Let $J^{r}T*M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r}T*M)*$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r}T*M)*$ is given.

We classify all $ℱ{ℳ}_{m,n}$-natural operators $:J²⟿J²V^A$ transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections $\left(\Gamma \right):V^{A}Y\to J²V^{A}Y$ on the vertical Weil bundle $V^{A}Y\to M$ corresponding to a Weil algebra A.

Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let ${\pi }_V$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an ${}^m$-measurable subset of ℝⁿ with ${}^m\left(A\right)<\infty$. Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $(V,v)\left|V\in G\right(n,m),v\in V$ such that, for all P ∈ A, one has ${}^{m(n-m)}\left(V\in G(n,m)\left|\right(V,{\pi }_V\left(P\right))\in Z\right)>0$. One can replace “for all P ∈ A” by “for...

We study the problem of how a map f:M → ℝ on an n-manifold M induces canonically an affinor $A\left(f\right):T{T}^{\left(r\right)}M\to T{T}^{\left(r\right)}M$ on the vector r-tangent bundle $T^{\left(r\right)}M=\left(J^r(M,ℝ)₀\right)*$ over M. This problem is reflected in the concept of natural operators $A:T_{|ℳfₙ}^{(0,0)}⇝T^{(1,1)}{T}^{\left(r\right)}$. For integers r ≥ 1 and n ≥ 2 we prove that the space of all such operators is a free (r+1)²-dimensional module over ${}^{\infty }\left(T^{\left(r\right)}ℝ\right)$ and we construct explicitly a basis of this module.

Let A = -Δ + V be a Schrödinger operator on ${ℝ}^d$, d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of $H_A^p$ if the maximal function $su{p}_{t>0}\left|T_{t}f\left(x\right)\right|$ belongs to $L^p\left({ℝ}^d\right)$, where ${T_t}_{t>0}$ is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space $H_A^p$ admits a special atomic decomposition.

It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions $p=2,4$ and $8$, respectively, but the procedure fails for $p=16$ in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are $n=2^7$. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower...

A complete characterization of bounded and unbounded norm hermitian operators on $H_E^{\infty }$ is given for the case when E is a complex Banach space with trivial multiplier algebra. As a consequence, the bi-circular projections on $H_E^{\infty }$ are determined. We also characterize a subclass of hermitian operators on $S_{}^{\infty }$ for a complex Hilbert space.

We prove that the problem of finding all $ℳf_m$-natural operators $B:Q⟿Q{T}^A$ lifting classical linear connections ∇ on m-manifolds M to classical linear connections $B_M\left(\nabla \right)$ on the Weil bundle $T^{A}M$ corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all $ℳf_m$-natural operators $C:Q⟿(T{¹}_{p-1},T*\otimes T*\otimes T)$ transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps $C_M\left(\nabla \right):T{¹}_{p-1}M={⨁}_M^{p-1}TM\to T*M\otimes T*M\otimes TM$.

In this paper we solve the problem of finding integrals of equations determining the Killing tensors on an $n$-dimensional differentiable manifold $M$ endowed with an equiaffine $SL(n,)$-structure and discuss possible applications of obtained results in Riemannian geometry.

Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space ${\Gamma }_X$ of any manifold $X$. The name comes from the fact that various elements of the geometry of ${\Gamma }_X$ are constructed via lifting of the corresponding elements of the geometry of $X$. In this note, we construct a general algebraic framework for lifted geometry which can be applied to various “infinite dimensional spaces” associated to $X$. In order to define...

This paper is about some geometric properties of the gluing of order $k$ in the category of Sikorski differential spaces, where $k$ is assumed to be an arbitrary natural number. Differential spaces are one of possible generalizations of the concept of an infinitely differentiable manifold. It is known that in many (very important) mathematical models, the manifold structure breaks down. Therefore it is important to introduce a more general concept. In this paper, in particular, the behaviour...

We present a large class of homogeneous 2-nondegenerate CR-manifolds $M$, both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CR-equivalence between domains $U$, $V$ in $M$ extends to a global real-analytic CR-automorphism of $M$. We show that this class contains $G$-orbits in Hermitian symmetric spaces $Z$ of compact type, where $G$ is a real form of the complex Lie group $Aut{\left(Z\right)}^0$ and $G$ has an open orbit that is a bounded symmetric domain of tube type. ...

A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ${\nabla }^{\omega }$ on the corresponding tractor bundle $V$, where $\omega$ is the normal Cartan connection. The first operator $D_0$ in the sequence is overdetermined and it is well known that ${\nabla }^{\omega }$ yields the prolongation of this operator in the homogeneous case $M=G/P$. Our first...

We describe all ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝Q{T}^{*}$ transforming projectable-projectable classical torsion-free linear connections $\nabla$ on fibred-fibred manifolds $Y$ into classical linear connections $D\left(\nabla \right)$ on cotangent bundles $T^{*}Y$ of $Y$. We show that this problem can be reduced to finding ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝(T^{*},{\otimes }^p{T}^{*}\otimes {\otimes }^{q}T)$ for $p=2$, $q=1$ and $p=3$, $q=0$.

Let $(M,g)$ be a compact Riemannian manifold and $P_g$ an elliptic, formally self-adjoint, conformally covariant operator of order $m$ acting on smooth sections of a bundle over $M$. We prove that if $P_g$ has no rigid eigenspaces (see Definition 2.2), the set of functions $f\in C^{\infty }(M,ℝ)$ for which $P_{e^{f}g}$ has only simple non-zero eigenvalues is a residual set in $C^{\infty }(M,ℝ)$. As a consequence we prove that if $P_g$ has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics...

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

Let n,r,k be natural numbers such that n ≥ k+1. Non-existence of natural operators $C^r₀⟿Q\left(reg{T}_k^r\to K_k^r\right)$ and $C^r₀⟿Q\left(reg{T}_k^{r*}\to K_k^{r*}\right)$ over n-manifolds is proved. Some generalizations are obtained.