Nonoscillation and oscillation of second-order linear dynamic equations via the sequence of functions technique

We study nonoscillation/oscillation of the dynamic equation

$${(rx^\Delta)}^{\Delta}(t) + p(t)x(t)= 0 \quad {\rm for} t \in[t_0, \infty)_{\mathbb{T}},$$

where \({t_0 \in \mathbb{T}}\), \({{\rm sup} \mathbb{T} = \infty}\), \({r \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, \mathbb{R}^+)}\), \({p \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, {\mathbb{R}^+_0})}\). By using the Riccati substitution technique, we construct a sequence of functions which yields a necessary and sufficient condition for the nonoscillation of the equation. In addition, our results are new in the theory of dynamic equations and not given in the discrete case either. We also illustrate applicability and sharpness of the main result with a general Euler equation on arbitrary time scales. We conclude the paper by extending our results to the equation

$${(rx^\Delta)}^{\Delta}(t) + p(t)x^\sigma(t)= 0 \quad {\rm for} t \in[t_0, \infty)_{\mathbb{T}},$$

which is extensively discussed on time scales.

     

Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

Let \({L_{w}}{:=-w^{-1}{\rm div}(A\nabla)}\) be the degenerate elliptic operator on the Euclidean space \({{\mathbb{R}^{n}}}\), where w is a Muckenhoupt \({A_{2}({\mathbb{R}^{n}})}\) weight. In this article, the authors establish the Riesz transform characterization of the Hardy space \({H^{p}_{L_{w}}({\mathbb{R}}^{n})}\) associated with L_w, for \({w \in A_{q}({\mathbb{R}}^{n}) \cap RH_{\frac{n}{n-2}}({\mathbb{R}^{n}})}\) with \({n \geq 3}\), \({q \in [1,2]}\) and \({p \in (q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]}\) if, for some \({r \in (1,\,2]}\), \({{\{tL_w e^{-tL_w}\}}_{t\geq 0}}\) satisfies the weighted \({L^{r}-L^{2}}\) full off-diagonal estimates.     

Homoclinic Solutions for <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)-Laplacian–Hamiltonian Systems Without Coercive Conditions

In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systems

$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$

where \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) with p(t) > 1, \({a \in C(\mathbb{R},\mathbb{R})}\), \({W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})}\) and \({\nabla W(t,u)}\) is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants \({0<\tau_1<\tau_2<\infty}\) such that \({\tau_1\leq a(t)\leq \tau_2 }\) for all \({t \in \mathbb{R}}\) and W(t, u) is of super-p(t) growth or sub-p(t) growth as \({|u|\rightarrow \infty}\), we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.

     

A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $${\mathbb R^{N}}$$

In this work we study the following class of problems in \({\mathbb R^{N}, N > 2s}\)

$$\varepsilon^{2s}(-\Delta)^{s}u + V(z)u = f(u), \,\,\,u(z) > 0$$

where \({0 < s < 1}\), \({(-\Delta)^{s}}\) is the fractional Laplacian, \({\varepsilon}\) is a positive parameter, the potential \({V : \mathbb{R}^N \to \mathbb{R}}\) and the nonlinearity \({f : \mathbb R \to \mathbb R}\) satisfy suitable assumptions; in particular it is assumed that \({V}\) achieves its positive minimum on some set \({M.}\) By using variational methods we prove existence and multiplicity of positive solutions when \({\varepsilon \to 0^{+}}\). In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the “topological complexity” of the set \({M}\).

     

Asymptotics for Erdős–Solovej Zero Modes in Strong Fields

We consider the strong field asymptotics for the occurrence of zero modes of certain Weyl–Dirac operators on \({\mathbb{R}^3}\). In particular, we are interested in those operators \({\mathcal{D}_B}\) for which the associated magnetic field \({B}\) is given by pulling back a two-form \({\beta}\) from the sphere \({\mathbb{S}^2}\) to \({\mathbb{R}^3}\) using a combination of the Hopf fibration and inverse stereographic projection. If \({\int_{\mathbb{s}^2} \beta \neq 0}\), we show that

$$\sum_{0 \leq t \leq T} {\rm dim Ker} \mathcal{D}{tB}=\frac{T^2}{8\pi^2}\,\Big| \int_{\mathbb{S}^2}\beta\Big|\,\int_{\mathbb{S}^2}|{\beta}| +o(T^2)$$

as \({T\to+\infty}\). The result relies on Erd?s and Solovej’s characterisation of the spectrum of \({\mathcal{D}_{tB}}\) in terms of a family of Dirac operators on \({\mathbb{S}^2}\), together with information about the strong field localisation of the Aharonov–Casher zero modes of the latter.

     

Open book structures on semi-algebraic manifolds

Given a C ² semi-algebraic mapping \({F} : {\mathbb{R}^N \rightarrow \mathbb{R}^p}\), we consider its restriction to \({W \hookrightarrow \mathbb{R^{N}}}\) an embedded closed semi-algebraic manifold of dimension \({n-1 \geq p \geq 2}\) and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection \({\frac{F}{\Vert F \Vert}:W{\setminus} F^{-1}(0) \to S^{p-1}}\). Moreover, we show that the well known local and global Milnor fibrations, in the real and complex settings, follow as a byproduct by considering W as spheres of small and big radii, respectively. Furthermore, we consider the composition mapping of F with the canonical projection \({\pi: \mathbb{R}^{p} \to \mathbb{R}^{p-1}}\) and prove that the fibers of \({\frac{F}{\Vert F \Vert}}\) and \({\frac{\pi \circ F}{\Vert \pi \circ F \Vert}}\) are homotopy equivalent. We also show several formulae relating the Euler characteristics of the fiber of the projection \({\frac{F}{\Vert F \Vert}}\) and \({W \cap F^{-1}(0)}\). Similar formulae are proved for mappings obtained after composition of F with canonical projections.     

An inverse spectral theorem for Kreĭn strings with a negative eigenvalue

A string is a pair \({(L, \mathfrak{m})}\) where \({L \in[0, \infty]}\) and \({\mathfrak{m}}\) is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \({\mathfrak{m}}\) as its mass density. To each string a differential operator acting in the space \({L^2(\mathfrak{m})}\) is associated. Namely, the Kre?n–Feller differential operator \({-D_{\mathfrak{m}}D_x}\) ; its eigenvalue equation can be written, e.g., as

$$f^{\prime}(x) + z \int_0^L f(y)\,d\mathfrak{m}(y) = 0,\quad x \in\mathbb R,\ f^{\prime}(0-) = 0.$$

A positive Borel measure τ on \({\mathbb R}\) is called a (canonical) spectral measure of the string \({\textsc S[L, \mathfrak{m}]}\) , if there exists an appropriately normalized Fourier transform of \({L^2(\mathfrak{m})}\) onto L ²(τ). In order that a given positive Borel measure τ is a spectral measure of some string, it is necessary that: (1) \({\int_{\mathbb R} \frac{d\tau(\lambda)}{1+|\lambda|} < \infty}\) . (2) Either \({{\rm supp} \tau \subseteq [0, \infty)}\) , or τ is discrete and has exactly one point mass in (?∞, 0). It is a deep result, going back to Kre?n in the 1950’s, that each measure with \({\int_{\mathbb R}\frac{d\tau(\lambda)}{1+|\lambda|} < \infty}\) and \({{\rm supp} \tau \subseteq [0, \infty)}\) is a spectral measure of some string, and that this string is uniquely determined by τ. The question remained open, which conditions characterize whether a measure τ with \({{\rm supp} \tau \not\subseteq [0, \infty)}\) is a spectral measure of some string. In the present paper, we answer this question. Interestingly, the solution is much more involved than the first guess might suggest.

     

Stability analysis of abstract systems of Timoshenko type

We consider an abstract system of Timoshenko type

$$\begin{aligned} \left\{ {\begin{array}{l} \rho_1{{\ddot \varphi}} + a A^{\frac12}(A^{\frac12}\varphi + \psi) =0\\\rho_2{{\ddot \psi}} + b A \psi + a (A^{\frac12}\varphi + \psi) -\delta A^\gamma {\theta} = 0\\\rho_3{{\dot \theta}} + c A\theta + \delta A^\gamma {{\dot \psi}} =0 \end{array}} \right. \end{aligned}$$

where the operator \({A}\) is strictly positive selfadjoint. For any fixed \({\gamma \in {\mathbb{R}}}\), the stability properties of the related solution semigroup \({S(t)}\) are discussed. In particular, a general technique is introduced in order to prove the lack of exponential decay of \({S(t)}\) when the spectrum of the leading operator \({A}\) does not consist of eigenvalues only.

     

A Sharp Weak-Type $$(\infty ,\infty )$$ Inequality for the Hilbert Transform

The paper is devoted to sharp weak type \((\infty ,\infty )\) estimates for \({\mathcal {H}}^{\mathbb {T}}\) and \({\mathcal {H}}^{\mathbb {R}}\), the Hilbert transforms on the circle and real line, respectively. Specifically, it is proved that

$$\begin{aligned} \left\| {\mathcal {H}}^{\mathbb {T}}f\right\| _{W({\mathbb {T}})}\le \Vert f\Vert _{L^\infty ({\mathbb {T}})} \end{aligned}$$

and

$$\begin{aligned} \left\| {\mathcal {H}}^{\mathbb {R}}f\right\| _{W({\mathbb {R}})}\le \Vert f\Vert _{L^\infty ({\mathbb {R}})}, \end{aligned}$$

where \(W({\mathbb {T}})\) and \(W({\mathbb {R}})\) stand for the weak-\(L^\infty \) spaces introduced by Bennett, DeVore and Sharpley. In both estimates, the constant \(1\) on the right is shown to be the best possible.

     

Intersection properties of radial solutions and global bifurcation diagrams for supercritical quasilinear elliptic equations

We study the positive solution \({u(r,\rho)}\) of the quasilinear elliptic equation

$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u^{\prime}|^{\beta-1}u^{\prime})^{\prime}+|u|^{p-1}u=0, & 0 < r < \infty,\\ u(0) = \rho > 0,\ u^{\prime}(0)=0.\end{cases}$$

This class of differential operators includes the usual Laplace, m-Laplace, and k-Hessian operators in the space of radial functions. The equation has a singular positive solution u ^*(r) under certain conditions on \({\alpha}\), \({\beta}\), \({\gamma}\), and p. A generalized Joseph–Lundgren exponent, which we denote by \({p^*_{JL}}\), is obtained. We study the intersection numbers between \({u(r,\rho)}\) and u ^*(r) and between \({u(r,\rho_0)}\) and \({u(r,\rho_1)}\), and see that \({p^*_{JL}}\) plays an important role. We also determine the bifurcation diagram of the problem

$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u^{\prime}|^{\beta-1}u^{\prime})^{\prime} + \lambda(u+1)^p=0, & 0 < r < 1,\\ u(r) > 0, & 0 \le r < 1,\\ u^{\prime}(0)=0,\ u(1)=0.\end{cases}$$

The main technique used in the proofs is a phase plane analysis.

     

Random approximation and the vertex index of convex bodies

We prove that there exists an absolute constant \({\alpha > 1}\) with the following property: if K is a convex body in \({{\mathbb R}^n}\) whose center of mass is at the origin, then a random subset \({X\subset K}\) of cardinality \({{\rm card}(X)=\lceil\alphan\rceil }\) satisfies with probability greater than \({1-e^{-c_1n}}\)

$$K\subseteq c_2n\, {\rm conv}(X),$$

where \({c_1, c_2 > 0}\) are absolute constants. As an application we show that the vertex index of any convex body K in \({{\mathbb R}^n}\) is bounded by \({c_3n^2}\), where \({c_3 > 0}\) is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symmetric case.

     

Optimal embeddings and compact embeddings of Bessel-potential-type spaces

First, we establish necessary and sufficient conditions for embeddings of Bessel potential spaces \({H^{\sigma}X(\mathbb R^n)}\) with order of smoothness less than one, modelled upon rearrangement invariant Banach function spaces \({X(\mathbb R^n)}\), into generalized Hölder spaces. To this end, we derive a sharp estimate of modulus of smoothness of the convolution of a function \({f\in X(\mathbb R^n)}\) with the Bessel potential kernel g _σ , 0 < σ < 1. Such an estimate states that if \({g_{\sigma}}\) belongs to the associate space of X, then

$\omega(f*g_{\sigma},t)\precsim \int\limits_0^{t^n}s^{\frac{\sigma}{n}-1}f^*(s)\,ds \quad {\rm for\,all} \quad t\in(0,1) \quad {\rm and\,every}\quad f\in X(\mathbb R^n).$

Second, we characterize compact subsets of generalized Hölder spaces and then we derive necessary and sufficient conditions for compact embeddings of Bessel potential spaces \({H^{\sigma}X(\mathbb R^n)}\) into generalized Hölder spaces. We apply our results to the case when \({X(\mathbb R^n)}\) is the Lorentz–Karamata space \({L_{p,q;b}(\mathbb R^n)}\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces \({H^{\sigma}L_{p,q;b}(\mathbb R^n)}\) into generalized Hölder spaces and also compact embeddings of spaces in question. Applications cover both superlimiting and limiting cases.

     

$${\varvec{\mathcal {}}}$$-minimaxness and local cohomology modules

Let R be a commutative Noetherian ring, \({\mathfrak {a}}\) an ideal of R, M a finitely generated R-module, and \({\mathcal {S}}\) a Serre subcategory of the category of R-modules. We introduce the concept of \({\mathcal {S}}\)-minimax R-modules and the notion of the \({\mathcal {S}}\)-finiteness dimension

$$\begin{aligned} f_{\mathfrak {a}}^{{\mathcal {S}}}(M):=\inf \lbrace f_{\mathfrak {a}R_{\mathfrak {p}}}(M_{\mathfrak {p}}) \vert \mathfrak {p}\in {\text {Supp}}_R(M/ \mathfrak {a}M) \text { and } R/\mathfrak {p}\notin {\mathcal {S}} \rbrace \end{aligned}$$

and we will prove that: (i) If \({\text {H}}_{\mathfrak {a}}^{0}(M), \cdots ,{\text {H}}_{\mathfrak {a}}^{n-1}(M)\) are \({\mathcal {S}}\)-minimax, then the set \(\lbrace \mathfrak {p}\in {\text {Ass}}_R( {\text {H}}_{\mathfrak {a}}^{n}(M)) \vert R/\mathfrak {p}\notin {\mathcal {S}}\rbrace \) is finite. This generalizes the main results of Brodmann–Lashgari (Proc Am Math Soc 128(10):2851–2853, 2000), Quy (Proc Am Math Soc 138:1965–1968, 2010), Bahmanpour–Naghipour (Proc Math Soc 136:2359–2363, 2008), Asadollahi–Naghipour (Commun Algebra 43:953–958, 2015), and Mehrvarz et al. (Commun Algebra 43:4860–4872, 2015). (ii) If \({\mathcal {S}}\) satisfies the condition \(C_{\mathfrak {a}}\), then

$$\begin{aligned} f_{\mathfrak {a}}^{{\mathcal {S}}}(M)= \inf \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not } {\mathcal {S}}\hbox {-}minimax\rbrace . \end{aligned}$$

This is a formulation of Faltings’ Local-global principle for the \({\mathcal {S}}\)-minimax local cohomology modules. (iii) \( \sup \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not } {\mathcal {S}}\text {-minimax} \rbrace = \sup \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not in } {\mathcal {S}} \rbrace \).

     

Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let \({\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}\) be a cocompact Fuchsian subgroup of first kind. For \({k \in \frac{1}{2} \mathbb{Z}}\) (or \({k \in 2\mathbb{Z}}\)), let \({S^{k}_{\nu}(\Gamma)}\) denote the complex vector space of cusp forms of weight-k and nebentypus \({\nu^{2k}}\) (\({\nu^{k\slash 2}}\), if \({k \in 2\mathbb{Z}}\)) with respect to \({\Gamma}\), where \({\nu}\) is a unitary character. Let \({\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}\) denote an orthonormal basis of \({S^{k}_{\nu}(\Gamma)}\). In this article, we show that as \({k \rightarrow \infty,}\) the sup-norm for \({\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}}\) is bounded by O(k), where the implied constant is independent of \({\Gamma}\). Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when \({\Gamma}\) is cofinite.     

Hecke algebras for {{\rm GL}_n} over local fields

We study the local Hecke algebra \({\mathcal{H}_{G}(K)}\) for \({G = {\rm GL}_{n}}\) and K a non-archimedean local field of characteristic zero. We show that for \({G = {\rm GL}_{2}}\) and any two such fields K and L, there is a Morita equivalence \({\mathcal{H}_{G}(K) \sim_{M} \mathcal{H}_{G}(L)}\), by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for \({G = {\rm GL}_{n}}\), there is an algebra isomorphism \({\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)}\) which is an isometry for the induced \({L^1}\)-norm if and only if there is a field isomorphism \({K \cong L}\).     

The generalized Korteweg-de Vries equation with time oscillating nonlinearity in scale critical Sobolev space

We consider the generalized Korteweg-de Vries (gKdV) equation with the time oscillating nonlinearity:

$${\partial}_t u+{\partial}_x^3 u+ g(\omega t) {\partial}_x (|u|^{p-1}u)= 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}.$$

Under the suitable assumption on g, we show that if the nonlinear term is mass critical or supercritical i.e., \({p \geq 5}\) and \({u(0) \in \dot{H}^{s_{p}}}\), where \({s_{p} = 1/2 - 2/(p-1)}\) is a scale critical exponent, then there exists a unique global solution to (gKdV) provided that \({|\omega|}\) is sufficiently large. We also obtain the behavior of the solution to (gKdV) as \({|\omega| \to \infty}\).

     

The Holditch-Type Theorem for the Polar Moment of Inertia of the Orbit Curve in the Generalized Complex Plane

In this study, we first calculate the polar moment of inertia of orbit curves under one-parameter planar motion in the generalized complex plane \({{\mathbb{C}_p}}\) and then give the Holditch-type theorem for \({{\mathbb{C}_p}}\): When the fixed points \({X}\) and \({Y}\) on the moving plane \({{\mathbb{K}_p} \subset {\mathbb{C}_p}}\) trace the same curve \({k}\) with the polar moment of inertia \({{T_X}}\), the different point \({Z}\) on this line segment \({XY}\) traces another curve \({{k_Z}}\) with the polar moment of inertia \({{T_Z}}\) during the one-parameter planar motion in the fixed plane \({{\mathbb{K}'_p} \subset {\mathbb{C}_p}}\). Thus, we obtain that the difference between the polar moments of inertia of these curves \({( {{T_Z} - {T_X}} )}\) depends on the only the \({p}\)-distances of this points and \({p}\)-rotation angle of the motion, \({{T_X} - {T_Z} = {\delta _p}ab.}\)     

Intrinsic Structures of Certain Musielak–Orlicz Hardy Spaces

For any \(p\in (0,\,1]\), let \(H^{\Phi _p}(\mathbb {R}^n)\) be the Musielak–Orlicz Hardy space associated with the Musielak–Orlicz growth function \(\Phi _p\), defined by setting, for any \(x\in \mathbb {R}^n\) and \(t\in [0,\,\infty )\),

$$\begin{aligned}&\Phi _{p}(x,\,t)\\&\quad := {\left\{ \begin{array}{ll} \displaystyle \frac{t}{\log {(e+t)}+[t(1+|x|)^n]^{1-p}}&{} \quad \text {when}\ n(1/p-1)\notin \mathbb N \cup \{0\},\\ \displaystyle \frac{t}{\log (e+t)+[t(1+|x|)^n]^{1-p}[\log (e+|x|)]^p}&{} \quad \text {when}\ n(1/p-1)\in \mathbb N\cup \{0\}, \end{array}\right. } \end{aligned}$$

which is the sharp target space of the bilinear decomposition of the product of the Hardy space \(H^p(\mathbb {R}^n)\) and its dual. Moreover, \(H^{\Phi _1}(\mathbb {R}^n)\) is the prototype appearing in the real-variable theory of general Musielak–Orlicz Hardy spaces. In this article, the authors find a new structure of the space \(H^{\Phi _p}(\mathbb {R}^n)\) by showing that, for any \(p\in (0,\,1]\), \(H^{\Phi _p}(\mathbb {R}^n)=H^{\phi _0}(\mathbb {R}^n) +H_{W_p}^p({{{\mathbb {R}}}^n})\) and, for any \(p\in (0,\,1)\), \(H^{\Phi _p}(\mathbb {R}^n)=H^{1}(\mathbb {R}^n) +H_{W_p}^p({{{\mathbb {R}}}^n})\), where \(H^1(\mathbb {R}^n)\) denotes the classical real Hardy space, \(H^{\phi _0}({{{\mathbb {R}}}^n})\) the Orlicz–Hardy space associated with the Orlicz function \(\phi _0(t):=t/\log (e+t)\) for any \(t\in [0,\infty )\), and \(H_{W_p}^p(\mathbb {R}^n)\) the weighted Hardy space associated with certain weight function \(W_p(x)\) that is comparable to \(\Phi _p(x,1)\) for any \(x\in \mathbb {R}^n\). As an application, the authors further establish an interpolation theorem of quasilinear operators based on this new structure.

     

Waist and trunk of knots

Let K be a knot in the 3-sphere S ³. We define the waist of K as

$waist (K) = \mathop{\rm max}\limits_{F\in\mathcal{F}} \mathop{\rm min}\limits_{D\in\mathcal{D}_{F}} |D \cap K|,$

where \({\mathcal{F}}\) is the set of all closed incompressible surfaces in S ³?K and \({\mathcal{D}_F}\) is the set of all compressing disks for F in S ³. We define the trunk of K as

$trunk(K) = \mathop{\rm min}\limits_{h\in\mathcal{H}} \mathop{\rm max}\limits_{t\in\mathbb{R}} |h^{-1}(t) \cap K|,$

where \({\mathcal{H}}\) is the set of all Morse function \({h : S^3 \to \mathbb{R}}\) with two critical points. We show that

$waist (K) \le \frac{trunk(K)}{3}$

.