Harmonic Analysis
http://hdl.handle.net/20.500.11824/3
2021-10-21T12:15:16ZA note on generalized Fujii-Wilson conditions and BMO spaces
http://hdl.handle.net/20.500.11824/1339
A note on generalized Fujii-Wilson conditions and BMO spaces
Ombrosi, S.; Pérez, C.; Rela, E.; Rivera-Ríos, I.
In this note we generalize the definition of the Fujii-Wilson condition providing quantitative characterizations of some interesting classes of weights, such as A∞, A∞weak and Cp, in terms of BMO type spaces suited to them. We will provide as well some self improvement properties for some of those generalized BMO spaces and some quantitative estimates for Bloom’s BMO type spaces.
2020-07-01T00:00:00ZDegenerate Poincare-Sobolev inequalities
http://hdl.handle.net/20.500.11824/1338
Degenerate Poincare-Sobolev inequalities
Pérez, C.; Rela, E.
Abstract. We study weighted Poincar ́e and Poincar ́e-Sobolev type in- equalities with an explicit analysis on the dependence on the Ap con- stants of the involved weights. We obtain inequalities of the form with different quantitative estimates for both the exponent q and the constant Cw . We will derive those estimates together with a large variety of related results as a consequence of a general selfimproving property shared by functions satisfying the inequality for all cubes Q ⊂ Rn and where a is some functional that obeys a specific
discrete geometrical summability condition. We introduce a Sobolev-
type exponent associated to the weight w and obtain further
improvementsinvolvingLp∗w normsonthelefthandsideoftheinequality
above. For the endpoint case of A1 weights we reach the classical critical
Sobolev exponent p∗ = pn which is the largest possible and provide n−p
different type of quantitative estimates for Cw. We also show that this best possible estimate cannot hold with an exponent on the A1 constant smaller than 1/p.
As a consequence of our results (and the method of proof) we obtain further extensions to two weights Poincar ́e inequalities and to the case of higher order derivatives. Some other related results in the spirit of the work of Keith and Zhong on the open ended condition of Poincar ́e inequality are obtained using extrapolation methods. We also apply our method to obtain similar estimates in the scale of Lorentz spaces.
We also provide an argument based on extrapolation ideas showing that there is no (p,p), p ≥ 1, Poincar ́e inequality valid for the whole class of RH∞ weights by showing their intimate connection with the failure of Poincar ́e inequalities, (p, p) in the range 0 < p < 1.
2021-01-01T00:00:00ZRegularity of maximal functions on Hardy–Sobolev spaces
http://hdl.handle.net/20.500.11824/1337
Regularity of maximal functions on Hardy–Sobolev spaces
Pérez, C.; Picón, T.; Saari, Olli; Sousa, Mateus
We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy–Sobolev spaces H1,p(Rd) when p > d/(d + 1). This range of exponents is sharp. As a by-product of the proof, we obtain similar results for the local Hardy–Sobolev spaces h1,p(Rd) in the same range of exponents.
2018-12-01T00:00:00ZBilinear Spherical Maximal Functions of Product Type
http://hdl.handle.net/20.500.11824/1329
Bilinear Spherical Maximal Functions of Product Type
Roncal, L.; Shrivastava, S.; Shuin, K.
In this paper we introduce and study a bilinear spherical maximal function of product type in the spirit of bilinear Calderón–Zygmund theory. This operator is different from the bilinear spherical maximal function considered by Geba et al. in (Math Res Lett20(4):675–694, 2013). We deal with lacunary and full versions of this operator, and prove weighted estimates with respect to genuine bilinear weights beyond the Banach range. Our results are implied by sharp sparse domination for both the operators,following ideas by Lacey (J Anal Math 139(2):613–635, 2019). In the case of the lacunary maximal operator we also use interpolation of analytic families of operators to address the weighted boundedness for the whole range of tuples
2021-08-12T00:00:00Z