Duke Math Research Articles

Hecke eigenvalues in 𝑝-modular representations of unramified 𝑈(2,1)

Let G G be the unramified unitary group U ( 2 , 1 ) ( E / F ) U(2, 1)(E/F) defined over a non-archimedean local field F F of odd residue characteristic p p , and let K K be a maximal compact open subgroup of G G . For an irreducible smooth F ¯ p \overline {\mathbf {F}}_p -representation π \pi of G G , and a weight σ \sigma of K K contained in π \pi , we prove that π \pi admits eigenvectors for the spherical Hecke algebra H ( K , σ ) \mathcal {H}(K, \sigma ) . Our approach is close to that of Barthel–Livné [Duke Math. J. 75 (1994), pp. 261–292] for G L 2 GL_2 . For G L n ( n ≥ 3 ) GL_n (n\geq 3) , we give a sketch of an essential case where the analogue of a crucial ingredient in loc.cit fails.

Irreducibility of some crystalline loci with irregular Hodge–Tate weights

We show that loci of crystalline representations of G K G_K for K / Q p K/\mathbb {Q}_p an unramified extension are irreducible when the Hodge–Tate weights are fixed and sufficiently small. This was previously known for weights in the interval [ − p , 0 ] [-p,0] and in this paper we show how that this bound can be relaxed provided the Hodge–Tate weights are sufficiently irregular at certain embeddings. This is motivated by the desire to extend the conjectures of Breuil–Mézard [Duke Math. J. 115 (2002), pp. 205–310] on loci of potentially crystalline representations to irregular weights.

Winning of inhomogeneous bad for curves

AbstractWe prove the absolute winning property of weighted inhomogeneous badly approximable vectors on nondegenerate analytic curves. This answers a question by Beresnevich, Nesharim, and Yang. In particular, our result is an inhomogeneous version of the main result in (Duke Math J 171(14), 2022) by Beresnevich, Nesharim, and Yang. Also, the generality of the inhomogeneous part that we considered extends the previous result in An et al. (Adv Math 324:148–202, 2018). Moreover, our results even contribute to classical results, namely establishing the inhomogeneous Schmidt’s conjecture in arbitrary dimensions.

ON THE GENERALIZATION OF SOME CLASSES OF CONVEX IN DIRECTION AND TYPICALLY REAL FUNCTIONS

In the article by M.O. Reade (Duke Math. Journal, 1956) using the condition |arg (f'(z)/ g'(z)) | ≤ γπ/2, where g(z) is a convex function,0≤γ≤1 , a class of functions close-to-convex (almost convex) of order γ is introduced. In our paper, we introduce a subclass of the class of close-to-convex (almost convex) order γ functions satisfying the condition |arg [(1-λzn ) f'(z)]| ≤ γπ/2, which, for different parameter values, gives a number of well-known subclasses of univalent (schlicht) functions. Based on this subclass, a class of close-to-starlike (almost star-shaped) functions is constructed, containing a number of subclasses that have been actively studied by many authors in recent years, as well as a classical class of typically real functions. For this classes exact theorems of distortion (growth) and radii of convexity (starlikeness) are obtained, generalizing previously known results. The case is also considered when the functions of the introduced classes have missing members in the power series expansion. The results obtained are accurate and not only generalize previously known results, but also reveal the properties of a number of new subclasses of univalent (schlicht) functions.

Semiclassical Equivalence of Two White Dwarf Models as Ground States of the Relativistic Hartree–Fock and Vlasov–Poisson energies

AbstractWe are concerned with the semi-classical limit for ground states of the relativistic Hartree–Fock energies (HF) under a mass constraint, which are considered as the quantum mean-field model of white dwarfs (Lenzmann and Lewin in Duke Math J 152:257–315, 2010). In Jang and Seok (Kinet Relat Models 15:605–620, 2022), fermionic ground states of the relativistic Vlasov–Poisson energy (VP) are constructed as a classical mean-field model of white dwarfs, and are shown to be equivalent to the classical Chandrasekhar model. In this paper, we prove that as the reduced Planck constant $$\hbar $$ ħ goes to the zero, the $$\hbar $$ ħ -parameter family of the ground energies and states of (HF) converges to the fermionic ground energy and state of (VP) with the same mass constraint.

Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity

In this paper we show that the geodesic flow of a Finsler metric is Anosov if and only if there exists a C 2 C^2 open neighborhood of Finsler metrics all of whose closed geodesics are hyperbolic. For surfaces this result holds also for Riemannian metrics. This follows from a recent result of Contreras and Mazzucchelli [Duke Math. J. 173 (2024), pp. 347–390]. Furthermore, geodesic flows of Riemannian or Finsler metrics on surfaces are C 2 C^2 stably ergodic if and only if they are Anosov.

A toy model for damped water waves

We consider a toy model for a damped water waves system in a domain [Formula: see text]. The toy model is based on the paradifferential formulation of the water waves system derived in the work of Alazard–Burq–Zuily [On the water-waves equations with surface tension, Duke Math. J. 158 (2011) 413–499]. The form of damping we consider is a modified sponge layer proposed for the [Formula: see text] water waves system by Clamond et al. in [An efficient model for three-dimensional surface wave simulations. Part II: Generation and absorption, J. Comput. Phys. 205 (2005) 686–705]. We show that, in the case of small Cauchy data, solutions to the toy model exhibit a quadratic lifespan. This is done via proving energy estimates with Klainerman vector fields.

On the uniqueness of variable coefficient Schrödinger equations

In this paper, we prove unique continuation properties for linear variable coefficient Schrödinger equations with bounded real potentials. Under certain smallness conditions on the leading coefficients, we prove that solutions decaying faster than any cubic exponential rate at two different times must be identically zero. Assuming a transversally anisotropic type condition, we recover the sharp Gaussian (quadratic exponential) rate in the series of works by Escauriaza–Kenig–Ponce–Vega [On uniqueness properties of solutions of Schrödinger equations, Comm. Partial Differential Equations 31(10–12) (2006) 1811–1823; Hardy’s uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. (JEMS) 10(4) (2008) 883–907; The sharp Hardy uncertainty principle for Schrödinger evolutions, Duke Math. J. 155(1) (2010) 163–187].

Pogorelov estimates for semi-convex solutions of 𝑘-curvature equations

In this paper, we consider k k -curvature equations σ k ( κ [ M u ] ) = f ( x , u , ∇ u ) \sigma _k(\kappa [M_u])=f(x,u,\nabla u) subject to ( k + 1 ) (k+1) -convex Dirichlet boundary data instead of affine Dirichlet data of Sheng, Urbas, and Wang [Duke Math. J. 123 (2004), pp. 235–264]. By using the crucial concavity inequality for Hessian operator of Lu [Calc. Var. Partial Differential Equations 62 (2023), p.23], we derive Pogorelov estimates of semi-convex admissible solutions for these k k -curvature equations.

Non-triviality of the phase transition for percolation on finite transitive graphs

We prove that if (G_{n})_{n \geq1}=((V_{n},E_{n}))_{n\geq 1} is a sequence of finite, vertex-transitive graphs with bounded degrees and |V_{n}|\to\infty that is at least (1+\varepsilon) -dimensional for some \varepsilon>0 in the sense that \operatorname{diam} (G_{n})=O(|V_{n}|^{1/(1+\varepsilon)}) \quad \text{as }n\to\infty then this sequence of graphs has a non-trivial phase transition for Bernoulli bond percolation. More precisely, we prove under these conditions that for each 0<\alpha <1 there exists p_{c}(\alpha)<1 such that for each p\geq p_{c}(\alpha) , Bernoulli- p bond percolation on G_{n} has a cluster of size at least \alpha |V_{n}| with probability tending to 1 as n\to \infty . In fact, we prove more generally that there exists a universal constant a such that the same conclusion holds whenever \operatorname{diam} (G_{n})=O\Big(\frac{|V_{n}|}{(\log |V_{n}|)^{a}}\Big) \quad \text{as }n\to\infty. This verifies a conjecture of Benjamini (2001) up to the value of the constant a , which he suggested should be 1 . We also prove a generalization of this result to quasitransitive graph sequences with a bounded number of vertex orbits. A key step in our argument is a direct proof of our result when the graphs G_{n} are all Cayley graphs of Abelian groups, in which case we show that one may indeed take a=1 . This result relies crucially on a new theorem of independent interest stating roughly that balls in arbitrary Abelian Cayley graphs can always be approximated by boxes in \Z^{d} with the standard generating set. Another key step is to adapt the methods of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin [Duke Math. J. 169 (2020)] from infinite graphs to finite graphs. This adaptation also leads to an isoperimetric criterion for infinite graphs to have a non-trivial uniqueness phase (i.e., to have p_{u}<1 ), which is of independent interest. We also prove that the set of possible values of the critical probability of an infinite quasitransitive graph has a gap at 1 in the sense that for every k,n<\infty there exists \varepsilon>0 such that every infinite graph G of degree at most k whose vertex set has at most n orbits under \operatorname{Aut}(G) has either p_c=1 or p_c\leq 1-\varepsilon .

A standard form for scattered linearized polynomials and properties of the related translation planes

In this paper, we present results concerning the stabilizer Gf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_f$$\\end{document} in GL(2,qn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{{GL}}\\,}}(2,q^n)$$\\end{document} of the subspace Uf={(x,f(x)):x∈Fqn}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_f=\\{(x,f(x)):x\\in \\mathbb {F}_{q^n}\\}$$\\end{document}, f(x) a scattered linearized polynomial in Fqn[x]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{q^n}[x]$$\\end{document}. Each Gf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_f$$\\end{document} contains the q-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q-1$$\\end{document} maps (x,y)↦(ax,ay)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(x,y)\\mapsto (ax,ay)$$\\end{document}, a∈Fq∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\in \\mathbb {F}_{q}^*$$\\end{document}. By virtue of the results of Beard (Duke Math J, 39:313–321, 1972) and Willett (Duke Math J 40(3):701–704, 1973), the matrices in Gf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_f$$\\end{document} are simultaneously diagonalizable. This has several consequences: (i) the polynomials such that |Gf|>q-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|G_f|>q-1$$\\end{document} have a standard form of type ∑j=0n/t-1ajxqs+jt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{j=0}^{n/t-1}a_jx^{q^{s+jt}}$$\\end{document} for some s and t such that (s,t)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(s,t)=1$$\\end{document}, t>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t>1$$\\end{document} a divisor of n; (ii) this standard form is essentially unique; (iii) for n>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n>2$$\\end{document} and q>3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q>3$$\\end{document}, the translation plane Af\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}_f$$\\end{document} associated with f(x) admits nontrivial affine homologies if and only if |Gf|>q-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|G_f|>q-1$$\\end{document}, and in that case those with axis through the origin form two groups of cardinality (qt-1)/(q-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(q^t-1)/(q-1)$$\\end{document} that exchange axes and coaxes; (iv) no plane of type Af\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}_f$$\\end{document}, f(x) a scattered polynomial not of pseudoregulus type, is a generalized André plane.

Density of mode-locking property for quasi-periodically forced Arnold circle maps

Abstract We show that the mode-locking region of the family of quasi-periodically forced Arnold circle maps with a topologically generic forcing function is dense. This gives a rigorous verification of certain numerical observations in [M. Ding, C. Grebogi and E. Ott. Evolution of attractors in quasiperiodically forced systems: from quasiperiodic to strange nonchaotic to chaotic. Phys. Rev. A 39(5) (1989), 2593–2598] for such forcing functions. More generally, under some general conditions on the base map, we show the density of the mode-locking property among dynamically forced maps (defined in [Z. Zhang. On topological genericity of the mode-locking phenomenon. Math. Ann. 376 (2020), 707–72]) equipped with a topology that is much stronger than the $C^0$ topology, compatible with smooth fiber maps. For quasi-periodic base maps, our result generalizes the main results in [A. Avila, J. Bochi and D. Damanik. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J.146 (2009), 253–280], [J. Wang, Q. Zhou and T. Jäger. Genericity of mode-locking for quasiperiodically forced circle maps. Adv. Math.348 (2019), 353–377] and Zhang (2020).

The ℓp$\ell ^p$ norm of the Riesz–Titchmarsh transform for even integer p$p$

AbstractThe long‐standing conjecture that for the norm of the Riesz–Titchmarsh discrete Hilbert transform is the same as the norm of the classical Hilbert transform, is verified when or , for . The proof, which is algebraic in nature, depends in a crucial way on the sharp estimate for the norm of a different variant of this operator for the full range of . The latter result was recently proved by the authors (Duke Math. J. 168 (2019), no. 3, 471–504).

On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation

We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold is a small divisor in a quadratic source term of the perturbation equation. This singularity is due to the threshold resonance of the linearized operator and the absence of null structure in the nonlinearity. The threshold resonance of the linearized operator produces a one-dimensional space of slowly decaying Klein-Gordon waves, relative to local norms. In contrast, the closely related perturbation equation for the sine-Gordon kink does exhibit null structure, which makes the corresponding quadratic source term amenable to normal forms (see Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820]). The main result of this work establishes decay estimates up to exponential time scales for small “codimension one type” perturbations of the soliton of the focusing cubic Klein-Gordon equation. The proof is based upon a super-symmetric approach to the study of modified scattering for 1D nonlinear Klein-Gordon equations with Pöschl-Teller potentials from Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820], and an implementation of a version of an adapted functional framework introduced by Germain and Pusateri [Forum Math. Pi 10 (2022), p. 172].

Limit shapes from harmonicity: dominos and the five vertex model

We discuss how to construct limit shapes for the domino tiling model (square lattice dimer model) and five-vertex model, in appropriate polygonal domains. Our methods are based on the harmonic extension method of Kenyon and Prause (2022 Duke Math J. 171 3003–22).

Growth rate of Dehn twist lattice points in Teichmüller space

Athreya et al. [Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J. 161 (2012) 1055–1111] have shown the number of mapping class group lattice points intersecting a closed ball of radius [Formula: see text] in Teichmüller space is asymptotic to [Formula: see text], where [Formula: see text] is the dimension of the Teichmüller space. In contrast we show the number of Dehn twist lattice points intersecting a closed ball of radius [Formula: see text] is coarsely asymptotic to [Formula: see text]. Moreover, we show the number multi-twist lattice points intersecting a closed ball of radius [Formula: see text] grows coarsely at least at the rate of [Formula: see text].

Noetherian spectrum condition and the ring 𝒜[X

Let [Formula: see text] be an ascending chain of commutative rings with identity and let [Formula: see text] be the ring of polynomials with coefficient of degree [Formula: see text] in [Formula: see text] for each [Formula: see text] In the first part of this paper, we give a necessary and sufficient conditions for the ring [Formula: see text] to satisfy Noetherian spectrum (a ring [Formula: see text] is said to have Noetherian spectrum if [Formula: see text] satisfies the Ascending Chain Condition (ACC) on radical ideals). In particular, we extend classical results proved by Ohm and Pendleton in [J. Ohm and R. Pendleton, Rings with Noetherian spectrum, Duke Math. J. 35 (1968) 631–639] and by Hizem in [S. Hizem, Chain conditions in rings of the form A + XB[X] and A + XI[X], in Commutative Algebra and its Applications (de Gruyter, 2009), pp. 259–274]. The second part of this paper is motivated by the interesting results proved on uniformly [Formula: see text]-Noetherian rings by Kim et al. in [W. Qi, H. Kim, F. Wang, M. Chen and W. Zhao, Uniformly S-Noetherian rings (submitted)]. We introduce the concept of uniformly [Formula: see text]-Noetherian spectrum and study its properties. Let [Formula: see text] be a commutative ring and [Formula: see text] a multiplicative subset of [Formula: see text] We say that [Formula: see text] has uniformly[Formula: see text]-Noetherian spectrum if that there exists an [Formula: see text] such that for any ideal [Formula: see text] of [Formula: see text], [Formula: see text] for some finitely generated subideal [Formula: see text] of [Formula: see text] We give the Eakin–Nagata–Formanek theorem for the uniformly [Formula: see text]-Noetherian spectrum condition. We also study the uniformly [Formula: see text]-Noetherian uniformly properties on several ring constructions (Nagata’s idealization and amalgamated algebras).

Unirationality of varieties described by families of projective hypersurfaces

Let \mathscr{X}\to W be a flat family of generically irreducible hypersurfaces of degree d\geq 2 in \mathbb{P}^n with singular locus of dimension t , with W unirational of dimension r . We prove that if n is large enough with respect to d , r and t , then \mathscr{X} is unirational. This extends results by J. Harris, B. Mazur and R. Pandharipande in [Duke Math. J. 95 (1998), 125–160] and A. Predonzan in [Rend. Sem. Mat. Univ. Padova 31 (1961), 281–293].

Deformation of the Weighted Scalar Curvature

Inspired by the work of Fischer-Marsden [Duke Math. J. 42 (1975), 519-547], we study in this paper the deformation of the weighted scalar curvature. By studying the kernel of the formal $L_\phi^2$-adjoint for the linearization of the weighted scalar curvature, we prove several geometric results. In particular, we define a weighted vacuum static space, and study locally conformally flat weighted vacuum static spaces. We then prove some stability results of the weighted scalar curvature on flat spaces. Finally, we consider the prescribed weighted scalar curvature problem on closed smooth metric measure spaces.

Holomorphic Hörmander-type functional calculus on sectors and strips

In this paper, the abstract multiplier theorems for 0 0 -sectorial and 0 0 -strip type operators by Kriegler and Weis [Math. Z. 289 (2018), pp. 405–444] are refined and generalized to arbitrary sectorial and strip-type operators. To this end, holomorphic Hörmander-type functions on sectors and strips are introduced, with even a finer scale of smoothness than the classical polynomial scale. Moreover, we establish alternative descriptions of these spaces involving Schwartz and “holomorphic Schwartz” functions. Finally, the abstract results are combined with a result by Carbonaro and Dragičević [Duke Math. J. 166 (2017), pp. 937–974] to obtain an improvement—with respect to the smoothness condition—of the known Hörmander-type multiplier theorem for general symmetric contraction semigroups.