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- Title
Weak type (1, 1) of some operators for the Laplacian with drift.
- Authors
Li, Hong-Quan; Sjögren, Peter; Wu, Yurong
- Abstract
Let $$v = (v_1, \ldots , v_n)$$ be a vector in $$\mathbb {R}^n {\setminus } \{ 0 \}$$ . Consider the Laplacian on $$\mathbb {R}^n$$ with drift $$\Delta _{v} = \sum _{i = 1}^n \Big ( \frac{\partial ^2}{\partial x_i^2} + 2 v_i \frac{\partial }{\partial x_i} \Big )$$ and the measure $$d\mu (x) = e^{2 \langle v, x \rangle } dx$$ , with respect to which $$\Delta _{v}$$ is self-adjoint. Let d and $$\nabla $$ denote the Euclidean distance and the gradient operator on $$\mathbb {R}^n$$ . Consider the space $$(\mathbb {R}^n, d, d\mu )$$ , which has the property of exponential volume growth. We obtain weak type (1, 1) for the Riesz transform $$\nabla (- \Delta _{v} )^{-\frac{1}{2}}$$ and for the heat maximal operator, with respect to $$d\mu $$ . Further, we prove that the uncentered Hardy-Littlewood maximal operator is bounded on $$L^p$$ for $$1 < p \le +\infty $$ but not of weak type (1, 1) if $$n \ge 2$$ .
- Publication
Mathematische Zeitschrift, 2016, Vol 282, Issue 3/4, p623
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-015-1555-z