Performance information

• A note on ubiquity of geometric Brascamp–Lieb data
  Neal Bez; Anthony Gauvan; Hiroshi Tsuji
  Bulletin of the London Mathematical Society, Volume：57, Number：1, First page：302, Last page：314, Dec. 2024
  Abstract
  
  Relying substantially on work of Garg, Gurvits, Oliveira and Wigderson, it is shown that geometric Brascamp–Lieb data are, in a certain sense, dense in the space of feasible Brascamp–Lieb data. This addresses a question raised by Bennett and Tao in their recent work on the adjoint Brascamp–Lieb inequality.
  Wiley, Scientific journal
  DOI：https://doi.org/10.1112/blms.13198
  DOI ID：10.1112/blms.13198, ISSN：0024-6093, eISSN：1469-2120
• Boundary Strichartz estimates and pointwise convergence for orthonormal systems
  Neal Bez; Shinya Kinoshita; Shobu Shiraki
  Transactions of the London Mathematical Society, Volume：11, Number：1, Dec. 2024
  Abstract
  
  We consider maximal estimates associated with fermionic systems. Firstly, we establish maximal estimates with respect to the spatial variable. These estimates are certain boundary cases of the many‐body Strichartz estimates pioneered by Frank, Lewin, Lieb and Seiringer. We also prove new maximal‐in‐time estimates, thereby significantly extending work of Lee, Nakamura and the first author on Carleson's pointwise convergence problem for fermionic systems.
  Wiley, Scientific journal
  DOI：https://doi.org/10.1112/tlm3.70002
  DOI ID：10.1112/tlm3.70002, ISSN：2052-4986, eISSN：2052-4986
• Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality　　　　　　　　　　　　　　　
  Neal Bez; Shohei Nakamura; Hiroshi Tsuji
  Journal of Functional Analysis, Volume：285, Number：10, First page：110121, Last page：110121, Nov. 2023, [Reviewed]
  Elsevier BV, Scientific journal
  DOI：https://doi.org/10.1016/j.jfa.2023.110121
  DOI ID：10.1016/j.jfa.2023.110121, ISSN：0022-1236
• Revisiting the Rellich inequality
  Neal Bez; Shuji Machihara; Tohru Ozawa
  Mathematische Zeitschrift, Volume：303, Number：2, Jan. 2023, [Reviewed]
  Springer Science and Business Media LLC, Scientific journal
  DOI：https://doi.org/10.1007/s00209-022-03203-4
  DOI ID：10.1007/s00209-022-03203-4, ISSN：0025-5874, eISSN：1432-1823
• Higher order transversality in harmonic analysis　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez
  RIMS Kôkyûroku Bessatsu, Volume：B88, First page：75, Last page：103, 2021, [Reviewed]
• Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates
  Neal Bez; Sanghyuk Lee; Shohei Nakamura
  Forum of Mathematics, Sigma, Volume：9, 2021, [Reviewed]
  
  We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein–Gordon and fractional Schrödinger equations. Our estimates extend those of Frank–Sabin in the case of the wave and Klein–Gordon equations, and generalize work of Frank et al. and Frank–Sabin for the Schrödinger equation. Due to a certain technical barrier, except for the classical Schrödinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results.
  
  
  The main novelty of this paper is our derivation and use of estimates for weighted oscillatory integrals, which we combine with an approach due to Frank and Sabin. Our weighted oscillatory integral estimates are, in a certain sense, rather delicate endpoint versions of known dispersive estimates with power-type weights of the form
  
  
  $|\xi |^{-\lambda }$
  
  or
  
  
  $(1 + |\xi |^2)^{-\lambda /2}$
  
  , where
  
  
  $\lambda \in \mathbb {R}$
  
  . We achieve optimal decay rates by considering such weights with appropriate
  
  
  $\lambda \in \mathbb {C}$
  
  . For the wave and Klein–Gordon equations, our weighted oscillatory integral estimates are new. For the fractional Schrödinger equation, our results overlap with prior work of Kenig–Ponce–Vega in a certain regime. Our contribution to the theory of weighted oscillatory integrals has also been influenced by earlier work of Carbery–Ziesler, Cowling et al., and Sogge–Stein.
  
  
  Finally, we provide some applications of our new Strichartz estimates for orthonormal families of data to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces.
  Cambridge University Press (CUP), Scientific journal
  DOI：https://doi.org/10.1017/fms.2020.64
  DOI ID：10.1017/fms.2020.64, eISSN：2050-5094
• On the nonlinear Brascamp-Lieb inequality　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez; Stefan Buschenhenke; Michael Cowling; Taryn Flock
  Duke Mathematical Journal, Volume：169, First page：3291, Last page：3338, 2020, [Reviewed]
• Maximal estimates for the Schrodinger equation with orthonormal initial data
  Neal Bez; Sanghyuk Lee; Shohei Nakamura
  Selecta Mathematica, Volume：26, Number：52, 2020, [Reviewed]
  Springer Science and Business Media LLC, Scientific journal
  DOI：https://doi.org/10.1007/s00029-020-00582-6
  DOI ID：10.1007/s00029-020-00582-6, ISSN：1022-1824, eISSN：1420-9020
• A supersolutions perspective on hypercontractivity　　　　　　　　　　　　　　　
  Yosuke Aoki; Jonathan Bennett; Neal Bez; Shuji Machihara; Kosuke Matsuura; Shobu Shiraki
  Annali di Matematica Pura ed Applicata, Volume：199, First page：2105, Last page：2116, 2020, [Reviewed]
• Inhomogeneous Strichartz estimates in some critical cases　　　　　　　　　　　　　　　
  Neal Bez; Jayson Cunanan; Sanghyuk Lee
  Proceedings of the American Mathematical Society, Volume：148, First page：639, Last page：652, 2020, [Reviewed]
• Hardy type inequalities with spherical derivatives　　　　　　　　　　　　　　　
  Neal Bez; Shuji Machihara; Tohru Ozawa
  SN Partial Differential Equations and Applications, Volume：1, Number：5, 2020, [Reviewed]
• On the Strichartz estimates for orthonormal systems of initial data with regularity　　　　　　　　　　　　　　　
  Neal Bez; Younghun Hong; Sanghyuk Lee; Shohei Nakamura; Yoshihiro Sawano
  Advances in Mathematics, Volume：354, Number：106736, First page：106736, Last page：106736, 2019, [Reviewed]
  Elsevier BV, Scientific journal
  DOI：https://doi.org/10.1016/j.aim.2019.106736
  DOI ID：10.1016/j.aim.2019.106736, ISSN：0001-8708
• Smoothing estimates for velocity averages with radial data　　　　　　　　　　　　　　　
  Neal Bez; Jayson Cunanan
  RIMS Kokyuroku Bessatsu, Volume：B74, First page：33, Last page：46, 2019, [Reviewed]
• Remarks on the Mizohata-Takeuchi conjecture and related problems　　　　　　　　　　　　　　　
  Neal Bez; Mitsuru Sugimoto
  Advanced Studies in Pure Mathematics, Volume：81, First page：1, Last page：12, 2019, [Reviewed]
• Generating monotone quantities for the heat equation　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez
  Journal fur die Reine und Angewandte Mathematik, Volume：756, First page：37, Last page：63, 2019, [Reviewed]
• Estimates for the kinetic transport equation in hyperbolic Sobolev spaces　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez; Susana Gutiérrez; Sanghyuk Lee
  Journal des Mathematiques Pures et Appliquees, Volume：114, First page：1, Last page：28, Jun. 2018, [Reviewed]
  We establish smoothing estimates in the framework of hyperbolic Sobolev spaces for the velocity averaging operator ρ of the solution of the kinetic transport equation. If the velocity domain is either the unit sphere or the unit ball, then, for any exponents q and r, we find a characterisation of the exponents β+ and β−, except possibly for an endpoint case, for which D+ β+ D− β− ρ is bounded from space–velocity Lx,v 2 to space–time Lt qLx r. Here, D+ and D− are the classical and hyperbolic derivative operators, respectively. In fact, we shall provide an argument which unifies these velocity domains and the velocity averaging estimates in either case are shown to be equivalent to mixed-norm bounds on the cone multiplier operator acting on L2. We develop our ideas further in several ways, including estimates for initial data lying in certain Besov spaces, for which a key tool in the proof is the sharp ℓp decoupling theorem recently established by Bourgain and Demeter. We also show that the level of permissible smoothness increases significantly if we restrict attention to initial data which are radially symmetric in the spatial variable.
  Elsevier Masson SAS, English, Scientific journal
  DOI：https://doi.org/10.1016/j.matpur.2018.03.007
  DOI ID：10.1016/j.matpur.2018.03.007, ISSN：0021-7824, SCOPUS ID：85045703051
• A sharp k-plane strichartz inequality for the schrödinger equation　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez; Taryn C. Flock; Susana Gutiérrez; Marina Iliopoulou
  Transactions of the American Mathematical Society, Volume：370, Number：8, First page：5617, Last page：5633, 2018, [Reviewed]
  We prove that ‖X(|u|2)‖L3 t,ℓ ≤ C‖f‖2L2(ℝ2),where u(x, t) is the solution to the linear time-dependent Schrödinger equation on ℝ2with initial datum f and X is the (spatial) X-ray transform on R2In particular, we identify the best constant C and show that a datum f is an extremiser if and only if it is a gaussian. We also establish bounds of this type in higher dimensions d, where the X-ray transform is replaced by the k-plane transform for any 1 ≤ k ≤ d − 1. In the process we obtain sharp L2(μ) bounds on Fourier extension operators associated with certain high-dimensional spheres involving measures μ supported on natural “co-k-planarity” sets.
  American Mathematical Society, English, Scientific journal
  DOI：https://doi.org/10.1090/tran/7309
  DOI ID：10.1090/tran/7309, ISSN：0002-9947, SCOPUS ID：85047105131
• Stability of the Brascamp-Lieb constant and applications　　　　　　　　　　　　　　　
  J. Bennett; N. Bez; T. Flock; S. Lee
  American Journal of Mathematics, Volume：140, First page：543, Last page：569, 2018, [Reviewed]
• Stability of trace theorems on the sphere　　　　　　　　　　　　　　　
  N. Bez; C. Jeavons; T. Ozawa; M. Sugimoto
  Journal of Geometric Analysis, Volume：28, First page：1456, Last page：1476, 2018, [Reviewed]
• Smoothing estimates for the kinetic transport equation at the critical regularity　　　　　　　　　　　　　　　
  Neal Bez; Jayson Cunanan; Sanghyuk Lee
  SIAM Journal on Mathematical Analysis, Volume：50, Number：2, First page：2280, Last page：2294, 2018, [Reviewed]
  We prove smoothing estimates for velocity averages of the kinetic transport equation in hyperbolic Sobolev spaces at the critical regularity, leading to a complete characterization of the allowable regularity exponents. Such estimates will be deduced from some mixed-norm estimates for the cone multiplier operator at a certain critical index. Our argument is not particular to the geometry of the cone and we illustrate this by establishing analogous estimates for the paraboloid.
  Society for Industrial and Applied Mathematics Publications, English, Scientific journal
  DOI：https://doi.org/10.1137/17M1148852
  DOI ID：10.1137/17M1148852, ISSN：1095-7154, SCOPUS ID：85047342233
• Behaviour of the Brascamp-Lieb constant　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez; Michael G. Cowling; Taryn C. Flock
  BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, Volume：49, Number：3, First page：512, Last page：518, Jun. 2017, [Reviewed]
  Recent progress in multilinear harmonic analysis naturally raises questions about the local behaviour of the best constant (or bound) in the general Brascamp-Lieb inequality as a function of the underlying linear transformations. In this paper we prove that this constant is continuous, but is not in general differentiable.
  WILEY, English, Scientific journal
  DOI：https://doi.org/10.1112/blms.12049
  DOI ID：10.1112/blms.12049, ISSN：0024-6093, eISSN：1469-2120, Web of Science ID：WOS:000402144800014
• On sharp bilinear Strichartz estimates of Ozawa-Tsutsumi type　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez; Chris Jeavons; Nikolaos Pattakos
  JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, Volume：69, Number：2, First page：459, Last page：476, Apr. 2017, [Reviewed]
  We provide a comprehensive analysis of sharp bilinear estimates of Ozawa Tsutsumi type for solutions u of the free Schrodinger equation, which give sharp control on vertical bar u vertical bar(2) in classical Sobolev spaces. In particular, we generalise their estimates in such a way that provides a unification with some sharp bilinear estimates proved by Carneiro and Planchon Vega, via entirely different methods, by seeing them all as special cases of a one-parameter family of sharp estimates. The extremal functions are solutions of the Maxwell-Boltzmann functional equation and hence Gaussian. For u(2) we argue that the natural analogous results involve certain dispersive Sobolev norms; in particular, despite the validity of the classical Ozawa Tsutsumi estimates for both vertical bar u vertical bar(2) and u(2) in the classical Sobolev spaces, we show that Gaussians are not extremisers in the latter case for spatial dimensions strictly greater than two.
  MATH SOC JAPAN, English, Scientific journal
  DOI：https://doi.org/10.2969/jmsj/06920459
  DOI ID：10.2969/jmsj/06920459, ISSN：0025-5645, Web of Science ID：WOS:000401401300001
• A conjecture regarding optimal Strichartz estimates for the wave equation　　　　　　　　　　　　　　　
  N. Bez; C. Jeavons; T. Ozawa; H. Saito
  Trends in Mathematics (New Trends in Analysis and Interdisciplinary Applications), 2017, [Reviewed]
• Optimal constants and extremisers for some smoothing estimates　　　　　　　　　　　　　　　
  N. Bez; M. Sugimoto
  Journal d'Analyse Mathematique, Volume：131, First page：159, Last page：187, 2017, [Reviewed]
• Sharpness of the brascamp-lieb inequality in lorentz spaces　　　　　　　　　　　　　　　
  Neal Bez; Sanghyuk Lee; Shohei Nakamura; Yoshihiro Sawano
  Electronic Research Announcements in Mathematical Sciences, Volume：24, First page：53, Last page：63, 2017, [Reviewed]
  We provide necessary conditions for the refined version of the Brascamp-Lieb inequality where the input functions are allowed to belong to Lorentz spaces, thereby establishing the sharpness of the range of Lorentz exponents in the subcritical case. Using similar considerations, some sharp refinements of the Strichartz estimates for the kinetic transport equation are established.
  American Mathematical Society, English, Scientific journal
  DOI：https://doi.org/10.3934/era.2017.24.006
  Scopus：https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85020907812&origin=inward
  Scopus Citedby：https://www.scopus.com/inward/citedby.uri?partnerID=HzOxMe3b&scp=85020907812&origin=inward
  DOI ID：10.3934/era.2017.24.006, ISSN：1935-9179, SCOPUS ID：85020907812
• SOME SHARP BILINEAR SPACE-TIME ESTIMATES FOR THE WAVE EQUATION　　　　　　　　　　　　　　　
  Neal Bez; Chris Jeavons; Tohru Ozawa
  MATHEMATIKA, Volume：62, Number：3, First page：719, Last page：737, 2016, [Reviewed]
  We prove a family of sharp bilinear space-time estimates for the halfwave propagator e(it root-Delta). As a consequence, for radially symmetric initial data, we establish sharp estimates of this kind for a range of exponents beyond the classical range.
  LONDON MATH SOC, English, Scientific journal
  DOI：https://doi.org/10.1112/S0025579316000012
  DOI ID：10.1112/S0025579316000012, ISSN：0025-5793, eISSN：2041-7942, Web of Science ID：WOS:000375948100005
• Extremisers for the trace theorem on the sphere　　　　　　　　　　　　　　　
  Neal Bez; Shuji Machihara; Mitsuru Sugimoto
  MATHEMATICAL RESEARCH LETTERS, Volume：23, Number：3, First page：633, Last page：647, 2016, [Reviewed]
  We find all extremisers for the trace theorem on the sphere. We also provide a sharp extension for functions belonging to certain Sobolev spaces with angular regularity.
  INT PRESS BOSTON, INC, English, Scientific journal
  ISSN：1073-2780, eISSN：1945-001X, Web of Science ID：WOS:000388457200003
• A survey on optimal smoothing estimates and trace theorems　　　　　　　　　　　　　　　
  N. Bez; M. Sugimoto
  Advances in Mathematics (China), Volume：45, First page：801, Last page：816, 2016, [Reviewed]
• Applications of the Funk-Hecke theorem to smoothing and trace estimates　　　　　　　　　　　　　　　
  Neal Bez; Hiroki Saito; Mitsuru Sugimoto
  ADVANCES IN MATHEMATICS, Volume：285, First page：1767, Last page：1795, Nov. 2015, [Reviewed]
  For a wide class of Kato-smoothing estimates with radial weights, the Funk-Hecke theorem is used to generate a new expression for the optimal constant in terms of the Fourier transform of the weight, from which several applications are given. For example, we are able to easily establish a unified theorem, assuming natural power-like asymptotic estimates for the Fourier transform of the weight, from which many well-studied smoothing estimates immediately follow, as well as sharpness of the decay and smoothness exponents. Furthermore, observing that the weight has an everywhere positive Fourier transform in many well-studied cases, our approach allows sharper information regarding the optimal constant and extremisers, substantially extending earlier work of Simon. These observations are very closely related to the Mizohata-Takeuchi conjecture regarding the equivalence of weighted L-2 bounds for the Fourier extension operator on the sphere and the uniform boundedness of the X-ray transform of the weight. For radial weights, this has been independently established by Barcelo-Ruiz-Vega and Carbery-Soria; we provide a short alternative proof in three and higher dimensions of this equivalence when the Fourier transform of the weight is positive, with the optimal relationship between constants. Finally, our approach works for the closely connected trace theorems on the sphere where analogous results are given, including the optimal constant and characterisation of extremisers for the inhomogeneous H-s (R-d) -> L-2 (Sd-1) trace theorem. (C) 2015 Elsevier Inc. All rights reserved.
  ACADEMIC PRESS INC ELSEVIER SCIENCE, English, Scientific journal
  DOI：https://doi.org/10.1016/j.aim.2015.08.025
  DOI ID：10.1016/j.aim.2015.08.025, ISSN：0001-8708, eISSN：1090-2082, Web of Science ID：WOS:000376417800048
• Optimal Forward and Reverse Estimates of Morawetz and Kato-Yajima Type with Angular Smoothing Index　　　　　　　　　　　　　　　
  Neal Bez; Mitsuru Sugimoto
  JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, Volume：21, Number：2, First page：318, Last page：341, Apr. 2015, [Reviewed]
  For the solution of the free Schrodinger equation, we obtain the optimal constants and characterise extremisers for forward and reverse smoothing estimates which are global in space and time, contain a homogeneous and radial weight in the space variable, and incorporate a certain angular regularity. This will follow from a more general result which permits analogous sharp forward and reverse smoothing estimates and a characterisation of extremisers for the solution of the free Klein-Gordon and wave equations. The nature of extremisers is shown to be sensitive to both the dimension and the size of the smoothing index relative to the dimension. Furthermore, in four spatial dimensions and certain special values of the smoothing index, we obtain an exact identity for each of these evolution equations.
  SPRINGER BIRKHAUSER, English, Scientific journal
  DOI：https://doi.org/10.1007/s00041-014-9371-0
  DOI ID：10.1007/s00041-014-9371-0, ISSN：1069-5869, eISSN：1531-5851, Web of Science ID：WOS:000351174800004
• Some Recent Progress on Sharp Kato-type Smoothing Estimates　　　　　　　　　　　　　　　
  Neal Bez; Mitsuru Sugimoto
  COMPLEX ANALYSIS AND DYNAMICAL SYSTEMS VI, PT 1: PDE, DIFFERENTIAL GEOMETRY, RADON TRANSFORM, Volume：653, First page：41, Last page：50, 2015, [Reviewed]
  Space-time estimates of the type
  parallel to Psi(x, del) exp(-it Delta)f parallel to L-t,x((RxRd))2 <= C parallel to f parallel to(L2(Rd)),
  where Psi(x, del) = < x >(-s) (s >= 1), Psi(x, del) = vertical bar x vertical bar(a-1) vertical bar del vertical bar(a) (1-d/2 <a<1/2) or Psi(x, del) = < x >(-s) vertical bar del vertical bar(1/2) (s>1/2), are often called smoothing estimates, and there is a vast literature surrounding these estimates. Simon [22] initiated the study of optimal constants for smoothing estimates by obtaining them in the special cases Psi(x, del) = vertical bar x vertical bar(-1) and Psi(x, del) - < x >(-1) vertical bar del vertical bar(1/2). In this article, recent progress on optimal constants and their extremisers for more general cases are discussed based on recent works [4] and [5] by the authors.
  AMER MATHEMATICAL SOC, English, International conference proceedings
  DOI：https://doi.org/10.1090/conm/653/13177
  DOI ID：10.1090/conm/653/13177, ISSN：0271-4132, Web of Science ID：WOS:000371648700004
• Sharp sobolev-strichartz estimates for the free Schrödinger propagator　　　　　　　　　　　　　　　
  Neal Bez; Chris Jeavons; Nikolaos Pattakos
  Trends in Mathematics, Volume：2, First page：281, Last page：288, 2015, [Reviewed]
  We consider gaussian extremisability of sharp linear Sobolev-Strichartz estimates and closely related sharp bilinear Ozawa-Tsutsumi estimates for the free Schrödinger equation.
  Springer International Publishing, English, International conference proceedings
  DOI：https://doi.org/10.1007/978-3-319-12577-0-33
  DOI ID：10.1007/978-3-319-12577-0-33, ISSN：2297-024X, SCOPUS ID：84959159490
• A SHARP SOBOLEV-STRICHARTZ ESTIMATE FOR THE WAVE EQUATION　　　　　　　　　　　　　　　
  Neal Bez; Chris Jeavons
  ELECTRONIC RESEARCH ANNOUNCEMENTS IN MATHEMATICAL SCIENCES, Volume：22, First page：46, Last page：54, 2015, [Reviewed]
  We calculate the the sharp constant and characterize the extremal initial data in (H) over dot(3/4) x H-1/4 for the L-4 Sobolev-Strichartz estimate for the wave equation in four spatial dimensions.
  AMER INST MATHEMATICAL SCIENCES, English, Scientific journal
  DOI：https://doi.org/10.3934/era.2015.22.46
  DOI ID：10.3934/era.2015.22.46, ISSN：1935-9179, Web of Science ID：WOS:000361819700005
• Flow Monotonicity and Strichartz Inequalities　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez; Marina Iliopoulou
  INTERNATIONAL MATHEMATICS RESEARCH NOTICES, Number：19, First page：9415, Last page：9437, 2015, [Reviewed]
  We identify complete monotonicity properties underlying a variety of well-known sharp Strichartz inequalities in euclidean space.
  OXFORD UNIV PRESS, English, Scientific journal
  DOI：https://doi.org/10.1093/imrn/rnu230
  DOI ID：10.1093/imrn/rnu230, ISSN：1073-7928, eISSN：1687-0247, Web of Science ID：WOS:000366499400008
• Optimal constant for a smoothing estimate of critical index　　　　　　　　　　　　　　　
  N. Bez; M. Sugimoto
  Trends in Mathematics (Fourier Analysis), First page：1, Last page：7, 2014, [Reviewed]
• On the Strichartz Estimates for the Kinetic Transport Equation　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez; Susana Gutierrez; Sanghyuk Lee
  COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, Volume：39, Number：10, First page：1821, Last page：1826, 2014, [Reviewed]
  We show that the endpoint Strichartz estimate for the kinetic transport equation is false in all dimensions. We also present an alternative approach to proving the non-endpoint cases using multilinear analysis.
  TAYLOR & FRANCIS INC, English, Scientific journal
  DOI：https://doi.org/10.1080/03605302.2013.850880
  DOI ID：10.1080/03605302.2013.850880, ISSN：0360-5302, eISSN：1532-4133, Web of Science ID：WOS:000341003700001
• A note on magnitude bounds for the mask coefficients of the interpolatory Dubuc-Deslauriers subdivision scheme　　　　　　　　　　　　　　　
  H. E. Bez; N. Bez
  LMS JOURNAL OF COMPUTATION AND MATHEMATICS, Volume：17, Number：1, First page：226, Last page：232, 2014, [Reviewed]
  We analyse the mask associated with the 2n-point interpolatory Dubuc-Deslauriers subdivision scheme S-a[n]. Sharp bounds are presented for the magnitude of the coefficients a(2i-1)([n]) of the mask. For scales i is an element of [1, root n] it is shown that vertical bar a(2i-1)([n])vertical bar is comparable to i(-1), and for larger power scales, exponentially decaying bounds are obtained. Using our bounds, we may precisely analyse the summability of the mask as a function of n by identifying which coefficients of the mask contribute to the essential behaviour in n, recovering and refining the recent result of Deng-Hormann Zhang that the operator norm of S-a[n] on l(infinity) grows logarithmically in n.
  CAMBRIDGE UNIV PRESS, English, Scientific journal
  DOI：https://doi.org/10.1112/S1461157013000363
  DOI ID：10.1112/S1461157013000363, ISSN：1461-1570, Web of Science ID：WOS:000349291700013
• New minimal bounds for the derivatives of rational Bezier paths and rational rectangular Bezier surfaces　　　　　　　　　　　　　　　
  H. E. Bez; N. Bez
  APPLIED MATHEMATICS AND COMPUTATION, Volume：225, First page：475, Last page：479, Dec. 2013, [Reviewed]
  New minimal bounds are derived for the magnitudes of the derivatives of the rational Bezier paths and the rational rectangular Bezier surface patches of arbitrary degree, which improve previous work of this type in many cases. Moreover, our new bounds are explicitly given by simple and closed-form expressions. An important advantage of the closed-form expressions is that they allow us to prove that our bounds are sharp under certain well-defined conditions. Some numerical examples, highlighting the potential of the new bounds in providing improved estimates, are given in an appendix. (C) 2013 Elsevier Inc. All rights reserved.
  ELSEVIER SCIENCE INC, English, Scientific journal
  DOI：https://doi.org/10.1016/j.amc.2013.09.039
  DOI ID：10.1016/j.amc.2013.09.039, ISSN：0096-3003, eISSN：1873-5649, Web of Science ID：WOS:000327765600041
• Global Nonlinear Brascamp-Lieb Inequalities　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez; Susana Gutierrez
  JOURNAL OF GEOMETRIC ANALYSIS, Volume：23, Number：4, First page：1806, Last page：1817, Oct. 2013, [Reviewed]
  We prove global versions of certain known nonlinear Brascamp-Lieb inequalities under a natural homogeneity assumption. We also establish a conditional theorem allowing one to generally pass from local to global nonlinear Brascamp-Lieb estimates under such a homogeneity assumption.
  SPRINGER, English, Scientific journal
  DOI：https://doi.org/10.1007/s12220-012-9307-3
  DOI ID：10.1007/s12220-012-9307-3, ISSN：1050-6926, Web of Science ID：WOS:000325065700009
• On derivative bounds for the rational quadratic Bezier paths　　　　　　　　　　　　　　　
  H. E. Bez; N. Bez
  COMPUTER AIDED GEOMETRIC DESIGN, Volume：30, Number：2, First page：254, Last page：261, Feb. 2013, [Reviewed]
  New derivative bounds for the rational quadratic Bezier paths are obtained, both for particular weight vectors and for classes of equivalent parametrisations. A comprehensive analysis of our bounds against existing bounds is made. (C) 2012 Elsevier B.V. All rights reserved.
  ELSEVIER SCIENCE BV, English, Scientific journal
  DOI：https://doi.org/10.1016/j.cagd.2012.12.003
  DOI ID：10.1016/j.cagd.2012.12.003, ISSN：0167-8396, Web of Science ID：WOS:000316037900006
• Transversal multilinear Radon-like transforms: local and global estimates　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez; Susana Gutierrez
  REVISTA MATEMATICA IBEROAMERICANA, Volume：29, Number：3, First page：765, Last page：788, 2013, [Reviewed]
  We prove local "L-P-improving" estimates for a class of multi-linear Radon-like transforms satisfying a strong transversality hypothesis. As a consequence, we obtain sharp multilinear convolution estimates for measures supported on fully transversal submanifolds of Euclidean space of arbitrary dimension. Motivated by potential applications in diffraction tomography, we also prove global estimates for the same class of Radon-like transforms under a natural homogeneity assumption.
  EUROPEAN MATHEMATICAL SOC, English, Scientific journal
  DOI：https://doi.org/10.4171/RMI/739
  DOI ID：10.4171/RMI/739, ISSN：0213-2230, Web of Science ID：WOS:000326990500002
• A sharp Strichartz estimate for the wave equation with data in the energy space　　　　　　　　　　　　　　　
  Neal Bez; Keith M. Rogers
  JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, Volume：15, Number：3, First page：805, Last page：823, 2013, [Reviewed]
  We prove a sharp bilinear estimate for the wave equation from which we obtain the sharp constant in the Strichartz estimate which controls the L-t, x(4)(R5+1) norm of the solution in terms of the energy. We also characterise the maximisers.
  EUROPEAN MATHEMATICAL SOC, English, Scientific journal
  DOI：https://doi.org/10.4171/JEMS/377
  DOI ID：10.4171/JEMS/377, ISSN：1435-9855, Web of Science ID：WOS:000317564700005
• A majorant problem for the periodic Schrodinger group　　　　　　　　　　　　　　　
  J. Bennett; N. Bez
  RIMS Kokyuroku Bessatsu, Volume：B33, First page：1, Last page：10, 2012, [Reviewed]
• Some nonlinear Brascamp-Lieb inequalities and applications to harmonic analysis　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez
  JOURNAL OF FUNCTIONAL ANALYSIS, Volume：259, Number：10, First page：2520, Last page：2556, Nov. 2010, [Reviewed]
  We use the method of induction-on-scales to prove certain diffeomorphism-invariant nonlinear Brascamp-Lieb inequalities. We provide applications to multilinear convolution inequalities and the restriction theory for the Fourier transform, extending to higher dimensions recent work of Bejenaru-Herr-Tataru and Bennett-Carbery-Wright. (C) 2010 Elsevier Inc. All rights reserved.
  ACADEMIC PRESS INC ELSEVIER SCIENCE, English, Scientific journal
  DOI：https://doi.org/10.1016/j.jfa.2010.07.015
  DOI ID：10.1016/j.jfa.2010.07.015, ISSN：0022-1236, Web of Science ID：WOS:000281532200002
• Heat-flow monotonicity related to the Hausdorff-Young inequality　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez; Anthony Carbery
  BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, Volume：41, First page：971, Last page：979, Dec. 2009, [Reviewed]
  It is known that if q is an even integer, then the L(q)(R(d)) norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres 'simultaneously slide' to the origin. We provide explicit examples to show that this monotonicity property fails dramatically if q > 2 is not an even integer. These results are equivalent, upon rescaling, to similar statements involving solutions to heat equations. Such considerations are natural given the celebrated theorem of Beckner concerning the gaussian extremisability of the Hausdorff-Young inequality.
  OXFORD UNIV PRESS, English, Scientific journal
  DOI：https://doi.org/10.1112/blms/bdp073
  DOI ID：10.1112/blms/bdp073, ISSN：0024-6093, Web of Science ID：WOS:000272924700002
• MAXIMAL OPERATORS AND HILBERT TRANSFORMS ALONG FLAT CURVES NEAR L-1　　　　　　　　　　　　　　　
  Neal Bez
  JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY, Volume：87, Number：3, First page：311, Last page：323, Dec. 2009, [Reviewed]
  For a class of convex curves in R-d we prove that the corresponding maximal operator and Hilbert transform are of weak type L log L. The point of interest here is that this class admits Curves which are infinitely flat at the origin. We also prove in analogous weak type result for a class of nonconvex hypersurfaces.
  CAMBRIDGE UNIV PRESS, English, Scientific journal
  DOI：https://doi.org/10.1017/S1446788709000111
  DOI ID：10.1017/S1446788709000111, ISSN：1446-7887, Web of Science ID：WOS:000273957500002
• Closure Properties of Solutions to Heat Inequalities　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez
  JOURNAL OF GEOMETRIC ANALYSIS, Volume：19, Number：3, First page：584, Last page：600, Jul. 2009, [Reviewed]
  We prove that if u(1), u(2) : (0, infinity) x R(d) -> (0, infinity) are sufficiently well-behaved solutions to certain heat inequalities on Rd then the function u : (0, infinity) x R(d) -> (0, infinity) given by u(1/p) = u(1)(1/p1) * u(2)(1/p2) also satisfies a heat inequality of a similar type provided 1/p1 + 1/p2 = 1 + 1/p. On iterating, this result leads to an analogous statement concerning n-fold convolutions. As a corollary, we give a direct heat-flow proof of the sharp n-fold Young convolution inequality and its reverse form.
  SPRINGER, English, Scientific journal
  DOI：https://doi.org/10.1007/s12220-009-9070-2
  DOI ID：10.1007/s12220-009-9070-2, ISSN：1050-6926, Web of Science ID：WOS:000265214800005
• Heat-flow monotonicity underlying some sharp inequalities in geometric and harmonic analysis　　　　　　　　　　　　　　　
  N. Bez
  RIMS Kokyuroku Bessatsu, Volume：B14, First page：1, Last page：16, 2009, [Reviewed]
  Kyoto University, English
  ISSN：1881-6193, CiNii Articles ID：110007480919, CiNii Books ID：AA12196120
• HEAT-FLOW MONOTONICITY OF STRICHARTZ NORMS　　　　　　　　　　　　　　　
  Jonathan Bennett; Neal Bez; Anthony Carbery; Dirk Hundertmark
  ANALYSIS & PDE, Volume：2, Number：2, First page：147, Last page：158, 2009, [Reviewed]
  Our main result is that for d = 1, 2 the classical Strichartz norm parallel to e(is Delta) f parallel to(2+4/d)(Ls,x)(RxR(d)) associated to the free Schrodinger equation in nondecreasing as the initial datum f evolves under a certain quadratic heat flow.
  MATHEMATICAL SCIENCE PUBL, English, Scientific journal
  ISSN：1948-206X, Web of Science ID：WOS:000281883500002
• Maximal Operators along Piecewise Linear Curves near L-1　　　　　　　　　　　　　　　
  Neal Bez
  INDIANA UNIVERSITY MATHEMATICS JOURNAL, Volume：58, Number：4, First page：1639, Last page：1657, 2009, [Reviewed]
  For certain piecewise linear plane curves F and convex functions Phi we address the question of whether the maximal operator along Gamma is of weak type Phi(L). For example, when F is the graph of the continuous map y for which Y(2(k)) = 2(2k) and y is linear on [2(k), 2(k+1)], k is an element of Z, then the maximal operator along Gamma is not of weak type L(logL)sigma for any sigma is an element of (0, 1/2).
  INDIANA UNIV MATH JOURNAL, English, Scientific journal
  DOI：https://doi.org/10.1512/iumj.2009.58.3606
  DOI ID：10.1512/iumj.2009.58.3606, ISSN：0022-2518, Web of Science ID：WOS:000269448000006
• Mixed-norm estimates for a class of nonisotropic directional maximal operators and Hilbert transforms　　　　　　　　　　　　　　　
  Neal Bez
  JOURNAL OF FUNCTIONAL ANALYSIS, Volume：255, Number：12, First page：3281, Last page：3302, Dec. 2008, [Reviewed]
  For all d >= 2 and P epsilon (1, max(2, (d + 1)/2)], we prove sharp L(p) to L(p)(L(q)) estimates (modulo an endpoint) for a directional maximal operator associated to curves generated by the dilation matrices exp((log t)P), where P has real entries and eigenvalues with positive real part. For the corresponding Hilbert transform we prove an analogous result for all d >= 2 and P epsilon (1, 2]. As corollaries, we prove L(p) bounds for variable kernel singular integral operators and Nikodym-type maximal operators taking averages over certain families of curved sets in R(d). (C) 2008 Elsevier Inc. All rights reserved.
  ACADEMIC PRESS INC ELSEVIER SCIENCE, English, Scientific journal
  DOI：https://doi.org/10.1016/j.jfa.2008.07.026
  DOI ID：10.1016/j.jfa.2008.07.026, ISSN：0022-1236, Web of Science ID：WOS:000261578900003
• L-p-boundedness for the Hilbert transform and maximal operator along a class of nonconvex curves　　　　　　　　　　　　　　　
  Neal Bez
  PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Volume：135, Number：1, First page：151, Last page：161, 2007, [Reviewed]
  Some sufficient conditions on a real polynomial P and a convex function gamma are given in order for the Hilbert transform and maximal operator along ( t, P gamma( t))) to be bounded on L-p, for all p in ( 1, infinity), with bounds independent of the coefficients of P. The same conclusion is shown to hold for the corresponding hypersurface in Rd+1 ( d >= 2) under weaker hypotheses on gamma.
  AMER MATHEMATICAL SOC, English, Scientific journal
  DOI：https://doi.org/10.1090/S0002-9939-06-08603-5
  DOI ID：10.1090/S0002-9939-06-08603-5, ISSN：0002-9939, Web of Science ID：WOS:000240542200020

• 滑らかさを加味した直交ストリッカーツ評価 (関数空間の一般化とその周辺)　　　　　　　　　　　　　　　
  BEZ NEAL; HONG YOUNGHUN; LEE SANGHYUK; 中村 昌平; 澤野 嘉宏
  Number：2143, First page：173, Last page：184, Dec. 2019
  直交ストリッカーツ評価と呼ばれる新しい評価を示す. 最初に, FrankとSabinによる関数の滑らかさを加味しない直交ストリッカーツ評価[5]を示し, 後で我々による関数の滑らかさを加味した直交ストリッカーツ評価のひとつの場合の証明する. この結果は我々の論文[2]として出版されている.
  Japanese
  ISSN：1880-2818, CiNii Articles ID：120006888263, CiNii Books ID：AN00061013

• The London Mathematical Society
• The Mathematical Society of Japan

• 基底状態の諸相に対する多角的探究の試み　　　　　　　　　　　　　　　
  01 Apr. 2022 - 31 Mar. 2027
  Grant amount(Total)：38740000, Direct funding：29800000, Indirect funding：8940000
  Grant number：22H00098
• Investigating the stability of the inverse Brascamp-Lieb inequality　　　　　　　　　　　　　　　
  JSPS, Grant-in-Aid for Scientific Research (B), Apr. 2023 - Mar. 2027
  Neal Bez, Principal investigator
  Grant number：23K25777
• Endpoint estimates for geometric maximal operators　　　　　　　　　　　　　　　
  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for JSPS Fellows, Nov. 2023 - Mar. 2026
  Saitama University
  Grant amount(Total)：1600000, Direct funding：1600000
  Grant number：23KF0188
• 古典場の理論における微分型相互作用の数学解析　　　　　　　　　　　　　　　
  01 Apr. 2019 - 31 Mar. 2024
  Grant amount(Total)：43680000, Direct funding：33600000, Indirect funding：10080000
  Grant number：19H00644
• Research on Dispersive Equations and Harmonic Analysis　　　　　　　　　　　　　　　
  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Fund for the Promotion of Joint International Research (Fostering Joint International Research (B)), 09 Oct. 2018 - 31 Mar. 2023
  Waseda University
  Grant amount(Total)：17810000, Direct funding：13700000, Indirect funding：4110000
  Grant number：18KK0073
• New perspectives on space-time estimates for dispersive equations　　　　　　　　　　　　　　　
  JSPS, Grant-in-Aid for Scientific Research (B), Apr. 2019 - Mar. 2023
  Neal Bez, Principal investigator
  Grant number：19H01796
• New developments on the restriction conjecture for the Fourier transform using multilinear analysis　　　　　　　　　　　　　　　
  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for JSPS Fellows, 25 Apr. 2018 - 31 Mar. 2020
  Saitama University
  Grant amount(Total)：1500000, Direct funding：1500000
  Grant number：18F18020
• Study on null forms in global space-time in the framework of equalities　　　　　　　　　　　　　　　
  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Challenging Exploratory Research, 01 Apr. 2016 - 31 Mar. 2020
  Ozawa Tohru, Waseda University
  Grant amount(Total)：3640000, Direct funding：2800000, Indirect funding：840000
  Stability of trace theorems on the sphere is studied as the most fundamental subject in the research of null forms in global space-time. We have established the desired optimal inequalities for the stability theory and given its characterization from the viewpoint of duality. Regarding the Hardy and Rellich inequalities, we have formulated their equality framework with explicit remainder terms, therely we were able to recast the associated best constants and extremizers in a direct and explicit understanding. This provides a new method, independent of implicit arguments of contradiction and compactness.
  Grant number：16K13771
• Conjectures associated with Brascamp-Lieb type inequalities　　　　　　　　　　　　　　　
  JSPS, Grant-in-Aid for Young Scientists (A), Apr. 2016 - Mar. 2019
  Neal Bez, Principal investigator
  Competitive research funding, Grant number：16H05995
• New frontiers in kinetic equation theory　　　　　　　　　　　　　　　
  JSPS, Grant-in-Aid for Research Activity Start-up, Aug. 2014 - Mar. 2016
  Neal Bez, Principal investigator
  Competitive research funding, Grant number：26887008