Harmonic analysis on and geometry of fractal sets

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Project objective

Research objectives and content It is proposed to tackle problems in the area of intersection of geometric measure theory and harmonic analysis, focussing particularly on the interaction between geometric and analytic properties of sets. Proposed problems include: 1. providing a geometric characterization of general 1-dimensional subsets of the plane of analytic capacity zero, 2. investigating implications of L2-boundedness of various singular integral operators (in particular, discovering for which such operators L2-boundedness implies rectifiability), 3. discovering whether a set in the plane which has projections onto lines of positive length in almost every direction must have bounded Melnikov curvature, 4. understanding asymptotic doubling properties of general sets, 5. investigating relationships between Poincari and Trudinger inequalities, doubling or strong doubling conditions, and the geometry of sets. Training content (objective, benefit and expected impact) The proposed collaboration would strengthen the preexisting links between NUIM and the applicant's current institution, the University of Jyvdskyld. (The two departments are currently part of a network, and a number of members of staff collaborate regularly.) The applicant's previous training in geometric measure theory and real analysis, and the host's expertise in harmonic analysis, make collaboration on such problems very natural. The proposed project would also broaden the applicant's range of expertise, while building on her current research knowledge. Links with industry / industrial relevance (22) None.

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Participants

• National University of Ireland - Maynooth · MAYNOOTHCoordinatorIreland

Links

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