Sharp nonexistence results of prescribed L 2-norm solutions for some class of Schrödinger–Poisson and quasi-linear equations

In this paper, we study the existence of minimizers for $$F(u) = \frac{1}{2} \int_{\mathbb{R}^3} |\nabla u|^{2} {\rm d}x + \frac{1}{4} \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \frac{| u(x)|^2 | u(y)|^2}{| x-y|} {\rm d}x{\rm d}y-\frac{1}{p} \int_{\mathbb{R}^3}|u|^p {\rm d}x$$ on the constraint $$S(c) = \{u \in H^1(\mathbb{R}^3) : \int_{\mathbb{R}^3}|u|^2 {\rm d}x = c\}$$ , where c >  0 is a given parameter. In the range ${p \in [3,\frac{10}{3}]}$ , we explicit a threshold value of c >  0 separating existence and nonexistence of minimizers. We also derive a nonexistence result of critical points of F(u) restricted to S(c) when c >  0 is sufficiently small. Finally, as a by-product of our approaches, we extend some results of Colin et al. (Nonlinearity 23(6):1353–1385, 2010) where a constrained minimization problem, associated with a quasi-linear equation, is considered.     

Extensions and their Minimizations on the Sierpinski Gasket

We study the extension problem on the Sierpinski Gasket (SG). In the first part we consider minimizing the functional \(\mathcal {E}_{\lambda }(f) = \mathcal {E}(f,f) + \lambda \int f^{2} d \mu \) with prescribed values at a finite set of points where \(\mathcal {E}\) denotes the energy (the analog of \(\int |\nabla f|^{2}\) in Euclidean space) and μ denotes the standard self-similiar measure on SG. We explicitly construct the minimizer \(f(x) = \sum _{i} c_{i} G_{\lambda }(x_{i}, x)\) for some constants c _i , where G _λ is the resolvent for λ≥0. We minimize the energy over sets in SG by calculating the explicit quadratic form \(\mathcal {E}(f)\) of the minimizer f. We consider properties of this quadratic form for arbitrary sets and then analyze some specific sets. One such set we consider is the bottom row of a graph approximation of SG. We describe both the quadratic form and a discretized form in terms of Haar functions which corresponds to the continuous result established in a previous paper. In the second part, we study a similar problem this time minimizing \(\int _{SG} |\Delta f(x)|^{2} d \mu (x)\) for general measures. In both cases, by using standard methods we show the existence and uniqueness to the minimization problem. We then study properties of the unique minimizers.     

A new approach to the description of one-parameter groups of formal power series in one indeterminate

The aim of the paper is to describe one-parameter groups of formal power series, that is to find a general form of all homomorphisms \({\Theta_G : G \to \Gamma}\) , \({\Theta_G(t) = \sum_{k=1}^{\infty} c_k(t)X^k}\) , \({c_1 : G \to \mathbb{K} \setminus\{0\}}\) , \({c_k : G \to \mathbb{K}}\) for k ≥ 2, from a commutative group (G, + ) into the group \({(\Gamma, \circ)}\) of invertible formal power series with coefficients in \({\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}}\) . Considering one-parameter groups of formal power series and one-parameter groups of truncated formal power series, we give explicit formulas for the coefficient functions c _k with more details in the case where either c ₁ = 1 or c ₁ takes infinitely many values. Here we give the results much more simply than they were presented in Jab?oński and Reich (Abh. Math. Sem. Univ. Hamburg 75:179–201, 2005; Result Math 47:61–68, 2005; Publ Math Debrecen 73(1–2):25–47, 2008). Also the case im c ₁ = E _m (here E _m stands for the group of all complex roots of order m of 1), not considered in Jab?oński and Reich (Abh. Math. Sem. Univ. Hamburg 75:179–201, 2005; Result Math 47:61–68, 2005; Publ Math Debrecen 73(1–2):25–47, 2008), will be discussed.     

Integral representations for some classes of functions holomorphic in a strip or in a half-plane

В данной работе рассм атриваются классы фу нкцийf(z), голоморфные в област иa (?∞<a<b≦+∞) приp≧1 иs≧0, и у довлетворяющие одному из следующих условий:

1. Еслиb≦+∞, то $$\int\limits_a^b {(\int\limits_{ - \infty }^{ + \infty } {\left| {f\left( {x + iy} \right)} \right|^p } dy)^s dx< + \infty .} $$
2. Еслиb=+∞, иa=0, то $$\int\limits_0^u {(\int\limits_{ - \infty }^{ + \infty } {\left| {f\left( {x + iy} \right)} \right|^p } dy)^s dx \leqq \varrho \left( u \right), u > 0,} $$ где?(u) — функция опред еленного роста.

Результаты работы су щественно обобщают т еорему Пэли—Винера о параме трическом представлений класс аH ₂ на полуплоскости.     

Extension of a theorem of Shi and Tam

In this note, we prove the following generalization of a theorem of Shi and Tam (J Differ Geom 62:79–125, 2002): Let (Ω, g) be an n-dimensional (n ≥ 3) compact Riemannian manifold, spin when n?>?7, with non-negative scalar curvature and mean convex boundary. If every boundary component Σ _i has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface ${{\hat \Sigma}_i \subset \mathbb{R}^n}$ , then $$ \int\limits_{\Sigma_i} H \ d \sigma \le \int\limits_{{\hat \Sigma}_i} \hat{H} \ d {\hat \sigma} $$ where H is the mean curvature of Σ _i in (Ω, g), ${\hat{H}}$ is the Euclidean mean curvature of ${{\hat \Sigma}_i}$ in ${\mathbb{R}^n}$ , and where d σ and ${d {\hat \sigma}}$ denote the respective volume forms. Moreover, equality holds for some boundary component Σ _i if, and only if, (Ω, g) is isometric to a domain in ${\mathbb{R}^n}$ . In the proof, we make use of a foliation of the exterior of the ${\hat \Sigma_i}$ ’s in ${\mathbb{R}^n}$ by the ${\frac{H}{R}}$ -flow studied by Gerhardt (J Differ Geom 32:299–314, 1990) and Urbas (Math Z 205(3):355–372, 1990). We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in Shi and Tam (J Differ Geom 62:79–125, 2002).     

L p -theory for second-order elliptic operators with unbounded coefficients

Second-order elliptic operators with unbounded coefficients of the form ${Au := -{\rm div}(a\nabla u) + F . \nabla u + Vu}$ in ${L^{p}(\mathbb{R}^{N}) (N \in \mathbb{N}, 1 < p < \infty)}$ are considered, which are the same as in recent papers Metafune et?al. (Z Anal Anwendungen 24:497–521, 2005), Arendt et?al. (J Operator Theory 55:185–211, 2006; J Math Anal Appl 338: 505–517, 2008) and Metafune et?al. (Forum Math 22:583–601, 2010). A new criterion for the m-accretivity and m-sectoriality of A in ${L^{p}(\mathbb{R}^{N})}$ is presented via a certain identity that behaves like a sesquilinear form over L ^p ×?L ^p'. It partially improves the results in (Metafune et?al. in Z Anal Anwendungen 24:497–521, 2005) and (Metafune et?al. in Forum Math 22:583–601, 2010) with a different approach. The result naturally extends Kato’s criterion in (Kato in Math Stud 55:253–266, 1981) for the nonnegative selfadjointness to the case of p ≠?2. The simplicity is illustrated with the typical example ${Au = -u\hspace{1pt}'' + x^{3}u\hspace{1pt}' + c |x|^{\gamma}u}$ in ${L^p(\mathbb{R})}$ which is dealt with in (Arendt et?al. in J Operator Theory 55:185–211, 2006; Arendt et?al. in J Math Anal Appl 338: 505–517, 2008).     

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m})$ . Our main results are:

• A general study of the relations between the Hopf–Lax semigroup and Hamilton–Jacobi equation in metric spaces (X,d).
• The equivalence of the heat flow in $L^{2}(X,\mathfrak {m})$ generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional $\mathrm {Ent}_{\mathfrak {m}}$ in the space of probability measures .
• The proof of density in energy of Lipschitz functions in the Sobolev space $W^{1,2}(X,\mathsf {d},\mathfrak {m})$ .
• A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper.

Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott and Villani (Ann. Math. 169:903–991, 2009) and Sturm (Acta Math. 196: 65–131, 2006, and Acta Math. 196:133–177, 2006) and require neither the doubling property nor the validity of the local Poincaré inequality.     

Some inequalities related to Sobolev norms

In this paper, we study some properties related to the new characterizations of Sobolev spaces introduced in Bourgain and Nguyen (C R Acad Sci, 343:75?C80, [2006]), Nguyen (J Funct Anal 237: 689?C720, [2006]; J Eur Math Soc 10:191?C229, [2008]). More precisely, we establish variants of the Poincaré inequality, the Sobolev inequality, and the Rellich?CKondrachov compactness theorem, where ${\int_{\mathbb{R}^N} |\nabla g|^p \;dx}$ is replaced by some quantity of the type $$I_{\delta} (g) =\mathop{\int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N}}_{|g(x) - g(y)| > \delta}\frac{\delta^p}{|x-y|^{N+p}}\, dx \, dy.$$      

The Idempotents of the TL n -Module {\otimes^{n}\mathbb{C}2} in Terms of Elements of {{\rm U}_{q} \mathfrak{sl}_2}

The vector space \({\otimes^{n}\mathbb{C}^2}\) upon which the XXZ Hamiltonian with n spins acts bears the structure of a module over both the Temperley–Lieb algebra \({{\rm TL}_{n}(\beta = q + q^{-1})}\) and the quantum algebra \({{\rm U}_{q} \mathfrak{sl}_2}\) . The decomposition of \({\otimes^{n}\mathbb{C}^2}\) as a \({{\rm U}_{q} \mathfrak{sl}_2}\) -module was first described by Rosso (Commun Math Phys 117:581–593, 1988), Lusztig (Cont Math 82:58–77, 1989) and Pasquier and Saleur (Nucl Phys B 330:523–556, 1990) and that as a TL _n -module by Martin (Int J Mod Phys A 7:645–673, 1992) (see also Read and Saleur Nucl Phys B 777(3):316–351, 2007; Gainutdinov and Vasseur Nucl Phys B 868:223–270, 2013). For q generic, i.e. not a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of \({\otimes^{n}\mathbb{C}^2}\) ) onto each of these irreducible modules as linear combinations of elements of \({{\rm U}_{q} \mathfrak{sl}_2}\) . When q = q _c is a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) (with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involves some new generators, whose action on \({\otimes^{n}\mathbb{C}^2}\) is that of the divided powers \({(S^{\pm})^{(r)} = \lim_{q \rightarrow q_{c}} (S^{\pm})^r/[r]!}\) .     

3-Nets realizing a group in a projective plane

In a projective plane $\mathit{PG}(2,\mathbb{K})$ defined over an algebraically closed field $\mathbb{K}$ of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614–1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672–688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzúa’s 3-nets (Adv. Geom. 10:287–310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds in characteristic p>0 apart from three possible exceptions $\rm{Alt}_{4}$ , $\rm{Sym}_{4}$ , and $\rm{Alt}_{5}$ . Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069–1088, 2007; Miguel and Buzunáriz in Graphs Comb. 25:469–488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672–688, 2008; Yuzvinsky in Compos. Math. 140:1614–1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641–1648, 2009).     

Some Recent Developments on Hopf��s Holomorphic Form

Hopf??s theorem on surfaces in ${\mathbb{R}^3}$ with constant mean curvature (Hopf in Math Nach 4:232?C249, 1950-51) was a turning point in the study of such surfaces. In recent years, Hopf-type theorems appeared in various ambient spaces, (Abresch and Rosenberg in Acta Math 193:141?C174, 2004 and Abresch and Rosenberg in Mat Contemp Sociedade Bras Mat 28:283-298, 2005). The simplest case is the study of surfaces with parallel mean curvature vector in ${M_k^n \times \mathbb{R}, n \ge 2}$ , where ${M_k^n}$ is a complete, simply-connected Riemannian manifold with constant sectional curvature k ?? 0. The case n?=?2 was solved in Abresch and Rosenberg 2004. Here we describe some new results for arbitrary n.     

Continuous horizontally rigid functions of two variables are affine

Cain, Clark and Rose defined a function ${{f\colon\mathbb{R}^n \to \mathbb{R}}}$ to be vertically rigid if graph(cf) is isometric to graph(f) for every c ≠?0. It is horizontally rigid if graph ${(f(c \vec{x}))}$ is isometric to graph(f) for every c ≠?0 (see Cain et?al., Real Anal. Exch. 31:515–518, 2005/2006). In Balka and Elekes (J. Math. Anal. Appl. 345:880–888, 2008) the authors of the present paper settled Jankovi?’s conjecture by showing that a continuous function of one variable is vertically rigid if and only if it is of the form a?+?bx or ${{a+be^{kx} (a,b,k \in \mathbb{R})}}$ . Later they proved in Balka and Elekes (Real. Anal. Exch. 35:139–156, 2009) that a continuous function of two variables is vertically rigid if and only if after a suitable rotation around the z-axis it is of the form ${a + bx + dy, a +s(y)e^{kx}}$ or ${{a + be^{kx} + dy (a,b,d,k \in \mathbb{R}, k \neq 0, s\colon \mathbb{R}\to \mathbb{R}\,{\rm is\, continuous})}}$ . The problem remained open in higher dimensions. The characterization in the case of horizontal rigidity is surprisingly simpler. Richter (Real Anal. Exch. 35:343–354, 2009) proved that a continuous function of one variable is horizontally rigid if and only if it is of the form ${{a+bx (a,b\in \mathbb{R})}}$ . The goal of the present paper is to prove that a continuous function of two variables is horizontally rigid if and only if it is of the form ${{a+ bx + dy (a,b,d \in \mathbb{R})}}$ . This problem also remains open in higher dimensions. The main new ingredient of the present paper is the use of functional equations.     

Characterizations of Strength Extremal Graphs

With graphs considered as natural models for many network design problems, edge connectivity κ′(G) and maximum number of edge-disjoint spanning trees τ(G) of a graph G have been used as measures for reliability and strength in communication networks modeled as graph G (see Cunningham, in J ACM 32:549–561, 1985; Matula, in Proceedings of 28th Symposium Foundations of Computer Science, pp 249–251, 1987, among others). Mader (Math Ann 191:21–28, 1971) and Matula (J Appl Math 22:459–480, 1972) introduced the maximum subgraph edge connectivity \({\overline{\kappa'}(G) = {\rm max} \{\kappa'(H) : H {\rm is} \, {\rm a} \, {\rm subgraph} \, {\rm of} G \}}\) . Motivated by their applications in network design and by the established inequalities $$\overline{\kappa'}(G) \ge \kappa'(G) \ge \tau(G),$$ we present the following in this paper:

1. For each integer k > 0, a characterization for graphs G with the property that \({\overline{\kappa'}(G) \le k}\) but for any edge e not in G, \({\overline{\kappa'}(G + e) \ge k+1}\) .
2. For any integer n > 0, a characterization for graphs G with |V(G)| = n such that κ′(G) = τ(G) with |E(G)| minimized.

     

A tableau formula for eta polynomials

We use the Pieri and Giambelli formulas of Buch et al. (Invent Math 178:345–405, 2009; J Reine Angew, 2013) and the calculus of raising operators developed in Buch et al. (A Giambelli formula for isotropic Grassmannians, arXiv:0811.2781, 2008) and Tamvakis (J Reine Angew Math 652, 207–244, 2011) to prove a tableau formula for the eta polynomials of Buch et al. (J Reine Angew, 2013) and the Stanley symmetric functions which correspond to Grassmannian elements of the Weyl group $\widetilde{W}_n$ of type $\text {D}_n$ . We define the skew elements of $\widetilde{W}_n$ and exhibit a bijection between the set of reduced words for any skew $w\in \widetilde{W}_n$ and a set of certain standard typed tableaux on a skew shape $\lambda /\mu $ associated to $w$ .     

On the limit superior of sequences of sets

qVЕРхНИИ пРЕДЕл пОслЕД ОВАтЕльНОстИ МНОжЕс тВA _n ОпРЕДЕльЕтсь сООтНО шЕНИЕМ \(\mathop {\lim sup}\limits_{n \to \infty } A_n = \mathop \cap \limits_{k = 1}^\infty \mathop \cup \limits_{n = k}^\infty A_n . B\) стАтьЕ РАссМАтРИВА Етсь слЕДУУЩИИ ВОпРО с: ЧтО МОжНО скАжАть О ВЕРхНИх пРЕДЕлАх \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) , еслИ ИжВЕстНО, ЧтО пРЕсЕЧЕНИь \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) «МАлы» Дль кАж-ДОИ пОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) ? ДОкАжыВАЕтсь, Ч тО

1. ЕслИ \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) — кОНЕЧНОЕ МНО жЕстВО Дль кАжДОИ пОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) , тО НАИДЕтсь тАкАь пОДпО слЕДОВАтЕльНОсть, Дл ь кОтОРОИ МНОжЕстВО \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) сЧЕтНО;
2. ЕслИ \(2^{\aleph _0 } = \aleph _1\) , тО сУЩЕстВУЕ т тАкАь пОслЕДОВАтЕл ьНОсть (A_n), ЧтО \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) — сЧЕтНОЕ МНОжЕстВО Дль лУБОИ п ОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) , НО \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) ИМЕЕт МОЩ-НОсть кОНтИНУУМА;
3. ЕслИA _n — БОРЕлЕ ВскИЕ МНОжЕстВА В НЕкОтОРО М пОлНОМ сЕпАРАБЕльНО М МЕтРИЧЕскОМ пРОстРАНстВЕ, И \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) — сЧЕт НОЕ МНОжЕстВО Дль кАж ДОИ пОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) , тО сУЩЕстВУЕт тАкАь п ОДпОслЕДОВАтЕльНОсть, ЧтО \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) — сЧЕтНОЕ МНОжЕстВО. кРОМЕ тОгО, ДОкАжАНО, Ч тО В слУЧАьх А) И В) В пОслЕДОВАтЕльНОстИ (A _n ) сУЩЕстВУЕт схОДьЩА ьсь пОДпОслЕДОВАтЕльНО сть.

кРОМЕ тОгО, ДОкАжАНО, Ч тО В слУЧАьх А) И В) В пОслЕДОВАтЕльНОстИ (А _n ) сУЩЕстВУЕт схОДьЩ Аьсь пОДпОслЕДОВАтЕльНО сть.     

An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

Let $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ be the empirical process associated to an ? ^d -valued stationary process (X _i ) _i≥0. In the present paper, we introduce very general conditions for weak convergence of $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ , which only involve properties of processes (f(X _i )) _i≥0 for a restricted class of functions $f\in\mathcal{G}$ . Our results significantly improve those of Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011) and provide new applications. The central interest in our approach is that it does not need the indicator functions which define the empirical process $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ to belong to the class  $\mathcal{G}$ . This is particularly useful when dealing with data arising from dynamical systems or functionals of Markov chains. In the proofs we make use of a new application of a chaining argument and generalize ideas first introduced in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011). Finally we will show how our general conditions apply in the case of multiple mixing processes of polynomial decrease and causal functions of independent and identically distributed processes, which could not be treated by the preceding results in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011).     

Generating sets of finite singular transformation semigroups

J.M. Howie proved that $\operatorname {Sing}_{n}$ , the semigroup of all singular mappings of {1,…,n} into itself, is generated by its idempotents of defect 1 (in J. London Math. Soc. 41, 707–716, 1966). He also proved that if n≥3 then a minimal generating set for $\operatorname {Sing}_{n}$ contains n(n?1)/2 transformations of defect 1 (in Gomes and Howie, Math. Proc. Camb. Philos. Soc. 101. 395–403, 1987). In this paper we find necessary and sufficient conditions for any set for transformations of defect 1 in $\operatorname {Sing}_{n}$ to be a (minimal) generating set for $\operatorname {Sing}_{n}$ .     

Changing-sign bubble solutions for an anisotropic sinh-Poisson equation

We consider the following anisotropic sinh-Poisson equation $${\rm div} (a(x) \nabla u)+ 2\varepsilon^2 a(x) {\rm sinh}\,u=0\ \ {\rm in}\ \Omega, \quad u=0 \ \ {\rm on}\ \partial \Omega,$$ where ${\Omega \subset \mathbb{R}^2}$ is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient ${a(x)}$ on the existence of bubbling solutions. We show that there exists a family of solutions u _?? concentrating positively and negatively at ${\bar{x}}$ , a given local critical point of a(x), for ?? sufficiently small, for which with the property $$2\varepsilon^2a(x){\rm sinh} u_\varepsilon \rightharpoonup 8\pi\sum\limits_{j=1}^{m}b_j\delta_{\bar{x}},$$ where ${b_j=\pm 1}$ . This result shows a striking difference with the isotropic case (a(x) ?? Constant) in Bartolucci and Pistoia (IMA J Appl Math 72(6):706?C729, 2007), Jost et?al. (Calc Var Partial Differ Equ 31:263?C276, 2008) and Esposito and Wei (Calc Var Partial Differ Equ 34:341?C375, 2009).     

On the Left and Right Brylinski-Kostant Filtrations

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ a Borel subalgebra, and $\mathfrak{h}\subset\mathfrak{b}$ a Cartan subalgebra. Let V be a finite dimensional simple $U(\mathfrak{g})$ module. Based on a principal s-triple (e,h,f) and following work of Kostant, Brylinski (J Amer Math Soc 2(3):517–533, 1989) defined a filtration $\mathcal{F}_e$ for all weight subspaces V _μ of V and calculated the dimensions of the graded subspaces for μ dominant. In Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) these dimensions were calculated for all μ. Let δM(0) be the finite dual of the Verma module of highest weight 0. It identifies with the global functions on a Weyl group translate of the open Bruhat cell and so inherits a natural degree filtration. On the other hand, up to an appropriate shift of weights, there is a unique $U(\mathfrak{b})$ module embedding of V into δM(0) and so the degree filtration induces a further filtration $\mathcal{F}$ on each weight subspace V _μ . A casual reading of Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) might lead one to believe that $\mathcal{F}$ and $\mathcal{F}_e$ coincide. However this is quite false. Rather one should view $\mathcal{F}_e$ as coming from a left action of $U(\mathfrak{n})$ and then there is a second (Brylinski-Kostant) filtration $\mathcal{F}'_e$ coming from a right action. It is $\mathcal{F}'_e$ which may coincide with $\mathcal{F}$ . In this paper the above claim is made precise. First it is noted that $\mathcal{F}$ is itself not canonical, but depends on a choice of variables. Then it is shown that a particular choice can be made to ensure that $\mathcal{F}=\mathcal{F}'_e$ . An explicit form for the unique left $U(\mathfrak{b})$ module embedding $V\hookrightarrow\delta M(0)$ is given using the right action of $U(\mathfrak{n})$ . This is used to give a purely algebraic proof of Brylinski’s main result in Brylinski (J Amer Math Soc 2(3):517–533, 1989) which is much simpler than Joseph et al. (J Amer Math Soc 13(4):945–970, 2000). It is noted that the dimensions of the graded subspaces of $\rm{gr}_{\mathcal{F}_e} V_{\!\mu}$ and $\rm{gr}_{\mathcal{F}'_e} V_{\!\mu}$ can be very different. Their interrelation may involve the Kashiwara involution. Indeed a combinatorial formula for multiplicities in tensor products involving crystal bases and the Kashiwara involution is recovered. Though the dimensions for the graded subspaces of $\rm{gr}_{\mathcal{F}'_e} V_{\!\mu}$ are determined by polynomial degree, their values remain unknown.