On continuous solutions of a problem of R.Schilling

Abstract. It is proved that if $$ q \ \in \lbrace({\sqrt 3}-1)/2,(3-{\sqrt 5})/2,{\sqrt 2}-1,({\sqrt 5}-1)/2\rbrace $$ then the zero function is the only solution ?: ? → ? of (1) satisfying (2) and right-hand-side or left-hand-side continuous at each point of the interval (?q/(1 ? q), ?q/(1 ? g) + δ) or of the interval (q/(1 ? q) ? δ, q/(1 ? q)) with some δ > 0.     

Relations between the K-loop and the defect of an absolute plane

Let ${\cal P}$ be the point set of an absolute plane, let ${\cal {\tilde P}}$ be the set of all point reflections, let ?, resp. ?⁺, be the group of all, resp. of all proper, motions and let $$^\sim:{\cal P\times P\rightarrow \tilde P};\ \ \ (a,\ b)\mapsto\ \widetilde {a,\ b}$$ be the map where ${\widetilde {a,\ b}}$ denotes the uniquely determined point-reflection interchanging a and b. Then $$\delta\:\ {\cal P}^{3}\rightarrow {\cal M}^{+};\ \ \ (a,b,c)\mapsto \delta_{a;b,c}\:=\ {\tilde a}\ {\rm o}\ \widetilde {a,\ b}\ {\rm o}\ \widetilde {b,\ c}\ {\rm o}\ \widetilde {c,\ a}$$ is called the defect function, or shortly the defect. We show that δ_a;b,c is a rotation around the point a where the angle of δ_a;b,c is exactly the angle defect of the triangle (a, b, c) (cfr. 3.5). After fixing a point $o\ \in {\cal P}$ and setting $a+b\:=\widetilde {o,\ a}\ {\rm o}\ {\tilde o}\ (b),\ ({\cal P},+)$ becomes a K-loop and the so called precession function $$\delta_{a,b}\:=\ \big((a+b)^{+}\big)^{-1}\ {\rm o}\ a^{+}\ {\rm o}\ b^+$$ of the loop ( ${\cal P}, +)$ coincides with the defect of the triangle (o, a, ?b) (cfr. (4.4.1)), hence δ_a,b = δ_o;a,?b for all $a, b \in {\cal P}$ . With the order relation of the absolute plane we associate an orientation function $$\Omega\:\ \Delta\ \times \Delta \rightarrow \lbrace -1,+1\rbrace$$ defined on the pairs of triangles (cfr. (2.8)). If (a, b, c) ∈ Δ is a triangle and d a point of the line $\overline {b,\ c}$ ¹, then (cfr. (3.9.2)): $$\delta_{a;b,c}\ {\rm o}\ \delta_{a;c,d}=\delta_{a;b,d}$$ and moreover, if d is even a point of the open segment ]b, c[ then (cfr. (2.8.5)): $$\Omega(a,\ b,\ c;\ a,\ b,\ d)=\Omega(a,\ b,\ d;\ a,\ d,\ c)=+1.$$ Thus the angle defect of the triangle (a, b, c) is the sum of the angle defects of the triangles (a, b, d) and (a, d, c).     

Conditionally oscillatory half-linear differential equations

We consider a nonoscillatory half-linear second order differential equation (*) $$ (r(t)\Phi (x'))' + c(t)\Phi (x) = 0,\Phi (x) = \left| x \right|^{p - 2} x,p > 1, $$ and suppose that we know its solution h. Using this solution we construct a function d such that the equation (**) $$ (r(t)\Phi (x'))' + [c(t) + \lambda d(t)]\Phi (x) = 0 $$ is conditionally oscillatory. Then we study oscillations of the perturbed equation (**). The obtained (non)oscillation criteria extend existing results for perturbed half-linear Euler and Euler-Weber equations.     

On a Functional Equation Associated with Simpson’s Rule

In this paper, we determine the general solution of the functional equation $$f(x)-g(y)=(x-y)\lbrack h(x+y)+\psi (x)+\phi (y)\rbrack$$ for all real numbers x and y. This equation arises in connection with Simpson’s Rule for the numerical evaluation of definite integrals. The solution of this functional equation is achieved through the functional equation $$g(x)-g(y)=(x-y)f(x+y)+(x+y)f(x-y).$$      

Sharp weak-type inequalities for Hilbert transform and Riesz projection

The paper is devoted to the study of the weak norms of the classical operators in the vector-valued setting.

1. Let S, H denote the singular integral involution operator and the Hilbert transform on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {\mathcal{S}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p,$$ $$\left\| {\mathcal{H}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p.$$ Both inequalities are sharp.
2. Let P ₊ and P _? stand for the Riesz projection and the co-analytic projection on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {P + f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p ,$$ $$\left\| {P - f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p .$$ Both inequalities are sharp.
3. We establish the sharp versions of the estimates above in the nonperiodic case.

The results are new even if the operators act on complex-valued functions. The proof rests on the construction of an appropriate plurisubharmonic function and probabilistic techniques.     

Gaussian Integral Means of Entire Functions

For an entire function \(f:\mathbb C\mapsto \mathbb C\) and a triple \((p,\alpha , r)\in (0,\infty )\times (-\infty ,\infty )\times (0,\infty ]\) , the Gaussian integral mean of \(f\) (with respect to the area measure \(dA\) ) is defined by $$\begin{aligned} {\mathsf M}_{p,\alpha }(f,r)=\left( \,\, {\int \limits _{|z| Via deriving a maximum principle for \({\mathsf M}_{p,\alpha }(f,r)\) , we establish not only Fock–Sobolev trace inequalities associated with \({\mathsf M}_{p,p/2}(z^m f(z),\infty )\) (as \(m=0,1,2,\ldots \) ), but also convexities of \(r\mapsto \ln {\mathsf M}_{p,\alpha }(z^m,r)\) and \(r\mapsto {\mathsf M}_{2,\alpha <0}(f,r)\) in \(\ln r\) with \(0 .     

A sequential implicit function theorem for the chords iteration

In this paper we study the local convergence of the method $$0 \in f\left( {p,x_k } \right) + A\left( {x_{k + 1} - x_k } \right) + F\left( {x_{k + 1} } \right),$$ in order to find the solution of the generalized equation $$find x \in X such that 0 \in f\left( {p,x} \right) + F\left( x \right).$$ We first show that under the strong metric regularity of the linearization of the associated mapping and some additional assumptions regarding dependence on the parameter and the relation between the operator A and the Jacobian $\nabla _x f\left( {\bar p,\bar x} \right)$ , we prove linear convergence of the method which is uniform in the parameter p. Then we go a step further and obtain a sequential implicit function theorem describing the dependence of the set of sequences of iterates of the parameter.     

Two results on arithmetical functions

Generalizing two results of Rieger [8] and Selberg [10] we give asymptotic formulas for sums of type $${\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n)\qquad {\rm and} {\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n),$$ where χ is a suitable multiplicative function, f₁,…, f _r are “small” additive, prime-independent arithmetical functions and k, l are coprime. The proofs are based on an analytic method which consists of considering the Dirichlet series generated by $ \chi(n)z_{1}^{f_{1}(n)}\cdot... \cdot z_{r}^{f_{r}(n)},z_{1}\dots z_{r} $ complex.     

On the functional equation x + f(y + f(x))=y + f(x + f(y))

For an abelian group (G, + ,0) we consider the functional equation $$f : G \to G, x + f(y + f(x)) = y + f(x + f(y)) \quad (\forall x, y \in G), \quad\quad\qquad (1)$$ most times together with the condition $$f(0) = 0.\qquad\qquad\qquad\qquad\qquad (0)$$ Our main question is whether a solution of ${(1) \wedge (0)}$ must be additive, i.e., an endomorphism of G. We shall answer this question in the negative (Example 3.14) Rätz (Aequationes Math 81:300, 2011).     

Pointwise estimates for Lagrange interpolation polynomials

Let f ∈ C[?1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes $ \left\{ {\cos \frac{{2k - 1}} {{2n}}\pi } \right\} \cup \{ - 1,1\} $ be Δ _{n + 2}(f, x). In this paper we study the estimate of Δ _{n + 2}(f,x), that keeps the interpolation property. As a result we prove that $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left| {T_n (x)} \right|\ln (n + 1) + \omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}\left| {T_n (x)} \right|} \right)} \right\}, $$ where T _n (x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if f ∈ C ^r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$      

On the stability of Hosszú’s functional equation

Two stability results are proved. The first one states that Hosszú’s functional equation $$f(x+y-xy)+f(xy)=f(x)-f(y)=0\ \ \ \ \ (x,y \in \rm R)$$ is stable. The second is a local stability theorem for additive functions in a Banach space setting.     

On The Generalized Cocycle Equation of Ebanks and Ng

I show that in order to solve the functional equation $$F_{1}(x+y,z)+F_{2}(y+z,x)F_{3}(z+x,\ y)+F_{4}(x,y)+F_{5}(y,z)+F_{6}(z,x)=0$$ for six unknown functions (x,y,z are elements of an abelian monoid, and the codomain of each F _j is the same divisible abelian group) it is necessary and sufficient to solve each of the following equations in a single unknown function $$\matrix{\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad G(x+y,\ z)- G(x,z)- G(y,z)=G(y+z,x)- G(y,x)- G(z,x)\cr \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad H(x+y,\ z)- H(x,z)- H(y,x)+H(y+z,\ x)- H(y,x)- H(z,x)\cr +H(z+x,\ y)- H(z,y)- H(x,y)=0.}$$      

An estimate for the sum of Legendre symbols

For the sum S of the Legendre symbols of a polynomial of odd degree n ≥ 3 modulo primes p ≥ 3, Weil’s estimate |S| ≤ (n ? 1) $ \sqrt p $ and Korobov’s estimate $$ \left| S \right| \leqslant (n - 1)\sqrt {p - \frac{{(n - 3)(n - 4)}} {4}} forp \geqslant \frac{{n^2 + 9}} {2} $$ are well known. In this paper, we prove a stronger estimate, namely, $$ \left| S \right| < (n - 1)\sqrt {p - \frac{{(n - 3)(n + 1)}} {4}} $$ .     

Parabolic power concavity and parabolic boundary value problems

This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem $$\begin{aligned} (P)\quad \left\{ \begin{array}{l@{\quad }l} \partial _t u=\varDelta u +f(x,t,u,\nabla u) &{} \text{ in }\quad \varOmega \times (0,\infty ),\\ u(x,t)=0 &{} \text{ on }\quad \partial \varOmega \times (0,\infty ),\\ u(x,0)=0 &{} \text{ in }\quad \varOmega , \end{array} \right. \end{aligned}$$ where $\varOmega $ is a bounded convex domain in $\mathbf{R}^n$ and $f$ is a nonnegative continuous function in $\varOmega \times (0,\infty )\times \mathbf{R}\times \mathbf{R}^n$ . We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\overline{\varOmega }\times [0,\infty )$ .     

Existence and extremal solutions of parabolic variational–hemivariational inequalities

We consider quasilinear parabolic variational–hemivariational inequalities in a cylindrical domain $Q=\Omega \times (0,\tau )$ of the form $$\begin{aligned} u\in K:\ \langle u_t+Au, v-u\rangle +\int _Q j^o(x,t, u;v-u)\,dxdt\ge 0,\ \ \forall \ v\in K, \end{aligned}$$ where $K\subset X_0=L^p(0,\tau ;W_0^{1,p}(\Omega ))$ is some closed and convex subset, $A$ is a time-dependent quasilinear elliptic operator, and $s\mapsto j(\cdot ,\cdot ,s)$ is assumed to be locally Lipschitz with $(s,r)\mapsto j^o(x,t, s;r)$ denoting its generalized directional derivative at $s$ in the direction $r$ . The main goal of this paper is threefold: first, an existence and comparison principle is proved; second, the existence of extremal solutions within some sector of appropriately defined sub-supersolutions is shown; third, the equivalence of the above parabolic variational–hemivariational inequality with an associated multi-valued parabolic variational inequality of the form $$\begin{aligned} u\in K:\ \langle u_t+Au, v-u\rangle +\int _Q \eta \, (v-u)\,dxdt\ge 0,\ \ \forall \ v\in K \end{aligned}$$ with $\eta (x,t)\in \partial j(x,t, u(x,t))$ is established, where $s\mapsto \partial j(x,t, s)$ denotes Clarke’s generalized gradient of the locally Lipschitz function $s\mapsto j(\cdot ,\cdot ,s)$ .     

New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

We obtain Hardy type inequalities $$\int_0^\infty {M\left( {\omega \left( r \right)\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr} \leqslant C_1 \int_0^\infty {M\left( {\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr + C_2 \int_0^\infty {M\left( {\left| {u'\left( r \right)} \right|} \right)\rho \left( r \right)dr,} }$$ and their Orlicz-norm counterparts $$\left\| {\omega u} \right\|_{L^M (\mathbb{R}_ + ,\rho )} \leqslant \tilde C_1 \left\| u \right\|_{L^M (\mathbb{R}_ + ,\rho )} + \tilde C_2 \left\| {u'} \right\|_{L^M (\mathbb{R}_ + ,\rho )} ,$$ with an N-function M, power, power-logarithmic and power-exponential weights ??, ??, holding on suitable dilation invariant supersets of C ₀ ^?? (?₊). Maximal sets of admissible functions u are described. This paper is based on authors?? earlier abstract results and applies them to particular classes of weights.     

A Barrier Principle for Surfaces with Prescribed Mean Curvature and Arbitrary Codimension

We consider two dimensional surfaces ${X : \Omega\to\mathbb R^{n+2}, \Omega\subset \mathbb C, w=u+iv\mapsto X(w)}$ with arbitrary codimension n and prove a barrier principle for strong (possibly branched) subsolutions ${X\in C^1(\Omega, \mathbb {R}^{n+2})\cap H_{2,{\rm loc}}^2(\Omega,\mathbb R^{n+2})}$ of the integral inequality $$\int_{\Omega} \Big\lbrace \langle \nabla X, \nabla \varphi\rangle +2W \sum_{k=1}^n H_k \langle N_k,\varphi \rangle \Big\rbrace \; dudv\ge 0$$ with mean curvature functions (H _k ) _k=1,...,n which lie locally on one side of a supporting hypersurface S. We show under suitable assumption on the 2-mean curvature of the supporting surface S that X is locally contained in S. This generalizes a corresponding result for surfaces in ${\mathbb R^3}$ , cf. (Dierkes et al., Regularity of Minimal Surfaces, §4.4, 2010).     

The equivalence of two functional equations involving the arithmetic mean, the geometric mean and their Gauss composition

In this paper, the equivalence of the two functional equations $$f\left(\frac{x+y}{2} \right)+f\left(\sqrt{xy} \right)=f(x)+f(y)$$ and $$2f\left(\mathcal{G}(x,y)\right)=f(x)+f(y)$$ will be proved by showing that the solutions of either of these equations are constant functions. Here I is a nonvoid open interval of the positive real half-line and ${\mathcal{G}}$ is the Gauss composition of the arithmetic and geometric means.     

On a functional equation related to competition

The functional equation $$f \left(\frac{x + y}{1 - xy}\right) = \frac{f\left(x\right) + f\left(y\right)} {1 + f\left(x\right) f\left(y\right)}, \quad xy < 1,$$ (introduced by the first author in a competition model) is considered. The main result says that a function \({f : \mathbb{R} \rightarrow \mathbb{R}}\) satisfies this equation if, and only if, \({f = {\rm tanh} \circ \, \alpha \circ {\rm tan}^{-1}}\) , where \({\alpha : \mathbb{R} \rightarrow \mathbb{R}}\) is an additive function.     

Applications of weighted estimates of Calderon’s commutators

The paper introduces singular integral operators of a new type defined in the space L ^p with the weight function on the complex plane. For these operators, norm estimates are derived. Namely, if V is a complex-valued function on the complex plane satisfying the condition |V(z) ? V(??)| ?? w|z ? ??| and F is an entire function, then we put $$P_F^* f(z) = \mathop {\sup }\limits_{\varepsilon > 0} \left| {\int\limits_{\left| {\zeta - z} \right| > \varepsilon } {F\left( {\frac{{V(\zeta ) - V(z)}} {{\zeta - z}}} \right)\frac{{f(\zeta )}} {{\left( {\zeta - z} \right)^2 }}d\sigma (\zeta )} } \right|.$$ It is shown that if the weight function ?? is a Muckenhoupt A _p weight for 1 < p < ??, then $$\left\| {P_F^* f} \right\|_{p,\omega } \leqslant C(F,w,p)\left\| f \right\|_{p,\omega } .$$ .