Weighted \(L^p\) estimates of Kato square roots associated to degenerate elliptic operators. (English) Zbl 06781947

Yang, Dachun; Zhang, Junqiang

Publ. Mat., Barc. 61, No. 2, 395-444 (2017).

Summary: Let \(w\) be a Muckenhoupt \(A_2({\mathbb R}^n)\) weight and \(L_w:=-w^{-1}\operatorname{div}(A\nabla)\) the degenerate elliptic operator on the Euclidean space \({\mathbb R}^n\), \(n\geq 2\). In this article, the authors establish some weighted \(L^p\) estimates of Kato square roots associated to the degenerate elliptic operators \(L_w\). More precisely, the authors prove that, for \(w\in A_{p}({\mathbb R}^n)\), \(p\in(\frac{2n}{n+1},2]\) and any \(f\in C^\infty_c({\mathbb R}^n)\), \(\|L_w^{1/2}(f)\|_{L^p(w,{\mathbb R}^n)} ~ \|\nabla f\|_{L^p(w,{\mathbb R}^n)}\), where \(C_c^\infty({\mathbb R}^n)\) denotes the set of all infinitely differential functions with compact supports and the implicit equivalent positive constants are independent of \(f\).

MSC:

47B06	Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
35J70	Degenerate elliptic equations
42B30	\(H^p\)-spaces
42B35	Function spaces arising in harmonic analysis

Keywords:

Kato square root; degenerate elliptic operator; Riesz transform; Lebesgue space; Hardy space; square function; Muckenhoupt weight