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1. Maximal Functions, Littlewood–Paley Theory, Riesz Transforms and Atomic Decomposition in the Multi-parameter Flag Setting

   https://doi.org/10.1090/memo/1373  

   Han, Yongsheng; Lee, Ming-Yi; Li, Ji; Wick, Brett (September 2022, Memoirs of the American Mathematical Society)

   In this paper, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood–Paley square function and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this paper include (1) establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the Plancherel–Pólya inequalities for flag quadratic forms; (2) introducing the maximal function and area function via flag Poisson kernels and flag version of harmonic functions; (3) developing an atomic decomposition via the finite speed propagation and area function in terms of flag heat semigroups. As a consequence of these real variable methods, we obtain the full characterisations of the multi-parameter Hardy space with the flag structure. 

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2. Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localization

   https://doi.org/10.1142/S0219530521500329  

   Du, Qiang; Mengesha, Tadele; Tian, Xiaochuan (May 2022, Analysis and Applications)

   This work aims to prove a Hardy-type inequality and a trace theorem for a class of function spaces on smooth domains with a nonlocal character. Functions in these spaces are allowed to be as rough as an [Formula: see text]-function inside the domain of definition but as smooth as a [Formula: see text]-function near the boundary. This feature is captured by a norm that is characterized by a nonlocal interaction kernel defined heterogeneously with a special localization feature on the boundary. Thus, the trace theorem we obtain here can be viewed as an improvement and refinement of the classical trace theorem for fractional Sobolev spaces [Formula: see text]. Similarly, the Hardy-type inequalities we establish for functions that vanish on the boundary show that functions in this generalized space have the same decay rate to the boundary as functions in the smaller space [Formula: see text]. The results we prove extend existing results shown in the Hilbert space setting with p = 2. A Poincaré-type inequality we establish for the function space under consideration together with the new trace theorem allows formulating and proving well-posedness of a nonlinear nonlocal variational problem with conventional local boundary condition. 

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3. Spectra of some weighted composition operators on the ball

   https://doi.org/10.1007/s44146-023-00089-4  

   Kaschner, Scott; Le, Trieu; Makdad, Chloe; Rempfer, Benjamin; Thompson, Derek; Winters, DeJuan (November 2023, Acta Scientiarum Mathematicarum)

   We find sufficient conditions for a self-map of the unit ball to converge uniformly under iteration to a fixed point or idempotent on the entire ball. Using these tools, we establish spectral containments for weighted composition operators on Hardy and Bergman spaces of the ball. When the compositional symbol is in the Schur–Agler class, we establish the spectral radii of these weighted composition operators. 

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4. Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

   https://doi.org/10.1142/S0129055X21500069  

   Bonheure, Denis; Dolbeault, Jean; Esteban, Maria J.; Laptev, Ari; Loss, Michael (November 2020, Reviews in Mathematical Physics)

   null (Ed.)

   This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions [Formula: see text] and [Formula: see text]. 

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5. Lebesgue Decomposition of Non-Commutative Measures

   https://doi.org/10.1093/imrn/rnaa231  

   Jury, Michael T; Martin, Robert T (September 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $$C^{\ast }-$$algebra, the $$C^{\ast }-$$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $$C^{\ast }-$$algebras; any *−representation of the Cuntz–Toeplitz $$C^{\ast }-$$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory. 

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