Harmonic Analysis
http://hdl.handle.net/20.500.11824/3
Sun, 23 Apr 2023 05:18:46 GMT2023-04-23T05:18:46ZNotes on $H^{\log}$: structural properties, dyadic variants, and bilinear $H^1$-$BMO$ mappings
http://hdl.handle.net/20.500.11824/1582
Notes on $H^{\log}$: structural properties, dyadic variants, and bilinear $H^1$-$BMO$ mappings
Bakas, O.; Pott, S.; Rodríguez-López, S.; Sola, A.
This article is devoted to a study of the Hardy space $H^{\log} (\mathbb{R}^d)$ introduced by Bonami, Grellier, and Ky. We present an alternative approach to their result relating the product of a function in the real Hardy space $H^1$ and a function in $BMO$ to distributions that belong to $H^{\log}$ based on dyadic paraproducts.
We also point out analogues of classical results of Hardy-Littlewood, Zygmund, and Stein for $H^{\log}$ and related Musielak-Orlicz spaces.
Sat, 01 Jan 2022 00:00:00 GMThttp://hdl.handle.net/20.500.11824/15822022-01-01T00:00:00ZPolynomial averages and pointwise ergodic theorems on nilpotent groups
http://hdl.handle.net/20.500.11824/1567
Polynomial averages and pointwise ergodic theorems on nilpotent groups
Ionescu, A. D.; Magyar, A.; Mirek, M.; Szarek, T.Z.
We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on $\sigma$-finite measure spaces. We also establish corresponding maximal inequalities on $L^p$ for $1<p\leq \infty$ and $\rho$-variational inequalities on $L^2$ for $2<\rho<\infty$. This gives an affirmative answer to the Furstenberg--Bergelson--Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two.
Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative, nilpotent setting.
In particular, we develop what we call a \textit{nilpotent circle method} that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.
Sat, 01 Jan 2022 00:00:00 GMThttp://hdl.handle.net/20.500.11824/15672022-01-01T00:00:00ZKato–Ponce estimates for fractional sublaplacians in the Heisenberg group
http://hdl.handle.net/20.500.11824/1552
Kato–Ponce estimates for fractional sublaplacians in the Heisenberg group
Fanelli, L.; Roncal, L.
We give a proof of commutator estimates for fractional powers of the sublaplacian
on the Heisenberg group. Our approach is based on pointwise and $L^p$ estimates involving square
fractional integrals and Littlewood--Paley square functions
Fri, 04 Nov 2022 00:00:00 GMThttp://hdl.handle.net/20.500.11824/15522022-11-04T00:00:00ZA∞ condition for general bases revisited: complete classification of definitions
http://hdl.handle.net/20.500.11824/1504
A∞ condition for general bases revisited: complete classification of definitions
Kosz, D.
We refer to the discussion on different characterizations of the
A∞ class of weights, initiated by Duoandikoetxea, Martín-Reyes, and Ombrosi
[Math. Z. 282 (2016), pp. 955–972]. Twelve definitions of the A∞ condition are
considered. For cubes in Rd every two conditions are known to be equivalent,
while for general bases we have a trichotomy: equivalence, one-way implication,
or no dependency may occur. In most cases the relations between different
conditions have already been established. Here all the unsolved cases are
treated and, as a result, a full diagram of the said relations is presented.
Fri, 27 May 2022 00:00:00 GMThttp://hdl.handle.net/20.500.11824/15042022-05-27T00:00:00Z