Journal
POTENTIAL ANALYSIS
Volume 57, Issue 4, Pages 545-569Publisher
SPRINGER
DOI: 10.1007/s11118-021-09927-y
Keywords
Critical function; Quatitative weighted estimate; Square function; Heat kernel
Categories
Funding
- Australian Research Council through the ARC [DP190100970]
- Vietnam National Foundation for Science and Technology Development (NAFOSTED) [101.020-2019.327]
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This paper proves quantitative weighted estimates for some square functions associated to L, including the vertical square function, the conical square function, and the g-functions. The novelty of our results is that the square functions associated to L may have rough kernels that do not belong to the Calderon-Zygmund class, and the class of weights is larger than the class of Muckenhoupt weights.
Let L be a non-negative self-adjoint operator on L-2(X) where X is a metric space with a doubling measure. Assume that the kernels of the semigroup generated by -L satisfy a suitable upper bound related to a critical function rho but these kernels are not assumed to satisfy any regularity conditions on spacial variables. In this paper, we prove the quantitative weighted estimates for some square functions associated to L which include the vertical square function, the conical square function and the g-functions. The novelty of our results is that the square functions associated to L might have rough kernels, hence do not belong to the Calderon-Zygmund class, and the class of weights is larger than the class of Muckenhoupt weights. Our results have applications in various settings of Schrodinger operators such as magnetic Schrodinger operators on the Euclidean space R-n and Schrodinger operators on doubling manifolds.
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