the Carnot - Caratheodory (C-C) metric and proved the same inequality with V in the Morrey space L^{1,λ} for some λ > 0. Later, Danielli in [2] generalized the paper [1] to the setting of C-C metric.

The aim of the talk is to prove inequality (*) when the gradient in the right-hand side is replaced by the energy associated to a non commuting system of vector fields, and the function V is taken in an appropriate Stummel-Kato class, defined using the Carnot-Carathéodory metric associated to the vector fields in a metric space and then to show applications to regularity for weak solutions to some quasilinear subelliptic equation.

References

1. F. Chiarenza and M. Frasca A remark on a paper by C. Fefferman Proc. Amer. Math. Soc. 108 (1990) 407–409;

2. D. Danielli A Fefferman - Phong inequality and applications to quasilinear subelliptic equations Potential Analysis 11 (1999) 387–413;

3. D. Danielli, N.Garofalo and D.Nhieu Trace inequalities for Carnot-Caratheodory spaces and applications Ann. Scuola Norm. Sup. Pisa XXVII (1998) 195–252;

4. C. Fefferman The uncertainty principle Bull. Amer. Math. Soc. 9 (1983) 129–206;