Boundedness of Generalized Riesz Transforms on Orlicz–Hardy Spaces Associated to Operators

Let ${\Phi}$ be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index ${p_\Phi^- \in(0,\,1]}$ . Let L be an injective operator of type ω having a bounded H _∞ functional calculus and satisfying the k-Davies–Gaffney estimates with ${k \in {\mathbb Z}_+}$ . In this paper, the authors first introduce an Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform ${D_{\gamma}L^{-\delta/(2k)}}$ is bounded from the Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz space ${L^{\widetilde{\Phi}}(\mathbb{R}^n)}$ when ${p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}$ , ${0 < \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz–Hardy space ${H^{\widetilde \Phi}(\mathbb{R}^n)}$ when ${p_\Phi^-\in (\frac{n}{n + \delta+ \lfloor \gamma \rfloor- \gamma},\,\frac{n}{n+ \delta- \gamma}]}$ , ${1\le \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the weak Orlicz–Hardy space ${WH^\Phi(\mathbb{R}^n)}$ when ${\gamma = \delta}$ and ${p_\Phi=n/(n + \lfloor \gamma \rfloor)}$ or ${p_\Phi^-=n/(n + \lfloor \gamma \rfloor)}$ with ${p_\Phi^-}$ attainable, where ${\widetilde{\Phi}}$ is an Orlicz function whose inverse function ${\widetilde{\Phi}^{-1}}$ is defined by ${\widetilde{\Phi}^{-1}(t):=\Phi^{-1}(t)t^{\frac{1}{n}(\gamma- \delta)}}$ for all ${t \in (0,\,\infty)}$ , ${p_\Phi}$ denotes the strictly critical lower type index of ${\Phi}$ , ${\lfloor \gamma \rfloor}$ the maximal integer not more than ${\gamma}$ and ${(p_-(L),\,p_+(L))}$ the range of exponents ${p \in[1,\, \infty]}$ for which the semigroup ${\{e^{-tL}\}_{t >0 }}$ is bounded on ${L^p(\mathbb{R}^n)}$ .     

Local behaviour of singular solutions for nonlinear elliptic equations in divergence form

We consider the following class of nonlinear elliptic equations $$\begin{array}{ll}{-}{\rm div}(\mathcal{A}(|x|)\nabla u) +u^q=0\quad {\rm in}\; B_1(0)\setminus\{0\}, \end{array}$$ where q > 1 and ${\mathcal{A}}$ is a positive C ¹(0,1] function which is regularly varying at zero with index ${\vartheta}$ in (2?N,2). We prove that all isolated singularities at zero for the positive solutions are removable if and only if ${\Phi\not\in L^q(B_1(0))}$ , where ${\Phi}$ denotes the fundamental solution of ${-{\rm div}(\mathcal{A}(|x|)\nabla u)=\delta_0}$ in ${\mathcal D'(B_1(0))}$ and δ₀ is the Dirac mass at 0. Moreover, we give a complete classification of the behaviour near zero of all positive solutions in the more delicate case that ${\Phi\in L^q(B_1(0))}$ . We also establish the existence of positive solutions in all the categories of such a classification. Our results apply in particular to the model case ${\mathcal{A}(|x|)=|x|^\vartheta}$ with ${\vartheta\in (2-N,2)}$ .     

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real ${L_\infty(\mathbb{R})}$ -functions ${\Phi}$ and ${\Psi}$ of compact support, we prove the norm resolvent convergence, as ${\varepsilon}$ and ${\nu}$ tend to 0, of a family ${S_{\varepsilon \nu}}$ of one-dimensional Schrödinger operators on the line of the form $$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$ provided the ratio ${\nu/\varepsilon}$ has a finite or infinite limit. The limit operator S ₀ depends on the shape of ${\Phi}$ and ${\Psi}$ as well as on the limit of ratio ${\nu/\varepsilon}$ . If the potential ${\alpha\Phi}$ possesses a zero-energy resonance, then S ₀ describes a non trivial point interaction at the origin. Otherwise S ₀ is the direct sum of the Dirichlet half-line Schrödinger operators.     

Uniqueness of integrable solutions to {\nabla \zeta=G \zeta, \zeta|_\Gamma = 0} for integrable tensor coefficients G and applications to elasticity

Let ${\Omega \subset \mathbb{R}^{N}}$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ${\partial\Omega}$ . We show that the solution to the linear first-order system $$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$ is unique if ${G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}$ and ${\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}$ . As a consequence, we prove $$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$ to be a norm for ${P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}$ with Curl ${P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}$ , Curl ${P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}$ for some p, q > 1 with 1/p + 1/q = 1 as well as det ${P \geq c^+ > 0}$ . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ${\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}$ satisfy sym ${(\nabla\Phi^\top\nabla\Psi) = 0}$ for some ${\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}$ with det ${\nabla\Psi \geq c^+ > 0}$ . Then, there exist a constant translation vector ${a \in \mathbb{R}^{3}}$ and a constant skew-symmetric matrix ${A \in \mathfrak{so}(3)}$ , such that ${\Phi = A\Psi + a}$ .     

On uniformly bounded spherical functions in Hilbert space

Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms ${G \ni a \mapsto ka \in G}$ , ${k \in K}$ . Let ${{\mathfrak{H}}}$ be a complex Hilbert space and let ${{\mathcal L}({\mathfrak{H}})}$ be the algebra of all bounded linear operators on ${{\mathfrak{H}}}$ . A mapping ${u \colon G \to {\mathcal L}({\mathfrak{H}})}$ is termed a K-spherical function if it satisfies (1) ${|K|^{-1} \sum_{k\in K} u (a+kb)=u (a) u (b)}$ for any ${a,b\in G}$ , where |K| denotes the cardinality of K, and (2) ${u (0) = {\rm id}_{\mathfrak {H}},}$ where ${{\rm id}_{\mathfrak {H}}}$ designates the identity operator on ${{\mathfrak{H}}}$ . The main result of the paper is that for each K-spherical function ${u \colon G \to {\mathcal {L}}({\mathfrak {H}})}$ such that ${\| u \|_{\infty} = \sup_{a\in G} \| u (a)\|_{{\mathcal L}({\mathfrak{H}})} < \infty,}$ there is an invertible operator S in ${{\mathcal L}({\mathfrak{H}})}$ with ${\| S \| \, \| S^{-1}\| \leq |K| \, \| u \|_{\infty}^2}$ such that the K-spherical function ${{\tilde{u}} \colon G \to {\mathcal L}({\mathfrak{H}})}$ defined by ${{\tilde{u}}(a) = S u (a) S^{-1},\,a \in G,}$ satisfies ${{\tilde{u}}(-a) = {\tilde{u}}(a)^*}$ for each ${a \in G}$ . It is shown that this last condition is equivalent to insisting that ${{\tilde{u}}(a)}$ be normal for each ${a \in G}$ .     

On functions whose symmetric part of gradient agree and a generalization of Reshetnyak’s compactness theorem

We consider the following question: Given a connected open domain ${\Omega \subset \mathbb{R}^n}$ , suppose ${u, v : \Omega \rightarrow \mathbb{R}^n}$ with det ${(\nabla u) > 0}$ , det ${(\nabla v) > 0}$ a.e. are such that ${\nabla u^T(x)\nabla u(x) = \nabla v(x)^T \nabla v(x)}$ a.e. , does this imply a global relation of the form ${\nabla v(x) = R\nabla u(x)}$ a.e. in Ω where ${R \in SO(n)}$ ? If u, v are C ¹ it is an exercise to see this true, if ${u, v\in W^{1,1}}$ we show this is false. In Theorem 1 we prove this question has a positive answer if ${v \in W^{1,1}}$ and ${u \in W^{1,n}}$ is a mapping of L ^p integrable dilatation for p > n ? 1. These conditions are sharp in two dimensions and this result represents a generalization of the corollary to Liouville’s theorem that states that the differential inclusion ${\nabla u \in SO(n)}$ can only be satisfied by an affine mapping. Liouville’s corollary for rotations has been generalized by Reshetnyak who proved convergence of gradients to a fixed rotation for any weakly converging sequence ${v_k \in W^{1,1}}$ for which $$\int \limits_{\Omega} {\rm dist}(\nabla v_k, SO(n))dz \rightarrow 0 \, {\rm as} \, k \rightarrow \infty.$$ Let S(·) denote the (multiplicative) symmetric part of a matrix. In Theorem 3 we prove an analogous result to Theorem 1 for any pair of weakly converging sequences ${v_k \in W^{1,p}}$ and ${u_k \in W^{1,\frac{p(n-1)}{p-1}}}$ (where ${p \in [1, n]}$ and the sequence (u _k ) has its dilatation pointwise bounded above by an L ^r integrable function, r > n ? 1) that satisfy ${\int_{\Omega} |S(\nabla u_k) - S(\nabla v_k)|^p dz \rightarrow 0}$ as k → ∞ and for which the sign of the det ${(\nabla v_k)}$ tends to 1 in L ¹. This result contains Reshetnyak’s theorem as the special case (u _k ) ≡ Id, p = 1.     

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

Regularity of the Monge–Ampère equation in Besov’s spaces

Let $\mu = e^{-V} \ dx$ be a probability measure and $T = \nabla \Phi $ be the optimal transportation mapping pushing forward $\mu $ onto a log-concave compactly supported measure $\nu = e^{-W} \ dx$ . In this paper, we introduce a new approach to the regularity problem for the corresponding Monge–Ampère equation $e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi )}$ in the Besov spaces $W^{\gamma ,1}_{loc}$ . We prove that $D^2 \Phi \in W^{\gamma ,1}_{loc}$ provided $e^{-V}$ belongs to a proper Besov class and $W$ is convex. In particular, $D^2 \Phi \in L^p_{loc}$ for some $p>1$ . Our proof does not rely on the previously known regularity results.     

BIB-Designs from Circular Nearrings

Let ${(N, \Phi)}$ be a finite circular Ferrero pair. We define the disk with center b and radius ${a, \mathcal{D}(a;b)}$ , as $$\mathcal{D} (a; b) = \{x \in \Phi(r)+c \mid r \neq 0, b\in \Phi (r)+c, |(\Phi (r)+c) \cap ( \Phi(a)+b)|=1\}.$$ Using this definition we introduce the concept of interior part of a circle, ${\Phi(a)+b}$ , as the set ${\mathcal{I}(\Phi (a)+b)=\mathcal{D} (a; b) \setminus (\Phi (a)+b)}$ . Moreover, if ${\mathcal{B}^{\mathcal{D}}}$ is the set of all disks, then, in some interesting cases, we show that the incidence structure ${(N, \mathcal{B}^{\mathcal{D}}, \in)}$ is actually a balanced incomplete block design and we are able to calculate its parameters depending on |N| and ${|\Phi|}$ .     

Admissible closures of polynomial time computable arithmetic

We propose two admissible closures ${\mathbb{A}({\sf PTCA})}$ and ${\mathbb{A}({\sf PHCA})}$ of Ferreira??s system PTCA of polynomial time computable arithmetic and of full bounded arithmetic (or polynomial hierarchy computable arithmetic) PHCA. The main results obtained are: (i) ${\mathbb{A}({\sf PTCA})}$ is conservative over PTCA with respect to ${\forall\exists\Sigma^b_1}$ sentences, and (ii) ${\mathbb{A}({\sf PHCA})}$ is conservative over full bounded arithmetic PHCA for ${\forall\exists\Sigma^b_{\infty}}$ sentences. This yields that (i) the ${\Sigma^b_1}$ definable functions of ${\mathbb{A}({\sf PTCA})}$ are the polytime functions, and (ii) the ${\Sigma^b_{\infty}}$ definable functions of ${\mathbb{A}({\sf PHCA})}$ are the functions in the polynomial time hierarchy.     

Young measures generated by sequences in Morrey spaces

Let ${\Omega\subset\mathbb{R}^n}$ be open and bounded. For 1 ≤ p < ∞ and 0 ≤ λ < n, we give a characterization of Young measures generated by sequences of functions ${\{{\bf f}_j\}_{j=1}^\infty}$ uniformly bounded in the Morrey space ${L^{p,\lambda}(\Omega;\mathbb{R}^N)}$ with ${\{\left|{{\bf f}_j}\right|^p\}_{j=1}^\infty}$ equiintegrable. We then treat the case that each f _j = ? u _j for some ${{\bf u}_j\in W^{1,p}(\Omega;\mathbb{R}^N)}$ . As an application of our results, we consider the functional $${\bf u} \mapsto \int\limits_{\Omega}f({\bf x}, {\bf u}({\bf x}), {\bf {\nabla}}{\bf u}({\bf x})){\rm d}{\bf x},$$ and provide conditions that guarantee the existence of a minimizing sequence with gradients uniformly bounded in ${L^{p,\lambda}(\Omega;\mathbb{R}^{N\times n})}$ .     

A maximum modulus theorem for the Oseen problem

Classical solutions of the Oseen problem are studied on an exterior domain Ω with Ljapunov boundary in R ³. It is proved a maximum modulus estimate of the following form: If ${{\bf u}\in C^2(\Omega)^3\cap C^0(\overline \Omega)^3}$ and ${p \in C^1(\Omega ), -\Delta {\bf u}+2\lambda \partial_1 {\bf u}+\nabla p=0, \nabla \cdot {\bf u}=0}$ in Ω, and if ${|{\bf u}| \le M}$ on ${\partial \Omega , \limsup |{\bf u}({\bf x})|\le M}$ as ${|{\bf x}|\to \infty }$ , then ${|{\bf u}({\bf x})|\le c M}$ in Ω. Here the constant c depends only on Ω and λ.     

{\mathcal{C}^{\alpha}}-regularity for a class of non-diagonal elliptic systems with p-growth

We consider weak solutions to nonlinear elliptic systems in a W ^1,p -setting which arise as Euler equations to certain variational problems. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the dependent and independent variables. We impose new structure conditions on the coefficients which yield everywhere ${\mathcal{C}^{\alpha}}$ -regularity and global ${\mathcal{C}^{\alpha}}$ -estimates for the solutions. These structure conditions cover variational integrals like ${\int F(\nabla u)\; dx}$ with potential ${F(\nabla u):=\tilde F (Q_1(\nabla u),\ldots, Q_N(\nabla u))}$ and positively definite quadratic forms in ${\nabla u}$ defined as ${Q_i(\nabla u)=\sum_{\alpha \beta} a_i^{\alpha \beta} \nabla u^\alpha \cdot \nabla u^\beta}$ . A simple example consists in ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{2}} + |\xi_2|^{\frac{p}{2}}}$ or ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{4}}|\xi_2|^{\frac{p}{4}}}$ . Since the Q _i need not to be linearly dependent our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. The proof uses a new weighted norm technique with singular weights in an L ^p -setting.     

Continuity of solutions to n-harmonic equations

In this paper, we study the nonhomogeneous n-harmonic equation $$-{\rm div}\,(|{\nabla} u|^{n-2}{\nabla} u)=f$$ in domains ${\Omega\subset {\mathbb {R}^n}}$ (n?≥?2), where ${f\in W^{-1,\frac{n}{n-1}}(\Omega)}$ . We derive a sharp condition to guarantee the continuity of solutions u. In particular, we show that when n?≥ 3, the condition that, for some ${\epsilon >0 ,}$ f belongs to $${\mathfrak{L}}({\rm log}\,{\mathfrak{L}})^{n-1}({\rm log}\,{\rm log}\,{\mathfrak{L}})^{n-2}\cdots({\rm log}\cdots{\rm log}\,{\mathfrak{L}})^{n-2}({\rm log}\cdots{\rm log}\,{\mathfrak{L}})^{n-2+\epsilon}(\Omega)$$ is sufficient for continuity of u, but not for ${\epsilon=0}$ .     

An Index Formula in Connection with Meromorphic Approximation

Let $\Phi $ be a continuous $n\times n$ matrix-valued function on the unit circle $\mathbb T $ such that the $(k-1)$ st singular value of the Hankel operator with symbol $\Phi $ is greater than the $k$ th singular value. In this case, it is well-known that $\Phi $ has a unique superoptimal meromorphic approximant $Q$ in $H^{\infty }_{(k)}$ ; that is, $Q$ has at most $k$ poles in the unit disc $\mathbb D $ (in the sense that the McMillan degree of $Q$ in $\mathbb D $ is at most $k$ ) and $Q$ minimizes the essential suprema of singular values $s_{j}\left((\Phi -Q)(\zeta )\right)\!, j\ge 0$ , with respect to the lexicographic ordering. For each $j\ge 0$ , the essential supremum of $s_{j}\left((\Phi -Q)(\zeta )\right)$ is called the $j$ th superoptimal singular value of degree $k$ of $\Phi $ . We prove that if $\Phi $ has $n$ non-zero superoptimal singular values of degree $k$ , then the Toeplitz operator $T_{\Phi -Q}$ with symbol $\Phi -Q$ is Fredholm and has index $$ \mathrm{ind}T_{\Phi -Q}=\dim \ker T_{\Phi -Q}=2k+\dim \mathcal E , $$ where $\mathcal E =\{ \xi \in \ker H_{Q}: \Vert H_{\Phi }\xi \Vert _{2}=\Vert (\Phi -Q)\xi \Vert _{2}\}$ and $H_{\Phi }$ denotes the Hankel operator with symbol $\Phi $ . This result can in fact be extended from continuous matrix-valued functions to the wider class of $k$ -admissible matrix-valued functions, i.e. essentially bounded $n\times n$ matrix-valued functions $\Phi $ on $\mathbb T $ for which the essential norm of the Hankel operator $H_{\Phi }$ is strictly less than the smallest non-zero superoptimal singular value of degree $k$ of $\Phi $ .     

Antibasis theorems for {\Pi^0_1} classes and the jump hierarchy

We prove two antibasis theorems for ${\Pi^0_1}$ classes. The first is a jump inversion theorem for ${\Pi^0_1}$ classes with respect to the global structure of the Turing degrees. For any ${P\subseteq 2^\omega}$ , define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists ${A \in P}$ of degree a. For any degree ${{\bf a \geq 0'}}$ , let ${\textrm{Jump}^{-1}({\bf a) = \{b : b' = a \}}}$ . We prove that, for any ${{\bf a \geq 0'}}$ and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}^{-1} ({\bf a}) \subseteq S(P)}$ then P contains a member of every degree. For any degree ${{\bf a \geq 0'}}$ such that a is recursively enumerable (r.e.) in 0', let ${Jump_{\bf \leq 0'} ^{-1}({\bf a)=\{b : b \leq 0' \textrm{and} b' = a \}}}$ . The second theorem concerns the degrees below 0'. We prove that for any ${{\bf a\geq 0'}}$ which is recursively enumerable in 0' and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}_{\bf \leq 0'} ^{-1}({\bf a)} \subseteq S(P)}$ then P contains a member of every degree.     

Yangians and quantum loop algebras

Let $\mathfrak{g }$ be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra $U_\hbar (L\mathfrak g )$ of $\mathfrak{g }$ degenerates to the Yangian ${Y_\hbar (\mathfrak g )}$ . We strengthen this result by constructing an explicit algebra homomorphism $\Phi $ from $U_\hbar (L\mathfrak g )$ to the completion of ${Y_\hbar (\mathfrak g )}$ with respect to its grading. We show moreover that $\Phi $ becomes an isomorphism when ${U_\hbar (L\mathfrak g )}$ is completed with respect to its evaluation ideal. We construct a similar homomorphism for $\mathfrak{g }=\mathfrak{gl }_n$ and show that it intertwines the actions of $U_\hbar (L\mathfrak gl _{n})$ and $Y_\hbar (\mathfrak gl _{n})$ on the equivariant $K$ -theory and cohomology of the variety of $n$ -step flags in ${\mathbb{C }}^d$ constructed by Ginzburg–Vasserot.     

Beyond the Trudinger-Moser supremum

Let Ω be a bounded, smooth domain in ${\mathbb{R}^2}$ . We consider the functional $$I(u) = \int_\Omega e^{u^2}\,dx$$ in the supercritical Trudinger-Moser regime, i.e. for ${\int_\Omega |\nabla u|^2dx > 4\pi}$ . More precisely, we are looking for critical points of I(u) in the class of functions ${u \in H_0^1 (\Omega )}$ such that ${\int_\Omega |\nabla u|^2 \, dx = 4\, \pi \, k\, (1+\alpha)}$ , for small α > 0. In particular, we prove the existence of 1-peak critical points of I(u) with ${\int_\Omega |\nabla u|^2dx = 4\pi(1 + \alpha)}$ for any bounded domain Ω, 2-peak critical points with ${\int_\Omega |\nabla u|^2dx = 8\pi(1 + \alpha)}$ for non-simply connected domains Ω, and k-peak critical points with ${\int_\Omega |\nabla u|^2 dx = 4k \pi(1 + \alpha)}$ if Ω is an annulus.