Hardy inequalities on metric measure spaces

Area of Science:

• Mathematical Analysis
• Geometric Measure Theory

Background:

• Hardy's inequality is a fundamental tool in analysis.
• Characterizing weights for Hardy inequalities is crucial for understanding function spaces.
• General metric measure spaces lack differentiable structures, necessitating integral forms of inequalities.

Purpose of the Study:

• To provide characterizations of weights for two-weight Hardy inequalities on general metric measure spaces with polar decompositions.
• To extend the applicability of Hardy inequalities to spaces lacking differentiable structures.

Main Methods:

• Integral form of Hardy's inequality.
• Analysis on general metric measure spaces with polar decompositions.
• Derivation of weight characterizations.

Main Results:

• Several characterizations for two-weight Hardy inequalities are established.
• New weighted Hardy inequalities are obtained for specific spaces like homogeneous groups, hyperbolic spaces, and Cartan-Hadamard manifolds.
• The analysis does not require doubling conditions on the measure.

Conclusions:

• The findings provide a generalized framework for weighted Hardy inequalities.
• The results are applicable to a broad range of metric measure spaces.
• The absence of doubling condition requirements broadens the scope of the inequalities.