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 $\sigma$.

We construct biharmonic non-harmonic maps between Riemannian manifolds $(M,g)$ and $(N,h)$ by first making the ansatz that $\varphi :(M,g)\to (N,h)$ be a harmonic map and then deforming the metric on $N$ by $${\tilde{h}}_{\alpha }=\alpha h+(1-\alpha )df\otimes df$$ to render $\varphi$ biharmonic, where $f$ is a smooth function with gradient of constant norm on $(N,h)$ and $\alpha \in (0,1)$. We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

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...

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...

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both $p(·)$-Harmonic and $p(·)$-biharmonic operators $$\begin{array}{c}{\Delta }_{p\left(x\right)}^{2}u-{\Delta }_{p\left(x\right)}u=\lambda w\left(x\right){\left|u\right|}^{q\left(x\right)-2}u\text{in}\Omega ,\\ u\in W^{2,p(·)}\left(\Omega \right)\cap W_0^{1,p(·)}\left(\Omega \right),\end{array}$$ is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^{p(·)}\left(\Omega \right)$ and $W^{m,p(·)}\left(\Omega \right)$.

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...

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 the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset $U$ of a Brelot $𝒫$-harmonic space $\Omega$ with countable base of open subsets and satisfying the axiom $D$. When $\Omega$ satisfies the hypothesis of uniqueness, we define the Martin boundary of $U$ and the Martin kernel $K$ and we obtain the integral representation of invariant functions by using the kernel $K$. As an application of the integral representation we extend to the cone...

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...

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).

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 $d+{\delta }_g$...

The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form $f\left(e^{it}\right)=e^{i\phi \left(t\right)}$, $0\le t\le 2\pi$ where $\phi$ is a continuously non-decreasing function that satisfies $\phi \left(2\pi \right)-\phi \left(0\right)=2N\pi$, assume every value finitely many times in the disc?

We deal with complete submanifolds with weighted Poincaré inequality. By assuming the submanifold is $\delta$-stable or has sufficiently small total curvature, we establish two vanishing theorems for $L^p$ harmonic $1$-forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitório.

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...

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 $(X,d,\mu )$ be a metric measure space satisfying the doubling condition and an $L^2$-Poincaré inequality. Consider the nonnegative operator $ℒ$ generalized by a Dirichlet form on $X$. We will show that a solution $u$ to $(-{\partial }_t^2+ℒ)u=0$ on $X\times {ℝ}_{+}$ satisfies an $\alpha$-Carleson condition if and only if $u$ can be represented as the Poisson integral of the operator $ℒ$ with the trace in the generalized Morrey space $L^{2,\alpha }\left(X\right)$, where $\alpha$ is a nonnegative function defined on a class of balls in $X$. This result extends the analogous characterization...

For k = 1,2,... let $H_k$ denote the harmonic number ${\sum }_{j=1}^{k}1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have ${\sum }_{k=1}^{p-1}\left(H_k\right)/\left(k{2}^k\right)\equiv 7/24p{B}_{p-3}\left(modp²\right)$, ${\sum }_{k=1}^{p-1}\left(H_{k,2}\right)/\left(k{2}^k\right)\equiv -3/8B_{p-3}\left(modp\right)$, and ${\sum }_{k=1}^{p-1}\left(H{²}_{k,2n}\right)/\left(k^{2n}\right)\equiv (\left(\genfrac{}{}{0pt}{}{6n+1}{2n-1}\right)+n)/(6n+1)p{B}_{p-1-6n}\left(modp²\right)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k,m}:={\sum }_{j=1}^{k}1/\left(j^m\right)$.