Researcher Ngô Thị Hồng with the two supervisors: Assoc. Prof. Lý Kim Hà (centre) and Assoc. Prof. Đào Văn Dương

On 6 July, at VNUHCM-University of Science, researcher Ngô Thị Hồng successfully defended a doctoral dissertation in Mathematical Analysis entitled: “Hausdorff Operators and Riesz Potentials on Certain Function Spaces”. The dissertation was supervised by Assoc. Prof. Lý Kim Hà and Assoc. Prof. Đào Văn Dương.

The dissertation focuses on the boundedness and norm estimates of various classes of Hausdorff operators, Riesz potentials, and their commutators on function spaces such as weighted Morrey spaces, weighted Herz spaces, and weighted Morrey-Herz spaces, with symbols belonging to weighted Lipschitz spaces and weighted BMO spaces. These operator classes are either inclusive of or closely related to numerous classical integral operators, such as the Hardy operator, Cesàro operator, Riemann–Liouville fractional integral, Hardy–Littlewood–Pólya operator, Hilbert operator, Hardy–Littlewood maximal operator, and singular integral operators, among others.

The dissertation is built upon five research articles published in reputable international journals: Journal of Pseudo-Differential Operators and Applications, Russian Journal of Mathematical Physics, Analysis and Mathematical Physics, Fractional Calculus and Applied Analysis, and p-Adic Numbers, Ultrametric Analysis and Applications. Key contributions of the dissertation include:

• Establishing necessary and sufficient conditions for the boundedness of Hausdorff operators associated with Opdam–Cherednik transforms and their commutators with symbols in the Lipschitz space on weighted Morrey and Morrey-Herz spaces.
• Introducing extended sublinear Hausdorff operators and studying their boundedness on product weighted Morrey and Morrey-Herz spaces.
• Providing necessary and sufficient conditions for the boundedness of multilinear p-adic Hausdorff operators on weighted Morrey and Herz spaces, along with applications.
• Establishing sufficient conditions for the boundedness of p-adic Riesz potentials on weighted Morrey-Herz spaces, and obtaining weighted Lipschitz and weighted BMO estimates for their commutators.
• Establishing sufficient conditions for the boundedness of p-adic Riesz potentials with rough kernels and their commutators with BMO-type symbols on weighted Morrey and Herz spaces.

The results of the dissertation have significant academic and practical relevance in the fields of harmonic analysis, partial differential equations, operator theory, function space theory, and related applications. Based on the current findings, several open problems can be explored in future studies, such as:

• Extending the investigation of boundedness for Hausdorff operators associated with Opdam–Cherednik transforms on function spaces with homogeneous weights, Muckenhoupt weights, or general weight classes in vector spaces.
• Studying the boundedness of commutators for extended sublinear Hausdorff operators and multilinear p-adic Hausdorff operators on more general function spaces.
• Establishing necessary conditions for the boundedness of p-adic Riesz potentials and p-adic Riesz potentials with rough kernels on various function spaces.

Additional images from the doctoral defence session

_{▪︎ Detailed information on the dissertation has been published on the official doctoral thesis portal of VNUHCM-University of Science.}

TMT