Dangyang He - Academia.edu (original) (raw)
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Papers by Dangyang He
Annali di matematica pura ed applicata, Jul 9, 2024
arXiv (Cornell University), Feb 27, 2023
We consider a class of non-doubling manifolds M defined by taking connected sum of finite Riemann... more We consider a class of non-doubling manifolds M defined by taking connected sum of finite Riemannian manifolds with dimension N which has the form R ni × M i and the Euclidean dimension n i are not necessarily all the same. In [21], Hassell and Sikora proved that the Riesz transform on M is weak type (1, 1), bounded on L p (M) for all 1 < p < n * where n * = min k n k and is unbounded for p ≥ n *. In this note we show that the Riesz transform is bounded from Lorentz space L n * ,1 (M) to L n * ,1 (M). This complete the picture by obtaining the end point results for p = n *. Our approach is based on parametrix construction described in [21] and a generalisation of Hardy-Hilbert type inequalities first studied by Hardy, Littlewood and Pólya. Contents 1. Introduction 1 2. Manifolds with ends 4 2.1. Resolvent of the Laplacian 5 2.2. The estimates of resolvent on the product space R n i × M i 5 2.3. Low energy parametrix 6 3. Hardy-Hilbert type inequalities and Lorentz spaces 8 3.1. Hardy-Littlewood inequality 8 3.2. Hardy-Hilbert type inequalities on homogeneous spaces 9 3.3. Hardy-Hilbert type inequalities on inhomogeneous spaces 12 3.4. The strong (p, q) estimates for G 3 and G 4 16 4. Proof of theorem 1.3 18 Acknowledgments 23 References 23
Annali di matematica pura ed applicata, Jul 9, 2024
arXiv (Cornell University), Feb 27, 2023
We consider a class of non-doubling manifolds M defined by taking connected sum of finite Riemann... more We consider a class of non-doubling manifolds M defined by taking connected sum of finite Riemannian manifolds with dimension N which has the form R ni × M i and the Euclidean dimension n i are not necessarily all the same. In [21], Hassell and Sikora proved that the Riesz transform on M is weak type (1, 1), bounded on L p (M) for all 1 < p < n * where n * = min k n k and is unbounded for p ≥ n *. In this note we show that the Riesz transform is bounded from Lorentz space L n * ,1 (M) to L n * ,1 (M). This complete the picture by obtaining the end point results for p = n *. Our approach is based on parametrix construction described in [21] and a generalisation of Hardy-Hilbert type inequalities first studied by Hardy, Littlewood and Pólya. Contents 1. Introduction 1 2. Manifolds with ends 4 2.1. Resolvent of the Laplacian 5 2.2. The estimates of resolvent on the product space R n i × M i 5 2.3. Low energy parametrix 6 3. Hardy-Hilbert type inequalities and Lorentz spaces 8 3.1. Hardy-Littlewood inequality 8 3.2. Hardy-Hilbert type inequalities on homogeneous spaces 9 3.3. Hardy-Hilbert type inequalities on inhomogeneous spaces 12 3.4. The strong (p, q) estimates for G 3 and G 4 16 4. Proof of theorem 1.3 18 Acknowledgments 23 References 23