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Riesz transform characterizations of Hardy spaces associated to degenerate elliptic operators. (English) Zbl 1355.42018

Let \(w\) be a Muckenhoupt \(A_2(\mathbb{R}^n)\) weight and let \(A(x)\), \(x\in\mathbb{R}^n\), be an \(n\times n\) matrix of complex-valued measurable functions satisfying a degenerate elliptic condition. The associated degenerate elliptic operator \(L_w\) is defined as \[ L_w f = -\frac{1}{w} \mathrm{div} (A \nabla f) \] for \(f\in \mathrm{Dom}(L_w)\subset \mathcal{H}_0^1(w, \mathbb{R}^n)\), where \(\mathcal{H}_0^1(w, \mathbb{R}^n)\) is a weighted Sobolev space. For \(0<p\leq 1\), the Hardy space \(H^p_{L_w}(\mathbb{R}^n)\) is defined as the completion of \[ \{f\in L^2(w, \mathbb{R}^n): \|S_{L_w} f\|_{L^p(w, \mathbb{R}^n)}\} \] with respect to the quasi-norm \(\|S_{L_w} f\|_{L^p(w, \mathbb{R}^n)}\), where \(S_{L_w} f\) is the square function associated with \(L_w\). Under additional restrictions on \(L_w\) and \(p\in (0, 1]\), the authors prove that \(H^p_{L_w}(\mathbb{R}^n)\), \(n\geq 3\), coincides with the Hardy space \(H^p_{L_w, \mathrm{Riesz}}(\mathbb{R}^n)\) defined in terms of the Riesz transform. For \(w\equiv 1\), this result was obtained by S. Hofmann et al. [Ann. Sci. Éc. Norm. Supér. (4) 44, No. 5, 723–800 (2011; Zbl 1243.47072)].

MSC:

42B30 \(H^p\)-spaces
35J70 Degenerate elliptic equations
42B35 Function spaces arising in harmonic analysis
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators

Citations:

Zbl 1243.47072
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References:

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