From Ginzburg-Landau to Gross-Pitaevskii

(w, c) ? R², u ? L_loc³ (R^N, C)\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}(\omega, c)\in {\Op R}^2, {\upsilon} \in L_{\rm loc}^3 ({\Op R}^N, {\bf C}) and x||j||_{L^￥(R^N×R)}² ￡ max{0, 1-w+[(c²)/4]}.\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}\Vert\varphi\Vert_{L^\infty({\Op R}^N\times{\Op R})}^2 \le \max\bigg\{0, 1-\omega+{c^2\over 4}\bigg\}.     

Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$

We prove the existence of a global heat flow u : Ω ×  \mathbbR⁺ ? \mathbbR^N {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR⁺ {\mathbb{R}^{+}}) ⊂  \mathbbRⁿ {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbR^N {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbR^N {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.     

Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}

In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem

{l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      4. Spectral theorem for multipliers on {L_{\omega}^{2}(\mathbb{R})}      Violeta Petkova 《Archiv der Mathematik》2009,93(4):357-368 We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces Lw2(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with f ? Cc(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces L2w(\mathbb Rk){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form.      5. On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$      Christophe Dupont 《Mathematische Annalen》2011,349(3):509-528 Let f be an endomorphism of \mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} and ν be an f-invariant measure with positive Lyapunov exponents (λ 1, . . . , λ k ). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dimHm 3 [(log d)/(l1)] + [(log d)/(l2)]{{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of f n in \mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f n .      6. The L p -Solvability of the Dirichlet Problem for Planar Elliptic Equations, Sharp Results      Carlo Sbordone  Gabriella Zecca 《Journal of Fourier Analysis and Applications》2009,15(6):871-903 Assume that the elliptic operator L=div (A(x)∇) is L p -resolutive, p>1, on the unit disc \mathbbD ì \mathbb R2\mathbb{D}\subset \mathbb {R}^{2} . This means that the Dirichlet problem $\left\{{l@{\quad}l}Lu=0&\mbox{in }\mathbb{D},\\[3pt]u=g&\mbox{on }\partial\mathbb{D}\right.$\left\{\begin{array}{l@{\quad}l}Lu=0&\mbox{in }\mathbb{D},\\[3pt]u=g&\mbox{on }\partial\mathbb{D}\end{array}\right.      7. Singular oscillatory integrals on {\mathbb{R}^n}      M. Papadimitrakis  I. R. Parissis 《Mathematische Zeitschrift》2010,266(1):169-179 Let ${\mathcal{P}_{d,n}}Let Pd,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRn{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? Pd,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      8. Exotic smooth structures on small 4-manifolds with odd signatures      Anar Akhmedov  B. Doug Park 《Inventiones Mathematicae》2010,181(3):577-603 Let M be (2n-1)\mathbbCP2#2n[`(\mathbbCP)]2(2n-1)\mathbb{CP}^{2}\#2n\overline{\mathbb{CP}}{}^{2} for any integer n≥1. We construct an irreducible symplectic 4-manifold homeomorphic to M and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to M. We also construct such exotic smooth structures when M is \mathbbCP2#4[`(\mathbbCP)]2\mathbb{CP}{}^{2}\#4\overline {\mathbb{CP}}{}^{2} or 3\mathbbCP2#k[`(\mathbbCP)]23\mathbb{CP}{}^{2}\#k\overline{\mathbb{CP}}{}^{2} for k=6,8,10.      9. Fourier Series with the Continuous Primitive Integral      Erik Talvila 《Journal of Fourier Analysis and Applications》2012,18(1):27-44 Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted Ac(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}) and is a Banach space under the Alexiewicz norm, ||f||\mathbbT=sup|I| ￡ 2p|òI f|\|f\|_{\mathbb{T}}=\sup_{|I|\leq2\pi}|\int_{I} f|, the supremum being taken over intervals of length not exceeding 2π. It contains the periodic functions integrable in the sense of Lebesgue and Henstock–Kurzweil. Many of the properties of L 1 Fourier series continue to hold for this larger space, with the L 1 norm replaced by the Alexiewicz norm. The Riemann–Lebesgue lemma takes the form [^(f)](n)=o(n)\hat{f}(n)=o(n) as |n|→∞. The convolution is defined for f ? Ac(\mathbbT)f\in{\mathcal{A}}_{c}(\mathbb{T}) and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate ||f*g||￥ ￡ ||f||\mathbbT ||g||BV\|f\ast g\|_{\infty}\leq\|f\|_{\mathbb{T}} \|g\|_{\mathcal{BV}}. For g ? L1(\mathbbT)g\in L^{1}(\mathbb{T}), ||f*g||\mathbbT ￡ ||f||\mathbb T ||g||1\|f\ast g\|_{\mathbb{T}}\leq\|f\|_{\mathbb {T}} \|g\|_{1}. As well, [^(f*g)](n)=[^(f)](n) [^(g)](n)\widehat{f\ast g}(n)=\hat{f}(n) \hat{g}(n). There are versions of the Salem–Zygmund–Rudin–Cohen factorization theorem, Fejér’s lemma and the Parseval equality. The trigonometric polynomials are dense in Ac(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}). The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let D n be the Dirichlet kernel and let f ? L1(\mathbbT)f\in L^{1}(\mathbb{T}). Then ||Dn*f-f||\mathbbT?0\|D_{n}\ast f-f\|_{\mathbb{T}}\to0 as n→∞. Fourier coefficients of functions of bounded variation are characterized. The Appendix contains a type of Fubini theorem.      10. Period functions for $ {\mathcal{C}^0} $- and $ {\mathcal{C}^1} $-flows      S. I. Maksymenko 《Ukrainian Mathematical Journal》2010,62(7):1109-1125 Let F:M ×\mathbbR ? M {\mathbf{F}}:M \times \mathbb{R} \to M be a continuous flow on a manifold M, let V ⊂ M be an open subset, and let x:V ? \mathbbR \xi :V \to \mathbb{R} be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any open connected subset V ⊂ M; the author has described the structure of the set P(V) of all period functions on V. Assume that F is topologically conjugate to some C1 {\mathcal{C}^1} -flow. It is shown in this paper that, in this case, the period functions of F satisfy some additional conditions that, generally speaking, are not satisfied for general continuous flows.      11. A generalisation of the Hopf construction and harmonic morphisms into {\mathbb {S}^2}      S. Montaldo  A. Ratto 《Annali di Matematica Pura ed Applicata》2010,189(4):605-613 In this paper, we construct a new family of harmonic morphisms ${\varphi:V^5\to\mathbb{S}^2}In this paper, we construct a new family of harmonic morphisms j:V5?\mathbbS2{\varphi:V^5\to\mathbb{S}^2}, where V 5 is a 5-dimensional open manifold contained in an ellipsoidal hypersurface of \mathbbC4 = \mathbbR8{\mathbb{C}^4\,=\,\mathbb{R}^8}. These harmonic morphisms admit a continuous extension to the completion V*5{{V^{\ast}}^5}, which turns out to be an explicit real algebraic variety. We work in the context of a generalization of the Hopf construction and equivariant theory.      12. New Besov-type spaces and Triebel–Lizorkin-type spaces including Q spaces      Dachun Yang  Wen Yuan 《Mathematische Zeitschrift》2010,265(2):451-480 Let ${s,\,\tau\in\mathbb{R}}Let s, t ? \mathbbR{s,\,\tau\in\mathbb{R}} and q ? (0,￥]{q\in(0,\infty]} . We introduce Besov-type spaces [(B)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} for p ? (0, ￥]{p\in(0,\,\infty]} and Triebel–Lizorkin-type spaces [(F)\dot]s, tp, q(\mathbbRn) for p ? (0, ￥){{{{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}\,{\rm for}\, p\in(0,\,\infty)} , which unify and generalize the Besov spaces, Triebel–Lizorkin spaces and Q spaces. We then establish the j{\varphi} -transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the j{\varphi} -transform characterization of [(B)\dot]s, tp, q(\mathbbRn) and [(F)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\, {\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} , we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and molecular decomposition characterizations of [(B)\dot]s, tp, q(\mathbbRn) and [(F)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\,{\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} . For s ? \mathbbR{s\in\mathbb{R}} , p ? (1, ￥), q ? [1, ￥){p\in(1,\,\infty), q\in[1,\,\infty)} and t ? [0, \frac1(max{p, q})￠]{\tau\in[0,\,\frac{1}{(\max\{p,\,q\})'}]} , via the Hausdorff capacity, we introduce certain Hardy–Hausdorff spaces B[(H)\dot]s, tp, q(\mathbbRn){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} and prove that the dual space of B[(H)\dot]s, tp, q(\mathbbRn){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} is just [(B)\dot]-s, tp￠, q￠(\mathbbRn){\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})} , where t′ denotes the conjugate index of t ? (1,￥){t\in (1,\infty)} .      13. Schlicht envelopes of holomorphy and topology      Christopher Hammond 《Mathematische Zeitschrift》2010,266(2):285-288 Let Ω be a domain in ${\mathbb{C}^{2}}Let Ω be a domain in \mathbbC2{\mathbb{C}^{2}}, and let p: [(W)\tilde]? \mathbbC2{\pi: \tilde{\Omega}\rightarrow \mathbb{C}^{2}} be its envelope of holomorphy. Also let W￠=p([(W)\tilde]){\Omega'=\pi(\tilde{\Omega})} with i: W\hookrightarrow W￠{i: \Omega \hookrightarrow \Omega'} the inclusion. We prove the following: if the induced map on fundamental groups i*:p1(W) ? p1(W￠){i_{*}:\pi_{1}(\Omega) \rightarrow \pi_{1}(\Omega')} is a surjection, and if π is a covering map, then Ω has a schlicht envelope of holomorphy. We then relate this to earlier work of Fornaess and Zame.      14. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      15. Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b      Pascual Lucas  H. Fabián Ramírez-Ospina 《Geometriae Dedicata》2011,153(1):151-175 We study hypersurfaces in the Lorentz-Minkowski space \mathbbLn+1{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L k ψ = Aψ + b, where L k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLn+1{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSn1(r){\mathbb{S}^n_1(r)} or \mathbbHn(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbSm1(r)×\mathbbRn-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbHm(-r)×\mathbbRn-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbLm×\mathbbSn-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).      16. Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients      Qi Zhang  Huaizhong Zhao 《Journal of Theoretical Probability》2012,25(2):396-423 In this paper we develop a new weak convergence and compact embedding method to study the existence and uniqueness of the Lr2p(\mathbbRd;\mathbbR1)×Lr2(\mathbbRd;\mathbbRd)L_{\rho}^{2p}({\mathbb{R}^{d}};{\mathbb{R}^{1}})\times L_{\rho}^{2}({\mathbb{R}^{d}};{\mathbb{R}^{d}}) valued solution of backward stochastic differential equations with p-growth coefficients. Then we establish the probabilistic representation of the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.      17. Measure of central symmetry for fields of convex figures and three-dimensional convex bodies      V. V. Makeev 《Journal of Mathematical Sciences》2011,175(5):562-568 Let g23:E2( \mathbbR3 ) ? G2( \mathbbR3 ) \gamma_2^3:{E_2}\left( {{\mathbb{R}^3}} \right) \to {G_2}\left( {{\mathbb{R}^3}} \right) be the tautological vector bundle over the Grassmann manifold of 2-planes in \mathbbR3 {\mathbb{R}^3} , where the fiber over a plane is the plane itself regarded as a two-dimensional subspace of \mathbbR3 {\mathbb{R}^3} . A field of convex figures is given in γ23 if a convex figure is distinguished in each fiber so that the figure continuously depends on the fiber. It is proved that each field of convex figures in γ23 contains a figure K containing a centrally symmetric convex figure of area ( 4 + 16?2 ) \left( {4 + 16\sqrt {2} } \right) S(K)/31 > 0.858 S(K) (S(K) denotes the area of K), and a figure K′ that is contained in a centrally symmetric convex figure of area ( 12?2 - 8 ) \left( {12\sqrt {2} - 8} \right) S(K′)/7 < 1.282 S(K′). It is also proved that each three-dimensional convex body K is contained in a centrally symmetric convex cylinder of volume ( 36?2 - 24 ) \left( {36\sqrt {2} - 24} \right) V(K)/7 < 3.845 V(K). (Here, V(K) denotes the volume of K.) Bibliography: 5 titles.      18. Compact Surfaces with Constant Gaussian Curvature in Product Spaces      Juan A. Aledo  Victorino Lozano  José A. Pastor 《Mediterranean Journal of Mathematics》2010,7(3):263-270 We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH2×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS2×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH2×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS2×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.      19. Triple Hilbert Transforms Along Polynomial Surfaces      Yong-Kum Cho  Sunggeum Hong  Joonil Kim  Chan Woo Yang 《Integral Equations and Operator Theory》2009,65(4):485-528 Given W ì \mathbbZ+3\Omega \subset {\mathbb{Z}}_{+}^{3}, we discuss a necessary and sufficient condition that the triple Hilbert transform associated with any polynomial of the form ($t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m$t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m) is bounded in Lp(\mathbbR4)L^p({\mathbb{R}}^4).      20. Solutions to Polynomial Generalized Bers–Vekua Equations in Clifford Analysis      Min Ku  Daoshun Wang  Lin Dong 《Complex Analysis and Operator Theory》2012,6(2):407-424 In this paper, we mainly study polynomial generalized Vekua-type equation _boxclose)w=0{p(\mathcal{D})w=0} and polynomial generalized Bers–Vekua equation p(D)w=0{p(\mathcal{\underline{D}})w=0} defined in W ì \mathbbRn+1{\Omega\subset\mathbb{R}^{n+1}} where D{\mathcal{D}} and D{\mathcal{\underline{D}}} mean generalized Vekua-type operator and generalized Bers–Vekua operator, respectively. Using Clifford algebra, we obtain the Fischer-type decomposition theorems for the solutions to these equations including (D-l)kw=0,(D-l)kw=0(k ? \mathbbN){\left(\mathcal{D}-\lambda\right)^{k}w=0,\left(\mathcal {\underline{D}}-\lambda\right)^{k}w=0\left(k\in\mathbb{N}\right)} with complex parameter λ as special cases, which derive the Almansi-type decomposition theorems for iterated generalized Bers–Vekua equation and polynomial generalized Cauchy–Riemann equation defined in W ì \mathbbRn+1{\Omega\subset\mathbb{R}^{n+1}}. Making use of the decomposition theorems we give the solutions to polynomial generalized Bers–Vekua equation defined in W ì \mathbbRn+1{\Omega\subset\mathbb{R}^{n+1}} under some conditions. Furthermore we discuss inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRn+1{\Omega\subset\mathbb{R}^{n+1}}, and develop the structure of the solutions to inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRn+1{\Omega\subset\mathbb{R}^{n+1}}.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Spectral theorem for multipliers on {L_{\omega}^{2}(\mathbb{R})}

We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces L_w²(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with f ? C_c(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces L²_w(\mathbb R^k){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form.     

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Let f be an endomorphism of \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and ν be an f-invariant measure with positive Lyapunov exponents (λ ₁, . . . , λ _k ). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dim_Hm 3 [(log d)/(l₁)] + [(log d)/(l₂)]{{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of f ⁿ in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f ⁿ .     

The L p -Solvability of the Dirichlet Problem for Planar Elliptic Equations, Sharp Results

Assume that the elliptic operator L=div (A(x)∇) is L ^p -resolutive, p>1, on the unit disc \mathbbD ì \mathbb R²\mathbb{D}\subset \mathbb {R}^{2} . This means that the Dirichlet problem

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      8. Exotic smooth structures on small 4-manifolds with odd signatures      Anar Akhmedov  B. Doug Park 《Inventiones Mathematicae》2010,181(3):577-603 Let M be (2n-1)\mathbbCP2#2n[`(\mathbbCP)]2(2n-1)\mathbb{CP}^{2}\#2n\overline{\mathbb{CP}}{}^{2} for any integer n≥1. We construct an irreducible symplectic 4-manifold homeomorphic to M and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to M. We also construct such exotic smooth structures when M is \mathbbCP2#4[`(\mathbbCP)]2\mathbb{CP}{}^{2}\#4\overline {\mathbb{CP}}{}^{2} or 3\mathbbCP2#k[`(\mathbbCP)]23\mathbb{CP}{}^{2}\#k\overline{\mathbb{CP}}{}^{2} for k=6,8,10.      9. Fourier Series with the Continuous Primitive Integral      Erik Talvila 《Journal of Fourier Analysis and Applications》2012,18(1):27-44 Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted Ac(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}) and is a Banach space under the Alexiewicz norm, ||f||\mathbbT=sup|I| ￡ 2p|òI f|\|f\|_{\mathbb{T}}=\sup_{|I|\leq2\pi}|\int_{I} f|, the supremum being taken over intervals of length not exceeding 2π. It contains the periodic functions integrable in the sense of Lebesgue and Henstock–Kurzweil. Many of the properties of L 1 Fourier series continue to hold for this larger space, with the L 1 norm replaced by the Alexiewicz norm. The Riemann–Lebesgue lemma takes the form [^(f)](n)=o(n)\hat{f}(n)=o(n) as |n|→∞. The convolution is defined for f ? Ac(\mathbbT)f\in{\mathcal{A}}_{c}(\mathbb{T}) and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate ||f*g||￥ ￡ ||f||\mathbbT ||g||BV\|f\ast g\|_{\infty}\leq\|f\|_{\mathbb{T}} \|g\|_{\mathcal{BV}}. For g ? L1(\mathbbT)g\in L^{1}(\mathbb{T}), ||f*g||\mathbbT ￡ ||f||\mathbb T ||g||1\|f\ast g\|_{\mathbb{T}}\leq\|f\|_{\mathbb {T}} \|g\|_{1}. As well, [^(f*g)](n)=[^(f)](n) [^(g)](n)\widehat{f\ast g}(n)=\hat{f}(n) \hat{g}(n). There are versions of the Salem–Zygmund–Rudin–Cohen factorization theorem, Fejér’s lemma and the Parseval equality. The trigonometric polynomials are dense in Ac(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}). The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let D n be the Dirichlet kernel and let f ? L1(\mathbbT)f\in L^{1}(\mathbb{T}). Then ||Dn*f-f||\mathbbT?0\|D_{n}\ast f-f\|_{\mathbb{T}}\to0 as n→∞. Fourier coefficients of functions of bounded variation are characterized. The Appendix contains a type of Fubini theorem.      10. Period functions for $ {\mathcal{C}^0} $- and $ {\mathcal{C}^1} $-flows      S. I. Maksymenko 《Ukrainian Mathematical Journal》2010,62(7):1109-1125 Let F:M ×\mathbbR ? M {\mathbf{F}}:M \times \mathbb{R} \to M be a continuous flow on a manifold M, let V ⊂ M be an open subset, and let x:V ? \mathbbR \xi :V \to \mathbb{R} be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any open connected subset V ⊂ M; the author has described the structure of the set P(V) of all period functions on V. Assume that F is topologically conjugate to some C1 {\mathcal{C}^1} -flow. It is shown in this paper that, in this case, the period functions of F satisfy some additional conditions that, generally speaking, are not satisfied for general continuous flows.      11. A generalisation of the Hopf construction and harmonic morphisms into {\mathbb {S}^2}      S. Montaldo  A. Ratto 《Annali di Matematica Pura ed Applicata》2010,189(4):605-613 In this paper, we construct a new family of harmonic morphisms ${\varphi:V^5\to\mathbb{S}^2}In this paper, we construct a new family of harmonic morphisms j:V5?\mathbbS2{\varphi:V^5\to\mathbb{S}^2}, where V 5 is a 5-dimensional open manifold contained in an ellipsoidal hypersurface of \mathbbC4 = \mathbbR8{\mathbb{C}^4\,=\,\mathbb{R}^8}. These harmonic morphisms admit a continuous extension to the completion V*5{{V^{\ast}}^5}, which turns out to be an explicit real algebraic variety. We work in the context of a generalization of the Hopf construction and equivariant theory.      12. New Besov-type spaces and Triebel–Lizorkin-type spaces including Q spaces      Dachun Yang  Wen Yuan 《Mathematische Zeitschrift》2010,265(2):451-480 Let ${s,\,\tau\in\mathbb{R}}Let s, t ? \mathbbR{s,\,\tau\in\mathbb{R}} and q ? (0,￥]{q\in(0,\infty]} . We introduce Besov-type spaces [(B)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} for p ? (0, ￥]{p\in(0,\,\infty]} and Triebel–Lizorkin-type spaces [(F)\dot]s, tp, q(\mathbbRn) for p ? (0, ￥){{{{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}\,{\rm for}\, p\in(0,\,\infty)} , which unify and generalize the Besov spaces, Triebel–Lizorkin spaces and Q spaces. We then establish the j{\varphi} -transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the j{\varphi} -transform characterization of [(B)\dot]s, tp, q(\mathbbRn) and [(F)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\, {\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} , we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and molecular decomposition characterizations of [(B)\dot]s, tp, q(\mathbbRn) and [(F)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\,{\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} . For s ? \mathbbR{s\in\mathbb{R}} , p ? (1, ￥), q ? [1, ￥){p\in(1,\,\infty), q\in[1,\,\infty)} and t ? [0, \frac1(max{p, q})￠]{\tau\in[0,\,\frac{1}{(\max\{p,\,q\})'}]} , via the Hausdorff capacity, we introduce certain Hardy–Hausdorff spaces B[(H)\dot]s, tp, q(\mathbbRn){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} and prove that the dual space of B[(H)\dot]s, tp, q(\mathbbRn){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} is just [(B)\dot]-s, tp￠, q￠(\mathbbRn){\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})} , where t′ denotes the conjugate index of t ? (1,￥){t\in (1,\infty)} .      13. Schlicht envelopes of holomorphy and topology      Christopher Hammond 《Mathematische Zeitschrift》2010,266(2):285-288 Let Ω be a domain in ${\mathbb{C}^{2}}Let Ω be a domain in \mathbbC2{\mathbb{C}^{2}}, and let p: [(W)\tilde]? \mathbbC2{\pi: \tilde{\Omega}\rightarrow \mathbb{C}^{2}} be its envelope of holomorphy. Also let W￠=p([(W)\tilde]){\Omega'=\pi(\tilde{\Omega})} with i: W\hookrightarrow W￠{i: \Omega \hookrightarrow \Omega'} the inclusion. We prove the following: if the induced map on fundamental groups i*:p1(W) ? p1(W￠){i_{*}:\pi_{1}(\Omega) \rightarrow \pi_{1}(\Omega')} is a surjection, and if π is a covering map, then Ω has a schlicht envelope of holomorphy. We then relate this to earlier work of Fornaess and Zame.      14. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      15. Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b      Pascual Lucas  H. Fabián Ramírez-Ospina 《Geometriae Dedicata》2011,153(1):151-175 We study hypersurfaces in the Lorentz-Minkowski space \mathbbLn+1{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L k ψ = Aψ + b, where L k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLn+1{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSn1(r){\mathbb{S}^n_1(r)} or \mathbbHn(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbSm1(r)×\mathbbRn-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbHm(-r)×\mathbbRn-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbLm×\mathbbSn-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).      16. Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients      Qi Zhang  Huaizhong Zhao 《Journal of Theoretical Probability》2012,25(2):396-423 In this paper we develop a new weak convergence and compact embedding method to study the existence and uniqueness of the Lr2p(\mathbbRd;\mathbbR1)×Lr2(\mathbbRd;\mathbbRd)L_{\rho}^{2p}({\mathbb{R}^{d}};{\mathbb{R}^{1}})\times L_{\rho}^{2}({\mathbb{R}^{d}};{\mathbb{R}^{d}}) valued solution of backward stochastic differential equations with p-growth coefficients. Then we establish the probabilistic representation of the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.      17. Measure of central symmetry for fields of convex figures and three-dimensional convex bodies      V. V. Makeev 《Journal of Mathematical Sciences》2011,175(5):562-568 Let g23:E2( \mathbbR3 ) ? G2( \mathbbR3 ) \gamma_2^3:{E_2}\left( {{\mathbb{R}^3}} \right) \to {G_2}\left( {{\mathbb{R}^3}} \right) be the tautological vector bundle over the Grassmann manifold of 2-planes in \mathbbR3 {\mathbb{R}^3} , where the fiber over a plane is the plane itself regarded as a two-dimensional subspace of \mathbbR3 {\mathbb{R}^3} . A field of convex figures is given in γ23 if a convex figure is distinguished in each fiber so that the figure continuously depends on the fiber. It is proved that each field of convex figures in γ23 contains a figure K containing a centrally symmetric convex figure of area ( 4 + 16?2 ) \left( {4 + 16\sqrt {2} } \right) S(K)/31 > 0.858 S(K) (S(K) denotes the area of K), and a figure K′ that is contained in a centrally symmetric convex figure of area ( 12?2 - 8 ) \left( {12\sqrt {2} - 8} \right) S(K′)/7 < 1.282 S(K′). It is also proved that each three-dimensional convex body K is contained in a centrally symmetric convex cylinder of volume ( 36?2 - 24 ) \left( {36\sqrt {2} - 24} \right) V(K)/7 < 3.845 V(K). (Here, V(K) denotes the volume of K.) Bibliography: 5 titles.      18. Compact Surfaces with Constant Gaussian Curvature in Product Spaces      Juan A. Aledo  Victorino Lozano  José A. Pastor 《Mediterranean Journal of Mathematics》2010,7(3):263-270 We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH2×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS2×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH2×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS2×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.      19. Triple Hilbert Transforms Along Polynomial Surfaces      Yong-Kum Cho  Sunggeum Hong  Joonil Kim  Chan Woo Yang 《Integral Equations and Operator Theory》2009,65(4):485-528 Given W ì \mathbbZ+3\Omega \subset {\mathbb{Z}}_{+}^{3}, we discuss a necessary and sufficient condition that the triple Hilbert transform associated with any polynomial of the form ($t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m$t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m) is bounded in Lp(\mathbbR4)L^p({\mathbb{R}}^4).      20. Solutions to Polynomial Generalized Bers–Vekua Equations in Clifford Analysis      Min Ku  Daoshun Wang  Lin Dong 《Complex Analysis and Operator Theory》2012,6(2):407-424 In this paper, we mainly study polynomial generalized Vekua-type equation _boxclose)w=0{p(\mathcal{D})w=0} and polynomial generalized Bers–Vekua equation p(D)w=0{p(\mathcal{\underline{D}})w=0} defined in W ì \mathbbRn+1{\Omega\subset\mathbb{R}^{n+1}} where D{\mathcal{D}} and D{\mathcal{\underline{D}}} mean generalized Vekua-type operator and generalized Bers–Vekua operator, respectively. Using Clifford algebra, we obtain the Fischer-type decomposition theorems for the solutions to these equations including (D-l)kw=0,(D-l)kw=0(k ? \mathbbN){\left(\mathcal{D}-\lambda\right)^{k}w=0,\left(\mathcal {\underline{D}}-\lambda\right)^{k}w=0\left(k\in\mathbb{N}\right)} with complex parameter λ as special cases, which derive the Almansi-type decomposition theorems for iterated generalized Bers–Vekua equation and polynomial generalized Cauchy–Riemann equation defined in W ì \mathbbRn+1{\Omega\subset\mathbb{R}^{n+1}}. Making use of the decomposition theorems we give the solutions to polynomial generalized Bers–Vekua equation defined in W ì \mathbbRn+1{\Omega\subset\mathbb{R}^{n+1}} under some conditions. Furthermore we discuss inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRn+1{\Omega\subset\mathbb{R}^{n+1}}, and develop the structure of the solutions to inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRn+1{\Omega\subset\mathbb{R}^{n+1}}.

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Exotic smooth structures on small 4-manifolds with odd signatures

Let M be (2n-1)\mathbbCP²#2n[`(\mathbbCP)]<sup>2</sup>(2n-1)\mathbb{CP}^{2}\#2n\overline{\mathbb{CP}}{}^{2} for any integer n≥1. We construct an irreducible symplectic 4-manifold homeomorphic to M and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to M. We also construct such exotic smooth structures when M is \mathbbCP<sup>2</sup>#4[`(\mathbbCP)]²\mathbb{CP}{}^{2}\#4\overline {\mathbb{CP}}{}^{2} or 3\mathbbCP²#k[`(\mathbbCP)]²3\mathbb{CP}{}^{2}\#k\overline{\mathbb{CP}}{}^{2} for k=6,8,10.     

Fourier Series with the Continuous Primitive Integral

Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted A_c(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}) and is a Banach space under the Alexiewicz norm, ||f||_\mathbbT=sup_{|I| ￡ 2p}|ò_I f|\|f\|_{\mathbb{T}}=\sup_{|I|\leq2\pi}|\int_{I} f|, the supremum being taken over intervals of length not exceeding 2π. It contains the periodic functions integrable in the sense of Lebesgue and Henstock–Kurzweil. Many of the properties of L ¹ Fourier series continue to hold for this larger space, with the L ¹ norm replaced by the Alexiewicz norm. The Riemann–Lebesgue lemma takes the form [^(f)](n)=o(n)\hat{f}(n)=o(n) as |n|→∞. The convolution is defined for f ? A_c(\mathbbT)f\in{\mathcal{A}}_{c}(\mathbb{T}) and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate ||f*g||_￥ ￡ ||f||_\mathbbT ||g||_BV\|f\ast g\|_{\infty}\leq\|f\|_{\mathbb{T}} \|g\|_{\mathcal{BV}}. For g ? L¹(\mathbbT)g\in L^{1}(\mathbb{T}), ||f*g||_\mathbbT ￡ ||f||_{\mathbb T} ||g||₁\|f\ast g\|_{\mathbb{T}}\leq\|f\|_{\mathbb {T}} \|g\|_{1}. As well, [^(f*g)](n)=[^(f)](n) [^(g)](n)\widehat{f\ast g}(n)=\hat{f}(n) \hat{g}(n). There are versions of the Salem–Zygmund–Rudin–Cohen factorization theorem, Fejér’s lemma and the Parseval equality. The trigonometric polynomials are dense in A_c(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}). The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let D _n be the Dirichlet kernel and let f ? L¹(\mathbbT)f\in L^{1}(\mathbb{T}). Then ||D_n*f-f||_\mathbbT?0\|D_{n}\ast f-f\|_{\mathbb{T}}\to0 as n→∞. Fourier coefficients of functions of bounded variation are characterized. The Appendix contains a type of Fubini theorem.     

Period functions for $ {\mathcal{C}^0} $- and $ {\mathcal{C}^1} $-flows

Let F:M ×\mathbbR ? M {\mathbf{F}}:M \times \mathbb{R} \to M be a continuous flow on a manifold M, let V ⊂ M be an open subset, and let x:V ? \mathbbR \xi :V \to \mathbb{R} be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any open connected subset V ⊂ M; the author has described the structure of the set P(V) of all period functions on V. Assume that F is topologically conjugate to some C¹ {\mathcal{C}^1} -flow. It is shown in this paper that, in this case, the period functions of F satisfy some additional conditions that, generally speaking, are not satisfied for general continuous flows.     

A generalisation of the Hopf construction and harmonic morphisms into {\mathbb {S}^2}

In this paper, we construct a new family of harmonic morphisms ${\varphi:V^5\to\mathbb{S}^2}In this paper, we construct a new family of harmonic morphisms j:V⁵?\mathbbS²{\varphi:V^5\to\mathbb{S}^2}, where V ⁵ is a 5-dimensional open manifold contained in an ellipsoidal hypersurface of \mathbbC⁴ = \mathbbR⁸{\mathbb{C}^4\,=\,\mathbb{R}^8}. These harmonic morphisms admit a continuous extension to the completion V^*5{{V^{\ast}}^5}, which turns out to be an explicit real algebraic variety. We work in the context of a generalization of the Hopf construction and equivariant theory.     

New Besov-type spaces and Triebel–Lizorkin-type spaces including Q spaces

Let ${s,\,\tau\in\mathbb{R}}Let s, t ? \mathbbR{s,\,\tau\in\mathbb{R}} and q ? (0,￥]{q\in(0,\infty]} . We introduce Besov-type spaces [(B)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} for p ? (0, ￥]{p\in(0,\,\infty]} and Triebel–Lizorkin-type spaces [(F)\dot]^s, t_p, q(\mathbbRⁿ) for p ? (0, ￥){{{{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}\,{\rm for}\, p\in(0,\,\infty)} , which unify and generalize the Besov spaces, Triebel–Lizorkin spaces and Q spaces. We then establish the j{\varphi} -transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the j{\varphi} -transform characterization of [(B)\dot]^s, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\, {\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} , we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and molecular decomposition characterizations of [(B)\dot]^s, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\,{\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} . For s ? \mathbbR{s\in\mathbb{R}} , p ? (1, ￥), q ? [1, ￥){p\in(1,\,\infty), q\in[1,\,\infty)} and t ? [0, \frac1(max{p, q})￠]{\tau\in[0,\,\frac{1}{(\max\{p,\,q\})'}]} , via the Hausdorff capacity, we introduce certain Hardy–Hausdorff spaces B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} and prove that the dual space of B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} is just [(B)\dot]^-s, t_p￠, q￠(\mathbbRⁿ){\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})} , where t′ denotes the conjugate index of t ? (1,￥){t\in (1,\infty)} .     

Schlicht envelopes of holomorphy and topology

Let Ω be a domain in ${\mathbb{C}^{2}}Let Ω be a domain in \mathbbC²{\mathbb{C}^{2}}, and let p: [(W)\tilde]? \mathbbC²{\pi: \tilde{\Omega}\rightarrow \mathbb{C}^{2}} be its envelope of holomorphy. Also let W￠=p([(W)\tilde]){\Omega'=\pi(\tilde{\Omega})} with i: W\hookrightarrow W￠{i: \Omega \hookrightarrow \Omega'} the inclusion. We prove the following: if the induced map on fundamental groups i_*:p₁(W) ? p₁(W￠){i_{*}:\pi_{1}(\Omega) \rightarrow \pi_{1}(\Omega')} is a surjection, and if π is a covering map, then Ω has a schlicht envelope of holomorphy. We then relate this to earlier work of Fornaess and Zame.     

On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}

In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbR^N{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients

In this paper we develop a new weak convergence and compact embedding method to study the existence and uniqueness of the L_r^2p(\mathbbR^d;\mathbbR¹)×L_r²(\mathbbR^d;\mathbbR^d)L_{\rho}^{2p}({\mathbb{R}^{d}};{\mathbb{R}^{1}})\times L_{\rho}^{2}({\mathbb{R}^{d}};{\mathbb{R}^{d}}) valued solution of backward stochastic differential equations with p-growth coefficients. Then we establish the probabilistic representation of the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.     

Measure of central symmetry for fields of convex figures and three-dimensional convex bodies

Let g₂³:E₂( \mathbbR³ ) ? G₂( \mathbbR³ ) \gamma_2^3:{E_2}\left( {{\mathbb{R}^3}} \right) \to {G_2}\left( {{\mathbb{R}^3}} \right) be the tautological vector bundle over the Grassmann manifold of 2-planes in \mathbbR³ {\mathbb{R}^3} , where the fiber over a plane is the plane itself regarded as a two-dimensional subspace of \mathbbR³ {\mathbb{R}^3} . A field of convex figures is given in γ₂³ if a convex figure is distinguished in each fiber so that the figure continuously depends on the fiber. It is proved that each field of convex figures in γ₂³ contains a figure K containing a centrally symmetric convex figure of area ( 4 + 16?2 ) \left( {4 + 16\sqrt {2} } \right) S(K)/31 > 0.858 S(K) (S(K) denotes the area of K), and a figure K′ that is contained in a centrally symmetric convex figure of area ( 12?2 - 8 ) \left( {12\sqrt {2} - 8} \right) S(K′)/7 < 1.282 S(K′). It is also proved that each three-dimensional convex body K is contained in a centrally symmetric convex cylinder of volume ( 36?2 - 24 ) \left( {36\sqrt {2} - 24} \right) V(K)/7 < 3.845 V(K). (Here, V(K) denotes the volume of K.) Bibliography: 5 titles.     

Compact Surfaces with Constant Gaussian Curvature in Product Spaces

We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.     

Triple Hilbert Transforms Along Polynomial Surfaces

Given W ì \mathbbZ₊³\Omega \subset {\mathbb{Z}}_{+}^{3}, we discuss a necessary and sufficient condition that the triple Hilbert transform associated with any polynomial of the form ($t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m$t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m) is bounded in L^p(\mathbbR⁴)L^p({\mathbb{R}}^4).     

Solutions to Polynomial Generalized Bers–Vekua Equations in Clifford Analysis

In this paper, we mainly study polynomial generalized Vekua-type equation _boxclose)w=0{p(\mathcal{D})w=0} and polynomial generalized Bers–Vekua equation p(D)w=0{p(\mathcal{\underline{D}})w=0} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}} where D{\mathcal{D}} and D{\mathcal{\underline{D}}} mean generalized Vekua-type operator and generalized Bers–Vekua operator, respectively. Using Clifford algebra, we obtain the Fischer-type decomposition theorems for the solutions to these equations including (D-l)^kw=0,(D-l)^kw=0(k ? \mathbbN){\left(\mathcal{D}-\lambda\right)^{k}w=0,\left(\mathcal {\underline{D}}-\lambda\right)^{k}w=0\left(k\in\mathbb{N}\right)} with complex parameter λ as special cases, which derive the Almansi-type decomposition theorems for iterated generalized Bers–Vekua equation and polynomial generalized Cauchy–Riemann equation defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}. Making use of the decomposition theorems we give the solutions to polynomial generalized Bers–Vekua equation defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}} under some conditions. Furthermore we discuss inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}, and develop the structure of the solutions to inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}.