2018/05/13 by Cruz Prisuelos-Arribas, Prisuelos-Arribas, Cruz
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1805.04853
openalex publication_date 2018/05/13 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We study the vertical and conical square functions defined via elliptic\noperators in divergence form. In general, vertical and conical square functions\nare equivalent operators just in L2. But when this square functions are\ndefined through the heat or Poisson semigroup that arise from an elliptic\noperator, we are able to find open intervals containing 2 where the\nequivalence holds. The intervals in question depend ultimately on the range\nwhere the semigroup is uniformly bounded or has off-diagonal estimates. As a\nconsequence we obtain new boundedness results for some square functions.\nBesides, we consider a non-tangential maximal function associated with the\nPoisson semigroup and extend the known range where that operator is bounded.\nOur methods are based on the use of extrapolation for Muckenhoupt weights and\nchange of angle estimates. All our results are obtained in the general setting\nof a degenerate elliptic operator, where the degeneracy is given by an A2\nweight, in weighted Lebesgue spaces. Of course they are valid in the unweighted\nand/or non-degenerate situations, which can be seen as special cases, and\nprovide new results even in those particular settings.\n We also consider the square root of a degenerate elliptic operator in\ndivergence form Lw and improve the lower bound of the interval where this\noperator is known to be bounded on Lp(vdw). To finish we give unweighted\nboundedness results for the degenerate operators under consideration.\n