Analysis of Partial Differential Equations (APDE)
http://hdl.handle.net/20.500.11824/1
2023-05-18T02:43:26ZNotes 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.
2022-01-01T00:00:00ZPolynomial 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.
2022-01-01T00:00:00ZOn the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols
http://hdl.handle.net/20.500.11824/1553
On the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols
Covi, G.; García-Ferrero, M.A.; Rüland, A.
In this article we consider direct and inverse problems for α-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of directional antilocality as introduced by Y. Ishikawa in the 80s. We consider the Dirichlet problem for these operators on the respective “domain of dependence of the operator” and in several, adapted function spaces. This formulation allows one to avoid natural “gauges” which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators we complement the investigation of the direct problem with infinite data and single measurement uniqueness results for the associated inverse problems. Here, due to the only directional antilocality, new geometric conditions arise on the measurement domains. We discuss both the setting of symmetric and a particular class of non-symmetric nonlocal elliptic operators, and contrast the corresponding results for the direct and inverse problems. In particular for only “one-sided operators” new phenomena emerge both in the direct and inverse problems: For instance, it is possible to study the problem in data spaces involving local and nonlocal data, the unique continuation property may not hold in general and further restrictions on the measurement set for the inverse problem arise.
2022-12-25T00:00:00ZKato–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
2022-11-04T00:00:00Z