Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equation

We analyze the well-posedness of the initial value problem for the dissipative quasi-geostrophic equations in the subcritical case. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions. While the only small self-similar solution in the strong space is the null solution, infinitely many self-similar solutions do exist in weak- spaces and in a recently introduced [7] space of tempered distributions. The asymptotic stability of solutions is obtained in both spaces, and as a consequence, a criterion of self-similarity persistence at large times is obtained.     

Approximate identities on some homogeneous Banach spaces

We study convergence of approximate identities on some complete semi-normed or normed spaces of locally L ^p functions where translations are isometries, namely Marcinkiewicz spaces M^p{\mathcal{M}^{p}} and Stepanoff spaces S^p{\mathcal{S}^p}, 1 ≤ p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M ^p ) or even unbounded (M^p₀{M^{p}_{0}}). We construct a function f that belongs to these spaces and has the property that all approximate identities f_e * f{\phi_\varepsilon * f} converge to f pointwise but they never converge in norm.     

Shift coordinates, stretch lines and polyhedral structures for Teichmüller space

This paper has two parts. In the first part, we study shift coordinates on a sphere S equipped with three distinguished points and a triangulation whose vertices are the distinguished points. These coordinates parametrize a space (S)\widetilde{{\cal T}}(S) that we call an unfolded Teichmüller space. This space contains Teichmüller spaces of the sphere with \frak b{\frak b} boundary components and \frak p{\frak p} cusps (which we call generalized pairs of pants), for all possible values of \frak b{\frak b} and \frak p{\frak p} satisfying \frak b+\frak p=3{\frak b}+{\frak p}=3 . The parametrization of [(T)\tilde](S)\widetilde{{\cal T}}(S) by shift coordinates equips this space with a natural polyhedral structure, which we describe more precisely as a cone over an octahedron in \Bbb R³{\Bbb {R}}^3 . Each cone over a simplex of this octahedron is interpreted as a Teichmüller space of the sphere with \frak b{\frak b} boundary components and \frak p{\frak p} cusps, for fixed \frak b{\frak b} and \frak p{\frak p} , the sphere being furthermore equipped with an orientation on each boundary component. There is a natural linear action of a finite group on [(T)\tilde](S)\widetilde{{\cal T}}(S) whose quotient is an augmented Teichmüller space in the usual sense. We describe several aspects of the geometry of the space [(T)\tilde](S)\widetilde{{\cal T}}(S) . Stretch lines and earthquakes can be defined on this space. In the second part of the paper, we use the shift coordinates to obtain estimates on the behaviour of stretch lines in the Teichmüller space of a surface obtained by gluing hyperbolic pairs of pants. We also use the shift coordinates to give formulae that express stretch lines in terms of Fenchel-Nielsen coordinates. We deduce the disjointness of some stretch lines in Teichmüller space. We study in more detail the case of a closed surface of genus 2.     

Resolvent estimates for the linearized operator in L p associated with motion of compressible viscous fluids

We consider the resolvent problem for the linearized system of equations that describe motion of compressible viscous barotropic fluids in a bounded domain with the Navier boundary condition. This problem has uniquely a solution in [(W)\dot]¹_p ×(W²_p)ⁿ{\dot{W}^{1}_{p} \times (W^{2}_{p})^{n}} satisfying L _p estimates for any 1 < p < ∞. Moreover, resolvent estimates for the linearized operator of the above system in [(W)\dot]¹_p ×(L_p)ⁿ{\dot{W}^{1}_{p} \times (L_{p})^{n}} are established. Our main results yield clearly that the linearized operator is the infinitesimal generator of a uniformly bounded analytic semigroup on [(W)\dot]¹_p ×(L_p)ⁿ{\dot{W}^{1}_{p} \times (L_{p})^{n}}.     

Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye–Hückel system

In this paper we establish some regularizing and decay rate estimates for mild solutions of the Debye–Hückel system. We prove that if the initial data belong to the critical Lebesgue space L^\fracn2(\mathbbRⁿ){L^{\frac{n}{2}}(\mathbb{R}^{n})} , then the L ^q -norm ( \fracn2 ￡ q ￡ ￥{\frac{n}{2} \leq q \leq \infty}) of the βth order spatial derivative of mild solutions are majorized by K₁(K₂|b|)^|b|t^{-\frac|b|2-1+\fracn2q}{K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2}-1+\frac{n}{2q}}} for some constants K ₁ and K ₂. These estimates particularly imply that mild solutions are analytic in the space variable, and provide decay estimates in the time variable for higher-order derivatives of mild solutions. We also prove that similar estimates also hold for mild solutions whose initial data belong to the critical homogeneous Besov space [(B)\dot]^-2+\fracnp_p,￥(\mathbbRⁿ){\dot{B}^{-2+\frac{n}{p}}_{p,\infty}(\mathbb{R}^n)} ( \fracn2 < p < n{\frac{n}{2} < p < n}).     

On the rationality of the moduli space of Lüroth quartics

We prove that the moduli space \mathfrakM_L{\mathfrak{M}_L} of Lüroth quartics in \mathbbP²{\mathbb{P}^2}, i.e. the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of PGL₃ (\mathbbC){\mathrm{PGL}_3 (\mathbb{C})} is rational, as is the related moduli space of Bateman seven-tuples of points in \mathbbP²{\mathbb{P}^2}.     

When is the Uvarov transformation positive definite?

Let L\cal{L} be a positive definite bilinear functional, then the Uvarov transformation of L\cal{L} is given by  U(p,q) = L(p,q) + m p(a)[`(q)](a<sup>-1</sup>) +[`(m)] p([`(a)]<sup>-1</sup>)\,\mathcal{U}(p,q) = \mathcal{L}(p,q) + m\,p(\alpha)\overline{q}(\alpha^{-1}) + \overline{m}\,p(\overline{\alpha}^{-1}) [`(q)]([`(a)])\overline{q}(\overline{\alpha}) where $|\alpha| > 1, m \in \mathbb{C}$|\alpha| > 1, m \in \mathbb{C}. In this paper we analyze conditions on m for U\cal{U} to be positive definite in the linear space of polynomials of degree less than or equal to n. In particular, we show that m has to lie inside a circle in the complex plane defined by α, n and the moments associated with L\cal{L}. We also give an upper bound for the radius of this circle that depends only on α and n. This and other conditions on m are visualized for some examples.     

Approximation by de la Vallée-Poussin operators on the classes of functions locally summable on the real axis

For the least upper bounds of deviations of the de la Vallée-Poussin operators on the classes [^(L)]_b^y \hat{L}_\beta^\psi of rapidly vanishing functions ψ in the metric of the spaces [^(L)]_p {\hat{L}_p} , 1 ≤ p ≤ ∞, we establish upper estimates that are exact on some subsets of functions from [^(L)]_p {\hat{L}_p} .     

Banach Spaces of Solutions of the Helmholtz Equation in the Plane

The purpose of this article is to study the Hilbert space W²\mathcal{ W}^2 consisting of all solutions of the Helmholtz equation Du+u=0\Delta u+u=0 in \BbbR²\Bbb{R}^2 that are the image under the Fourier transform of L²L^2 densities in the unit circle. We characterize this space as a close subspace of the Hilbert space H²\mathcal{ H}^2 of all functions belonging to L²( | x | ^-3dx) L^2( | x | ^{-3}dx) jointly with their angular and radial derivatives, in the complement of the unit disk in \BbbR²\Bbb{R}^2. We calculate the reproducing kernel of W²\mathcal{ W}^2 and study its reproducing properties in the corresponding spaces H^p\mathcal{H}^p, for $p>1$p>1.     

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}}.     

Completely periodic directions and orbit closures of many pseudo-Anosov Teichmueller discs in Q(1,1,1,1)

In this paper, we investigate the closure of a large class of Teichmüller discs in the stratum Q(1, 1, 1, 1){\mathcal{Q}(1, 1, 1, 1)} or equivalently, in a GL⁺₂(\mathbbR){{\rm GL}^+_2(\mathbb{R})} -invariant locus L{\mathcal{L}} of translation surfaces of genus three. We describe a systematic way to prove that the GL⁺₂(\mathbbR){{\rm GL}^+_2(\mathbb{R})} -orbit closure of a translation surface in L{\mathcal{L}} is the whole locus L{\mathcal{L}} . The strategy of the proof is an analysis of completely periodic directions on such a surface and an iterated application of Ratner’s theorem to unipotent subgroups acting on an “adequate” splitting. This analysis applies for example to all Teichmüller discs obtained by the Thurston–Veech’s construction with a trace field of degree three which are moreover “obviously not Veech”. We produce an infinite series of such examples and show moreover that the favourable splitting situation does not arise everywhere on L{\mathcal{L}} , contrary to the situation in genus two. We also study completely periodic directions on translation surfaces in L{\mathcal{L}} . For instance, we prove that completely periodic directions are dense on surfaces obtained by the Thurston–Veech’s construction.     

New spaces of functions and hyperfunctions for Hankel transforms and convolutions

In this paper we study the Hankel transformation and convolution on certain spaces G_e{\cal G}_{e} of entire functions and its dual G_e￠{\cal G}_{e}{\prime} that is a space of hyperfunctions and contains the (even)-Schwartz space S _e ′. We prove that the Hankel transform is an automorphism of G_e￠{\cal G}_{e}{\prime} . Also the Hankel convolutors of G_e￠{\cal G}_{e}{\prime} are investigated.     

The （L^p, Fp^β,∞）-Boundedness of Commutators of Multipliers

In this paper, we study the commutator generalized by a multiplier and a Lipschitz function. Under some assumptions, we establish the boundedness properties of it from L^P（R^n） into Fp^β,∞（R^n）, the Triebel Lizorkin spaces.     

An Algorithm to Find the Lineality Space of the Positive Hull of a Set of Vectors

The positive hull of a finite set of vectors, V{\cal V}, in d-dimensional space may or may not contain a lineality space L{\cal L}. This article presents an algorithm that identifies the vectors of V{\cal V} that belong to L{\cal L}. This is done by means of a sequence of supporting hyperplanes because every supporting hyperplane of the positive hull of V{\cal V} contains L{\cal L}. Computational results show the effectiveness of the algorithm, which is compared to the best procedure currently available (to the best knowledge of the author) that solves the same problem. The algorithm introduced here is especially efficient in the case of large problems, where cardinality and/or dimensions are large.     

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.     

A Noncommutative Version of <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and Characterizations of Subdiagonal Algebras

Let M{\mathcal M} be a σ-finite von Neumann algebra and \mathfrak A{\mathfrak A} a maximal subdiagonal algebra of M{\mathcal M} with respect to a faithful normal conditional expectation F{\Phi} . Based on Haagerup’s noncommutative L ^p space L^p(M){L^p(\mathcal M)} associated with M{\mathcal M} , we give a noncommutative version of H ^p space relative to \mathfrak A{\mathfrak A} . If h ₀ is the image of a faithful normal state j{\varphi} in L¹(M){L^1(\mathcal M)} such that j°F = j{\varphi\circ \Phi=\varphi} , then it is shown that the closure of {\mathfrak Ah₀^\frac1p}{\{\mathfrak Ah_0^{\frac1p}\}} in L^p(M){L^p(\mathcal M)} for 1 ≤ p < ∞ is independent of the choice of the state preserving F{\Phi} . Moreover, several characterizations for a subalgebra of the von Neumann algebra M{\mathcal M} to be a maximal subdiagonal algebra are given.     

Hardy and Carleson Measure Spaces Associated with Operators on Spaces of Homogeneous Type

Let (X, d, μ) be a metric measure space with doubling property. The Hardy spaces associated with operators L were introduced and studied by many authors. All these spaces, however, were first defined by L ²(X) functions and finally the Hardy spaces were formally defined by the closure of these subspaces of L ²(X) with respect to Hardy spaces norms. A natural and interesting question in this context is to characterize the closure. The purpose of this paper is to answer this question. More precisely, we will introduce \({CMO}_{L}^{p}(X)\), the Carleson measure spaces associated with operators L, and characterize the Hardy spaces associated with operators L via \(({CMO}_{L}^{p}(X))'\), the distributions of \({CMO}_{L}^{p}(X)\).     

An Lp-Lq-Version of Morgan’s Theorem for the Dunkl-Bessel Transform

In this paper, we give an L^p-L^q-version of Morgans theorem for the Dunkl-Bessel transform on More precisely, we prove that for all and then for all measurable function f on the conditions and imply f = 0, if and only if where are the Lebesgue spaces associated with the Dunkl-Bessel transform.Received: November 21, 2003 Revised: April 26, 2004 Accepted: May 28, 2004     

On Singular Integral Operators with Rough Kernel Along Surface

In this note we give a simple method to transfer the effect of the surface to the radial function in the kernel of singular integral along surface. Using this idea, we give some continuity of the singular integrals along surface with Hardy space function kernels on some function spaces, such as L^p(\mathbb Rⁿ),L^p(\mathbb Rⁿ,w){L^p({\mathbb R}^n),L^p({\mathbb R}^n,\omega)}, Triebel–Lizorkin spaces [(F)\dot]_p^s,q(\mathbb Rⁿ){{\dot F}_{p}^{s,q}({\mathbb R}^n)}, Besov spaces [(B)\dot]_p^s,q(\mathbb Rⁿ){{\dot B}_{p}^{s,q}({\mathbb R}^n)}, generalized Morrey spaces L^p,f(\mathbb Rⁿ){L^{p,\phi}({\mathbb R}^n)} and Herz spaces [(K)\dot]_p^{a, q}(\mathbb Rⁿ){\dot K_p^{\alpha, q}({\mathbb R}^n)}. Our results improve and extend substantially some known results on the singular integral operators along surface.     

(Almost) primitivity of Hecke L -functions

Let f be a cusp form of the Hecke space \frak M₀(l,k,e){\frak M}_0(\lambda,k,\epsilon) and let L _f be the normalized L-function associated to f. Recently it has been proved that L _f belongs to an axiomatically defined class of functions [`(S)]<sup>\sharp</sup>\bar{\cal S}^\sharp . We prove that when λ ≤ 2, L <sub>f</sub> is always almost primitive, i.e., that if L <sub>f</sub> is written as product of functions in [`(S)]^\sharp\bar{\cal S}^\sharp , then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if l ? {?2,?3,2}\lambda\notin\{\sqrt{2},\sqrt{3},2\} then L _f is also primitive, i.e., that if L _f = F ₁ F ₂ then F ₁ (or F ₂) is constant; for l ? {?2,?3,2}\lambda\in\{\sqrt{2},\sqrt{3},2\} the factorization of non-primitive functions is studied and examples of non-primitive functions are given. At last, the subset of functions f for which L _f belongs to the more familiar extended Selberg class S^\sharp{\cal S}^\sharp is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in S^\sharp{\cal S}^\sharp .