We study local reflections ${\varphi }_{\sigma }$ with respect to a curve $\sigma$ in a Riemannian manifold and prove that $\sigma$ is a geodesic if ${\varphi }_{\sigma }$ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if ${\varphi }_{\sigma }$ is harmonic for all geodesies $\sigma$.

Let $\Omega \subset {ℝ}^n,n\ge 3$, and let $p,1<p<\infty ,p\ne 2$, be given. In this paper we study the dimension of $p$-harmonic measures that arise from non-negative solutions to the $p$-Laplace equation, vanishing on a portion of $\partial \Omega$, in the setting of $\delta$-Reifenberg flat domains. We prove, for $p\ge n$, that there exists $\tilde{\delta }=\tilde{\delta }(p,n)>0$ small such that if $\Omega$ is a $\delta$-Reifenberg flat domain with $\delta <\tilde{\delta }$, then $p$-harmonic measure is concentrated on a set of $\sigma$-finite $H^{n-1}$-measure. We prove, for $p\ge n$, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$-harmonic...

If $(M,\nabla )$ is a manifold with a symmetric linear connection, then $T^{*}M$ can be endowed with the natural Riemann extension $\overline{g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\overline{g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $𝒫$ on $(T^{*}M,\overline{g})$ and prove that $𝒫$ is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\overline{g}$ reduces...

The harmonic Cesàro operator is defined for a function f in $L^p\left(ℝ\right)$ for some 1 ≤ p < ∞ by setting $\left(f\right)\left(x\right):={\int }_x^{\infty }(f\left(u\right)/u)du$ for x > 0 and $\left(f\right)\left(x\right):=-{\int }_{-\infty }^x(f\left(u\right)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L{¹}_{loc}\left(ℝ\right)$ by setting $*\left(f\right)\left(x\right):=(1/x){\int }^{x₀}f\left(u\right)du$ for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If $f\in L^p\left(ℝ\right)$ for some 1 ≤ p ≤ 2, then ${\left(\left(f\right)\right)}^{\wedge }\left(t\right)=*\left(f̂\right)\left(t\right)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f\in L^p\left(ℝ\right)$ for some 1 < p ≤ 2, then...

We consider the class ${\mathscr{H}}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $\left|z\right|<1$ and normalized so that $h\left(0\right)=0=h^{\text{'}}\left(0\right)-1$ and $g\left(0\right)=0=g^{\text{'}}\left(0\right)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes ${𝒫}_H^0\left(\alpha \right)$ and ${𝒢}_H^0\left(\beta \right)$ of functions from ${\mathscr{H}}_0$ and show that if $f\in {𝒫}_H^0\left(\alpha \right)$ and $F\in {𝒢}_H^0\left(\beta \right)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $\alpha$ and $\beta$ are satisfied. In the second part we study the harmonic sections...

Let us consider two closed surfaces $ℳ$, $𝒩$ of class $C^1$ and two functions $\varphi :ℳ\to ℝ$, $\psi :𝒩\to ℝ$ of class $C^1$, called measuring functions. The natural pseudodistance $d$ between the pairs $(ℳ,)$, $(𝒩,\psi )$ is defined as the infimum of $\Theta \left(f\right):={max}_{P\in ℳ}\left|\varphi \left(P\right)-\psi \left(f\left(P\right)\right)\right|$ as $f$ varies in the set of all homeomorphisms from $ℳ$ onto $𝒩$. In this paper we prove that the natural pseudodistance equals either $|c_1-c_2|$, $\frac{1}{2}\left|c_1-c_2\right|$, or $\frac{1}{3}\left|c_1-c_2\right|$, where $c_1$ and $c_2$ are two suitable critical values of the measuring functions. This shows that a previous relation between the natural pseudodistance and...

Let ${\left(h_k\right)}_{k\ge 0}$ be the Haar system on [0,1]. We show that for any vectors $a_k$ from a separable Hilbert space and any ${\epsilon }_k\in [-1,1]$, k = 0,1,2,..., we have the sharp inequality $\left|\right|{\sum }_{k=0}^n{\epsilon }_k{a}_k{h}_k{\left|\right|}_{W\left(\right[0,1\left]\right)}\le 2\left|\right|{\sum }_{k=0}^n{a}_k{h}_k{\left|\right|}_{L^{\infty }\left([0,1]\right)}$, n = 0,1,2,..., where W([0,1]) is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ${\left|\right|Y\left|\right|}_{W\left(\Omega \right)}{\le 2\left|\right|X\left|\right|}_{L^{\infty }\left(\Omega \right)}$, where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

For odd-dimensional Poincaré–Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for $k<n/2$) which are $C^m$ (with $m\in \mathbb{N}$) on the conformal compactification $\overline{X}$ of $X$. This set is infinite-dimensional for small $m$ but it becomes finite-dimensional if $m$ is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology $H^k(\overline{X},\partial \overline{X})$ and the kernel of the Branson–Gover [3] differential operators $(L_k,G_k)$ on the conformal infinity $(\partial \overline{X},\left[h_0\right])$. We also relate the set of $C^{n-2k+1}\left({\Lambda }^k\left(\overline{X}\right)\right)$ forms in the kernel of $d+{\delta }_g$...

If $(M,g)$ is a Riemannian manifold, we have the well-known base preserving   vector bundle isomorphism $TM\widetilde{=}{T}^{*}M$ given by $v\to g(v,-)$ between the tangent $TM$ and the cotangent $T^{*}M$ bundles of $M$. In the present note, we generalize this isomorphism to the one $T^{\left(r\right)}M\widetilde{=}{T}^{r*}M$ between the $r$-th order vector tangent $T^{\left(r\right)}M={\left(J^r{(M,R)}_0\right)}^{*}$ and the $r$-th order cotangent $T^{r*}M=J^r{(M,R)}_0$ bundles of $M$. Next, we describe all base preserving  vector bundle maps $C_M\left(g\right):T^{\left(r\right)}M\to T^{r*}M$ depending on a Riemannian metric $g$ in terms of natural (in $g$) tensor fields on $M$.

We determine in ${ℝ}^n$ the form of curves $C$ corresponding to strictly monotone functions as well as the components of affine connections $\nabla$ for which any image of $C$ under a compact-free group $\Omega$ of affinities containing the translation group is a geodesic with respect to $\nabla$. Special attention is paid to the case that $\Omega$ contains many dilatations or that $C$ is a curve in ${ℝ}^3$. If $C$ is a curve in ${ℝ}^3$ and $\Omega$ is the translation group then we calculate not only the components of the curvature and the Weyl...

The relationship between weighted Lipschitz functions and analytic Bloch spaces has attracted much attention. In this paper, we define harmonic $\omega$-$\alpha$-Bloch space and characterize it in terms of $$\omega \left(\right(1-{\left|x\right|}^2{)}^{\beta }{(1-|y|}^2{)}^{\alpha -\beta })\left|\frac{f\left(x\right)-f\left(y\right)}{x-y}\right|$$ and $$\omega \left(\right(1-{\left|x\right|}^2{)}^{\beta }{(1-|y|}^2{)}^{\alpha -\beta })\left|\frac{f\left(x\right)-f\left(y\right)}{\left|x\right|y-x^{\text{'}}}\right|$$ where $\omega$ is a majorant. Similar results are extended to harmonic little $\omega$-$\alpha$-Bloch and Besov spaces. Our results are generalizations of the corresponding ones in G. Ren, U. Kähler (2005).

In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping $F$ in the unit disk $𝔻$, if $F\left(𝔻\right)$ is a convex domain, then the inequality $|G\left(z_2\right)-G\left(z_1\right)|<|H\left(z_2\right)-H\left(z_1\right)|$ holds for all distinct points $z_1,z_2\in 𝔻$. Here $H$ and $G$ are holomorphic mappings in $𝔻$ determined by $F=H+\overline{G}$, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain $\Omega$ in $ℂ$ and improve it provided $F$ is additionally a quasiconformal mapping...

We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from $S^1$ to the plane modulo the group of diffeomorphisms of $S^1$, acting as reparametrizations. In particular we investigate the metric, for a constant $A>0$, $G_c^A(h,k):={\int }_{S^1}(1+A{\kappa }_c{\left(\theta \right)}^2)\langle h\left(\theta \right),k\left(\theta \right)\rangle \left|c^{\text{'}}\left(\theta \right)\right|d\theta$ where ${\kappa }_c$ is the curvature of the curve $c$ and $h$, $k$ are normal vector fields to $c$. The term $A{\kappa }^2$ is a sort of geometric Tikhonov regularization because, for $A=0$, the geodesic distance between any two distinct curves is 0, while...

In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1}u=-{\mu }_{1}v$, $L_{2}v=-{\mu }_{2}u$, on a domain $D$ of ${ℝ}^d$, where ${\mu }_1$ and ${\mu }_2$ are suitable measures on $D$, and $L_1$, $L_2$ are two second order linear differential elliptic operators on $D$ with coefficients of class ${𝒞}^{\infty }$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_1$ and $L_2$, and...

For $n=2m\ge 4$, let $\Omega \in {ℝ}^n$ be a bounded smooth domain and $𝒩\subset {ℝ}^L$ a compact smooth Riemannian manifold without boundary. Suppose that $\left\{u_k\right\}\in W^{m,2}(\Omega ,𝒩)$ is a sequence of weak solutions in the critical dimension to the perturbed $m$-polyharmonic maps $$\frac{\mathrm{d}}{\mathrm{d}t}{|}_{t=0}{E}_m\left(\Pi (u+t\xi )\right)=0$$ with ${\Phi }_k\to 0$ in ${\left(W^{m,2}(\Omega ,𝒩)\right)}^{*}$ and $u_k\rightharpoonup u$ weakly in $W^{m,2}(\Omega ,𝒩)$. Then $u$ is an $m$-polyharmonic map. In particular, the space of $m$-polyharmonic maps is sequentially compact for the weak-$W^{m,2}$ topology.

Si costruiscono curve di genere $g=4n\mathrm{—}3$, $n\ge 3$ che hanno $2^{n-3}(2^{n-2}-1)$ fasci semicanonici $L$ tali che $h^0(L)=4$. Per $n+3$ si dimostra che gli $L$ sono molto ampi.

We prove that a connected Riemannian manifold admitting a pair of non-trivial Einstein-Weyl structures $(g,\pm \omega )$ with constant scalar curvature is either Einstein, or the dual field of $\omega$ is Killing. Next, let $(M^n,g)$ be a complete and connected Riemannian manifold of dimension at least $3$ admitting a pair of Einstein-Weyl structures $(g,\pm \omega )$. Then the Einstein-Weyl vector field $E$ (dual to the $1$-form $\omega$) generates an infinitesimal harmonic transformation if and only if $E$ is Killing.

Given a quasisymmetric automorphism $\gamma$ of the unit circle $𝕋$ we define and study a modification $P_{\gamma }$ of the classical Poisson integral operator in the case of the unit disk $𝔻$. The modification is done by means of the generalized Fourier coefficients of $\gamma$. For a Lebesgue’s integrable complexvalued function $f$ on $𝕋$, $P_{\gamma }\left[f\right]$ is a complex-valued harmonic function in $𝔻$ and it coincides with the classical Poisson integral of $f$ provided $\gamma$ is the identity mapping on $𝕋$. Our considerations are motivated by...

A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation ${\Delta }^{p}f=0$, where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f:\Omega \to {ℂ}^m$ is said to be p-harmonic in Ω if each component function $f_i$ (i∈ 1,...,m) of $f=(f₁,...,f_m)$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings...

Siano $d$, $g$, $t$ interi con $0\le t\le g$; se esiste in ${\mathbb{P}}_3(ℂ)$ una curva connessa, non singolare di grado $d$ e genere $g$, allora esiste in ${\mathbb{P}}_3(ℂ)$ una curva irriducibile di grado $d$, genere aritmetico $g$ e $t$ nodi.

Ogni logica $L$ genera canonicamente la $L$-equivalenza ${\equiv }_L$e la $L$-immersione $\underset{𝐿}{\to }$ proprio come la logica del primo ordine genera l’equivalenza elementare $\equiv$ e l’immersione elementare $\lesssim$. Astraendo da $L$, è interessante studiare in sè relazioni d’equivalenza e di immersione generali tra strutture. Mostriamo che esiste una corrispondenza biunivoca tra relazioni d’equivalenza con la proprietà di Robinson e relazioni d’immersione con la proprietà di Amalgamazione Forte ($A{P}^{+}$). Caratterizziamo algebricamente...

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.

Given any compact manifold $M$, we construct a non-empty open subset $𝒪$ of the space ${\mathrm{Diff}}^1\left(M\right)$ of $C^1$-diffeomorphisms and a dense subset $𝒟\subset 𝒪$ such that the centralizer of every diffeomorphism in $𝒟$ is uncountable, hence non-trivial.