Boundedness of the anisotropic maximal and anisotropic singular integral operators in generalized Morrey spaces

In this paper we give the conditions on the pair (ω ₁, ω ₂) which ensures the boundedness of the anisotropic maximal operator and anisotropic singular integral operators from one generalized Morrey space M_p,w1 \mathcal{M}_{p,\omega _1 } to another M_p,w2 \mathcal{M}_{p,\omega _2 }, 1 < p < g8, and from the space M_1,w1 \mathcal{M}_{1,\omega _1 } to the weak space WM_1,w2 W\mathcal{M}_{1,\omega _2 }.     

Micro-Local Analysis with Fourier Lebesgue Spaces. Part?I

Let ω,ω ₀ be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, WF_FL^q(w)(f)\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f) of f ? S￠f\in \mathcal{S}' with respect to weighted Fourier Lebesgue space FL^q_(w)\mathcal{F}L^{q}_{(\omega )}. We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in S^(w₀)_r,0S^{(\omega _{0})}_{\rho ,0} hold for such wave-front sets. Especially we prove that

Iterates and Hypoellipticity of Partial Differential Operators on Non-Quasianalytic Classes

Let P be a linear partial differential operator with constant coefficients. For a weight function ω and an open subset Ω of \mathbbR^N{\mathbb{R}^N} , the class E_P,{w}(W){\mathcal{E}_{P,\{\omega\}}(\Omega)} of Roumieu type involving the successive iterates of the operator P is considered. The completeness of this space is characterized in terms of the hypoellipticity of P. Results of Komatsu and Newberger-Zielezny are extended. Moreover, for weights ω satisfying a certain growth condition, this class coincides with a class of ultradifferentiable functions if and only if P is elliptic. These results remain true in the Beurling case E_P,(w)(W){\mathcal{E}_{P,(\omega)}(\Omega)}.     

Maximal Chains in Positive Subfamilies of P(ω)

A family P ì [w]^w{\mathcal P} \subset [\omega]^\omega is called positive iff it is the union of some infinite upper set in the Boolean algebra P(ω)/Fin. For example, if I ì P(w){\mathcal I} \subset P(\omega) is an ideal containing the ideal Fin of finite subsets of ω, then P(w) \IP(\omega) \setminus {\mathcal I} is a positive family and the set Dense(\mathbb Q)\mbox{Dense}({\mathbb Q}) of dense subsets of the rational line is a positive family which is not the complement of some ideal on P(\mathbb Q)P({\mathbb Q}). We prove that, for a positive family P{\mathcal P}, the order types of maximal chains in the complete lattice áP è{?}, ì ?\langle {\mathcal P} \cup \{\emptyset\}, \subset \rangle are exactly the order types of compact nowhere dense subsets of the real line having the minimum non-isolated. Also we compare this result with the corresponding results concerning maximal chains in the Boolean algebras P(ω) and Intalg[0,1)_{\mathbb R}\mbox{Intalg}[0,1)_{{\mathbb R}} and the poset E(\mathbb Q)E({\mathbb Q}), where E(\mathbb Q)E({\mathbb Q}) is the set of elementary submodels of the rational line.     

Around splitting and reaping for partitions of ω

We investigate splitting number and reaping number for the structure (ω) ^ω of infinite partitions of ω. We prove that \mathfrakr_d ￡ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and \mathfraks_d 3 \mathfrakb{\mathfrak{s}_{d}\geq\mathfrak{b}} . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})} . To prove the consistency \mathfrakr_d < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and \mathfraks_d < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants \mathfrakr_pair{\mathfrak{r}_{pair}} and \mathfraks_pair{\mathfrak{s}_{pair}} . We also study the relation between \mathfrakr_pair, \mathfraks_pair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that cov(M),cov(N) ￡ \mathfrakr_pair ￡ \mathfraks_d,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and \mathfraks ￡ \mathfraks_pair ￡ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} .     

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}}.     

Desingularization of singular Riemannian foliation

Let F{\mathcal{F}} be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F{\mathcal{F}} we construct a regular Riemannian foliation [^(F)]{\hat{\mathcal{F}}} on a compact Riemannian manifold [^(M)]{\hat{M}} and a desingularization map [^(r)]:[^(M)]? M{\hat{\rho}:\hat{M}\rightarrow M} that projects leaves of [^(F)]{\hat{\mathcal{F}}} into leaves of F{\mathcal{F}}. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F{\mathcal{F}} are compact, then, for each small ${\epsilon >0 }${\epsilon >0 }, we can find [^(M)]{\hat{M}} and [^(F)]{\hat{\mathcal{F}}} so that the desingularization map induces an e{\epsilon}-isometry between M/F{M/\mathcal{F}} and [^(M)]/[^(F)]{\hat{M}/\hat{\mathcal{F}}}. This implies in particular that the space of leaves M/F{M/\mathcal{F}} is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {([^(M)]_n/[^(F)]_n)}{\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}.     

Singularly perturbed 1D Cahn–Hilliard equation revisited

We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors ${\{{\mathcal M}_\epsilon\}, \epsilon\geq 0}We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors {M_e}, e 3 0{\{{\mathcal M}_\epsilon\}, \epsilon\geq 0} being the perturbation parameter, such that the map e? M_e{\epsilon \mapsto {\mathcal M}_\epsilon} is H?lder continuous. Besides, the continuity at e = 0{\epsilon=0} is obtained with respect to a metric independent of e.{\epsilon.} Continuity properties of global attractors and inertial manifolds are also examined.     

Projection decomposition in multiplier algebras

In this paper we present new structural information about the multiplier algebra M (A ){\mathcal M (\mathcal A )} of a σ-unital purely infinite simple C*-algebra A{\mathcal {A}}, by characterizing the positive elements A ? M (A ){A\in \mathcal M (\mathcal A )} that are strict sums of projections belonging to A{\mathcal A } . If A ? A{A\not\in \mathcal {A}} and A itself is not a projection, then the necessary and sufficient condition for A to be a strict sum of projections belonging to A{\mathcal {A} } is that ${\|A\| >1 }${\|A\| >1 } and that the essential norm ||A||_ess 3 1{\|A\|_{ess} \geq 1}. Based on a generalization of the Perera–Rordam weak divisibility of separable simple C*-algebras of real rank zero to all σ-unital simple C*-algebras of real rank zero, we show that every positive element of A{\mathcal {A}} with norm >1 can be approximated by finite sums of projections. Based on block tri-diagonal approximations, we decompose any positive element A ? M (A ){A\in \mathcal M (\mathcal {A} )} with ${\| A\| >1 }${\| A\| >1 } and || A||_ess 3 1{\| A\|_{ess} \geq 1} into a strictly converging sum of positive elements in A{\mathcal A} with norm >1.     

Localization and Locality for Resistance Forms

For a resistance form ${(X, \mathcal{D}(\varepsilon),\varepsilon)}For a resistance form (X, D(e),e){(X, \mathcal{D}(\varepsilon),\varepsilon)} and a point x₀ ? X{x_0 \in X} as boundary, on the space X₀:=X \{x₀}{X_0:=X {\setminus}\{x_0\}} we consider the Dirichlet space D_x0:={f ? D(e) | f(x₀)=0}{\mathcal{D}_{x_0}:=\{f\in\mathcal{D}(\varepsilon)\, |\, f(x_0)=0\}} and we develop a good potential theory. For any finely open subset D of X ₀ we consider a localized resistance form (D_{X0 \ D},e_D{\mathcal{D}_{X_0 {\setminus} D},\varepsilon_{D}}) where D_{X0 \ D}:={f ? D_x0 | f=0{\mathcal{D}_{X_0 {\setminus} D}:=\{f\in\mathcal{D}_{x_0}\, |\, f=0} on X₀ \ D}, e_D(f,g):=e(f,g){X_0 {\setminus} D\},\, \varepsilon_D(f,g):=\varepsilon(f,g)} for all f,g ? D_{X0 \ D}{f,g\in\mathcal{D}_{X_0 {\setminus} D}}. The main result is the equivalence between the local property of the resistance form and the sheaf property for the excessive elements on finely open sets.     

Boundedness of maximal,potential type,and singular integral operators in the generalized variable exponent Morrey type spaces

We consider generalized Morrey type spaces M^{p( ·),q( ·),w( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets W ì \mathbbRⁿ \Omega \subset {\mathbb{R}^n} , we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with standard kernel. We prove a Sobolev–Adams type embedding theorem M^{p( ·),q₁( ·),w₁( ·)}( W) ? M^{q( ·),q₂( ·),w₂( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) for the potential type operator I ^α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles.     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

Composition Followed by Differentiation Between H ∞ and Zygmund Spaces

Let j{\varphi} be an analytic self-map of the unit disk \mathbbD{\mathbb{D}}, H(\mathbbD){H(\mathbb{D})} the space of analytic functions on \mathbbD{\mathbb{D}} and g ? H(\mathbbD){g \in H(\mathbb{D})}. The boundedness and compactness of the operator DC_j : H^￥ ? Z{DC_\varphi : H^\infty \rightarrow { \mathcal Z}} are investigated in this paper.     

Measure preserving transformations of jump Lévy processes

Let x(t),t ? [ 0,1 ] \xi (t),t \in \left[ {0,1} \right] , be a jump Lévy process. By P_x {\mathcal{P}_\xi } we denote the law of in the Skorokhod space \mathbbD {\mathbb{D}} [0, 1]. Under some nondegeneracy condition on the Lévy measure Λ of the process, we construct a group of P_x {\mathcal{P}_\xi } -preserving transformations of the space \mathbbD {\mathbb{D}} [0, 1]. Bibliography: 10 titles.     

Weighted Non-Trivial Multiply Intersecting Families

Let n and r be positive integers. Suppose that a family satisfies F₁∩···∩F_r ≠∅ for all F₁, . . .,F_r ∈ and . We prove that there exists ε=ε(r) >0 such that holds for 1/2≤w≤1/2+ε if r≥13.     

An abstract setting for hamiltonian actions

In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous 2-cochain ω on a Lie algebra \mathfrak h{\mathfrak h} with values in an \mathfrak h{\mathfrak h}-module V, we associate subalgebras \mathfrak sp(\mathfrak h,w) ê \mathfrak ham(\mathfrak h,w){\mathfrak {sp}(\mathfrak h,\omega) \supseteq \mathfrak {ham}(\mathfrak h,\omega)} of symplectic, resp., hamiltonian elements. Then \mathfrak ham(\mathfrak h,w){\mathfrak {ham}(\mathfrak h,\omega)} has a natural central extension which in turn is contained in a larger abelian extension of \mathfrak sp(\mathfrak h,w){\mathfrak {sp}(\mathfrak h,\omega)}. In this setting, we study linear actions of a Lie group G on V which are compatible with a homomorphism \mathfrak g ? \mathfrak ham(\mathfrak h,w){\mathfrak g \to \mathfrak {ham}(\mathfrak h,\omega)}, i.e., abstract hamiltonian actions, corresponding central and abelian extensions of G and momentum maps J : \mathfrak g ? V{J : \mathfrak g \to V}.     

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.     

Actions of symbolic dynamical systems on C*-algebras. II. Simplicity of C*-symbolic crossed products and some examples

A C*-symbolic dynamical system ${(\mathcal{A}, \rho, \Sigma)}A C*-symbolic dynamical system (A, r, S){(\mathcal{A}, \rho, \Sigma)} consists of a unital C*-algebra A{\mathcal{A}} and a finite family { r_a }_{a ? S}{\{ \rho_\alpha \}_{\alpha \in \Sigma}} of endomorphisms ρ _α of A{\mathcal{A}} indexed by symbols α of Σ satisfying some conditions. The endomorphisms r_a, a ? S{\rho_\alpha, \alpha \in \Sigma } yield both a subshift Λ and a C*-algebra of a Hilbert C*-bimodule. The obtained C*-algebra is regarded as a crossed product of A{\mathcal{A}} by the subshift Λ. We will study simplicity condition of these C*-algebras. Some examples such as irrational rotation Cuntz–Krieger algebras will be studied.     

Regularity of the Anosov distributions of Euler-Lagrange deformations of geodesic flows

Let g be a negatively curved Riemannian metric of a closed C ^∞ manifold M of dimension at least three. Let L _λ be a C ^∞ one-parameter convex superlinear Lagrangian on TM such that L₀(v) = \frac12 g(v, v){L_0(v)= \frac{1}{2} g(v, v)} for any v ∈ TM. We denote by j^l{\varphi^\lambda} the restriction of the Euler-Lagrange flow of L _λ on the \frac12{\frac{1}{2}} -energy level. If λ is small enough then the flow j^l{\varphi^\lambda} is Anosov. In this paper we study the geometric consequences of different assumptions about the regularity of the Anosov distributions of j^l{\varphi^\lambda} . For example, in the case that the initial Riemannian metric g is real hyperbolic, we prove that for λ small, j^l{\varphi^\lambda} has C ³ weak stable and weak unstable distributions if and only if j^l{\varphi^\lambda} is C ^∞ orbit equivalent to the geodesic flow of g.     

On the Right (Left) Invertible Completions for Operator Matrices

Let ${\mathcal {H}_{1}}Let H₁{\mathcal {H}_{1}} and H₂{\mathcal {H}_{2}} be separable Hilbert spaces, and let A ? B(H₁), B ? B(H₂){A \in \mathcal {B}(\mathcal {H}_{1}),\, B \in \mathcal {B}(\mathcal {H}_{2})} and C ? B(H₂, H₁){C \in \mathcal {B}(\mathcal {H}_{2},\, \mathcal {H}_{1})} be given operators. A necessary and sufficient condition is given for ${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)}${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)} to be a right (left) invertible operator for some X ? B(H₁, H₂){X \in \mathcal {B}(\mathcal {H}_{1},\, \mathcal {H}_{2})}. Furthermore, some related results are obtained.