skip to main content


Title: On the Korányi spherical maximal function on Heisenberg groups
Abstract

We prove$$L^p\rightarrow L^q$$LpLqestimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp$$L^p\rightarrow L^q$$LpLqestimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds in (Ganguly and Thangavelu in J Funct Anal 280(3):108832, 2021) for the associated lacunary maximal operator.

 
more » « less
Award ID(s):
2054220
NSF-PAR ID:
10487395
Author(s) / Creator(s):
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Mathematische Annalen
Volume:
388
Issue:
1
ISSN:
0025-5831
Format(s):
Medium: X Size: p. 191-247
Size(s):
p. 191-247
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    We prove that the solutions to the discrete nonlinear Schrödinger equation with non-local algebraically decaying coupling converge strongly in$$L^2({\mathbb {R}}^2)$$L2(R2)to those of the continuum fractional nonlinear Schrödinger equation, as the discretization parameter tends to zero. The proof relies on sharp dispersive estimates that yield the Strichartz estimates that are uniform in the discretization parameter. An explicit computation of the leading term of the oscillatory integral asymptotics is used to show that the best constants of a family of dispersive estimates blow up as the non-locality parameter$$\alpha \in (1,2)$$α(1,2)approaches the boundaries.

     
    more » « less
  2. Abstract

    We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard exclusive$$\rho ^0$$ρ0meson muoproduction at COMPASS using 160 GeV/cpolarised$$ \mu ^{+}$$μ+and$$ \mu ^{-}$$μ-beams impinging on a liquid hydrogen target. The measurement covers the kinematic range 5.0 GeV/$$c^2$$c2$$< W<$$<W<17.0 GeV/$$c^2$$c2, 1.0 (GeV/c)$$^2$$2$$< Q^2<$$<Q2<10.0 (GeV/c)$$^2$$2and 0.01 (GeV/c)$$^2$$2$$< p_{\textrm{T}}^2<$$<pT2<0.5 (GeV/c)$$^2$$2. Here,Wdenotes the mass of the final hadronic system,$$Q^2$$Q2the virtuality of the exchanged photon, and$$p_{\textrm{T}}$$pTthe transverse momentum of the$$\rho ^0$$ρ0meson with respect to the virtual-photon direction. The measured non-zero SDMEs for the transitions of transversely polarised virtual photons to longitudinally polarised vector mesons ($$\gamma ^*_T \rightarrow V^{ }_L$$γTVL) indicate a violation ofs-channel helicity conservation. Additionally, we observe a dominant contribution of natural-parity-exchange transitions and a very small contribution of unnatural-parity-exchange transitions, which is compatible with zero within experimental uncertainties. The results provide important input for modelling Generalised Parton Distributions (GPDs). In particular, they may allow one to evaluate in a model-dependent way the role of parton helicity-flip GPDs in exclusive$$\rho ^0$$ρ0production.

     
    more » « less
  3. Abstract

    A recent generalization of the Erdős Unit Distance Problem, proposed by Palsson, Senger, and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in 3-space. Studying a variant of this question, we prove sharp bounds on the number of unit distance paths and cycles on the sphere of radius$$1/{\sqrt{2}}$$1/2. We also consider a similar problem about 3-regular unit distance graphs in $$\mathbb {R}^3$$R3.

     
    more » « less
  4. Abstract

    We introduce a family of Finsler metrics, called the$$L^p$$Lp-Fisher–Rao metrics$$F_p$$Fp, for$$p\in (1,\infty )$$p(1,), which generalizes the classical Fisher–Rao metric$$F_2$$F2, both on the space of densities$${\text {Dens}}_+(M)$$Dens+(M)and probability densities$${\text {Prob}}(M)$$Prob(M). We then study their relations to the Amari–C̆encov$$\alpha $$α-connections$$\nabla ^{(\alpha )}$$(α)from information geometry: on$${\text {Dens}}_+(M)$$Dens+(M), the geodesic equations of$$F_p$$Fpand$$\nabla ^{(\alpha )}$$(α)coincide, for$$p = 2/(1-\alpha )$$p=2/(1-α). Both are pullbacks of canonical constructions on$$L^p(M)$$Lp(M), in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of$$\alpha $$α-geodesics as being energy minimizing curves. On$${\text {Prob}}(M)$$Prob(M), the$$F_p$$Fpand$$\nabla ^{(\alpha )}$$(α)geodesics can still be thought as pullbacks of natural operations on the unit sphere in$$L^p(M)$$Lp(M), but in this case they no longer coincide unless$$p=2$$p=2. Using this transformation, we solve the geodesic equation of the$$\alpha $$α-connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman–Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of$$F_p$$Fp, and study their relation to$$\nabla ^{(\alpha )}$$(α).

     
    more » « less
  5. Abstract

    We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$C2($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$Hilbk[C2]) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$Cħ×-action. First, we find a two-parameter family$$X_{k,l}$$Xk,lof self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$Hilbk[C2]is obtained via direct limit$$l\longrightarrow \infty $$land by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$ħ-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$P2with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

     
    more » « less