Asymptotic formula for the mean value of a multiple trigonometric sum

When k≥k₀=10 Mr²n log (rn) we have for the trigonometric integral $$J_n (k,P) = \int_E {|S(A)|^{2k} dA,} $$ where $$\begin{gathered} S(A) = \sum _{x_1 = 1}^P \cdots \sum _{x_r = 1}^P \exp (2\pi if_A (x_1 , \ldots ,x_r )), \hfill \\ f_A (x_1 , \ldots ,x_r ) = \sum _{t_1 = 0}^n \cdots \sum _{t_r = 0}^n \alpha _{t_1 \cdots l_r } x_1^{t_1 } \cdots x_{r^r }^t \hfill \\ \end{gathered} $$ and E is the M-dimensional unit cube, the asymptotic formula $$J_n (k,P) = \sigma \theta P^{2kr - rnM/2} + O(P^{2kr - rnM/2 - 1/(2M)} ) + O(P^{2kr - rnM/2 - 1/(500r^2 \log (rn))} ),$$ where σ is a singular series and θ is a singular integral.     

Nearly Quartic Mappings in β-Homogeneous F-Spaces

In this paper, we investigate the Hyers–Ulam stability of the following quartic equation $$\begin{array}{ll} {\sum\limits^{n}_{k=2}}\left({\sum\limits^{k}_{i_{1}=2}}{\sum\limits^{k+1}_{i_{2}=i_{1}+1}} \ldots {\sum\limits^{n}_{i_{n-k+1}=i_{n-k}+1}}\right)\\ \quad\times f \left({\sum\limits^{n}_{i=1,i \neq i_{1},\ldots,i_{n-k+1}}} x_{i}-{\sum\limits^{n-k+1}_{r=1}}x_{i_{r}}\right) + f \left({\sum\limits^{n}_{i=1}}x_{i}\right)\\ \quad-2^{n-2}{\sum\limits^{}_{1 \leq{i} \leq{j} \leq{n}}}(f(x_{i} + x_{j}){+f(x_{i} - x_{j})){+2^{n-5}(n - 2){\sum\limits^{n}_{i=1}}f(2x_{i})}} = \theta \end{array} $$ $({n \in \mathbb{N}, n \geq 3})$ in β-homogeneous F-spaces.     

Hyperbolic Laplace Operator and the Weinstein Equation in {\mathbb{R}^3}

We study the Weinstein equation $$\Delta u - \frac{k}{{x}_{2}} \frac{\partial}{\partial{x}_{2}} + \frac{l}{x^{2}_{2}}u = 0$$ , on the upper half space ${\mathbb{R}^3_{+} = \{ (x_{0}, x_{1}, x_{2}) \in \mathbb{R}^{3} | x_2 > 0\}}$ in case ${4l \leq (k + 1)^{2}}$ . If l =  0 then the operator ${x^{2k}_{2} (\Delta - \frac{k}{x_{2}} \frac{\partial}{\partial{x}_{2}})}$ is the Laplace- Beltrami operator of the Riemannian metric ${ds^2 = x^{-2k}_{2} (\sum^{2}_{i = 0} dx^{2}_{i})}$ . The general case ${\mathbb{R}^{n}_{+}}$ has been studied earlier by the authors, but the results are improved in case ${\mathbb{R}^3_{+}}$ . If k =  1 then the Riemannian metric is the hyperbolic distance of Poincaré upper half-space. The Weinstein equation is connected to the axially symmetric potentials. We compute solutions of the Weinstein equation depending only on the hyperbolic distance and x ₂. The solutions of the Weinstein equation form a socalled Brelot harmonic space and therefore it is known that they satisfy the mean value properties with respect to the harmonic measure. However, without using the theory of Brelot harmonic spaces, we present the explicit mean value properties which give a formula for a harmonic measure evaluated in the center point of the hyperbolic ball. Earlier these results were proved only for k =  1 and l =  0 or k =  1 and l =  1. We also compute the fundamental solutions. The main tools are the hyperbolic metric and its invariance properties. In the consecutive papers, these results are applied to find explicit kernels for k-hypermonogenic functions that are higher dimensional generalizations of complex holomorphic functions.     

Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators

Let λkbe the k-th Dirichlet eigenvalue of totally characteristic degenerate elliptic operator-ΔB defined on a stretched cone B0 ■ [0,1) × X with boundary on {x1 = 0}. More precisely,ΔB=(x1αx1)2+ α2x2+ + α2xnis also called the cone Laplacian. In this paper,by using Mellin-Fourier transform,we prove thatλk Cnk2 n for any k 1,where Cn=(nn+2)(2π)2(|B0|Bn)-2n,which gives the lower bounds of the Dirchlet eigenvalues of-ΔB. On the other hand,by using the Rayleigh-Ritz inequality,we deduce the upper bounds ofλk,i.e.,λk+1 1 +4n k2/nλ1. Combining the lower and upper bounds of λk,we can easily obtain the lower bound for the first Dirichlet eigenvalue λ1 Cn(1 +4n)-12n2.     

Harmonic divisors and rationality of zeros of Jacobi polynomials

Let $P_{n}^{ ( \alpha,\beta ) } ( x ) $ be the Jacobi polynomial of degree n with parameters α,β. The main result of the paper states the following: If b≠1,3 and c are non-zero relatively prime natural numbers then $P_{n}^{ ( k+ ( d-3 ) /2,k+ ( d-3 ) /2 ) } ( \sqrt{b/c} ) \neq0$ for all natural numbers d,n and $k\in\mathbb{N}_{0}$ . Moreover, under the above assumption, the polynomial $Q ( x ) = \frac{b}{c} ( x_{1}^{2}+\cdots+x_{d-1}^{2} ) + ( \frac{b}{c}-1 ) x_{d}^{2}$ is not a harmonic divisor, and the Dirichlet problem for the cone {Q(x)<0} has polynomial harmonic solutions for polynomial data functions.     

On Uniform Approximation in R2 of Continuous Functions of Two Variables with Bounded Variation in the Sense of Hardy

Introduce the notation: $\mathbb{Z}$ is the set of integers, $\bar {\mathbb{Z}}={\mathbb{Z}} \cup \{-\infty, +\infty\},{\mathbb{R}}_+^2 =\{x=(x_1,x_2) \in {\mathbb{R}}^2; x_1>0,x_2>0\}$ , $g_{k,m} (x,\alpha,h)= \int\limits_0^1 {g_1 (\frac{(k+u)h_1 - x_1}{\alpha_1})g_2(\frac{(m+u)h_2 - x_2}{\alpha_2})}du$ , where $g_i :\mathbb{R} \to \mathbb{R},x \in \mathbb{R}^2 ,\alpha ,h \in \mathbb{R}_ + ^2 $ . Under certain conditions on the functions g ₁, g ₂, we prove that the system of functions $g_{k,m} (x,\alpha^(n), h^(n)) (k,m \in \bar {\mathbb{Z}})$ , where $\alpha ^{\left( n \right)} ,h^{\left( n \right)} \in \mathbb{R}_ + ^2 $ are arbitrary infinitesimal sequences, is complete in the space C $\mathbb{R}^2 $ of uniformly continuous bounded functions f equipped with the norm $||f|| = \mathop {\sup }\limits_{x \in \mathbb{R}^2 } |f(x)|$ . Starting with the functions g _k,m , it is possible to construct a method for uniform approximating in $\mathbb{R}^2 $ any continuous function of bounded variation in the sense of Hardy. An error estimate is derived in terms of the second order moduli of continuity. Based on the obtained results, we discuss in detail the accuracy of uniform approximation of functions of several variables by linear functions. The error estimates are derived by using second order moduli of continuity. We pay a particular attention to sharpness of constants. Bibliography: 8 titles.     

Properties of finite unrefinable chains of ring topologies

Let R(+, ·) be a nilpotent ring and $ \left( {\mathfrak{M}, < } \right) $ be the lattice of all ring topologies on R(+, ·) or the lattice of all such ring topologies on R(+, ·) in each of which the ring R possesses a basis of neighborhoods of zero consisting of subgroups. Let ?? and ??? be ring topologies from $ \mathfrak{M} $ such that $ \tau = {\tau_0}{ \prec_\mathfrak{M}}{\tau_1}{ \prec_\mathfrak{M}} \cdots { \prec_\mathfrak{M}}{\tau_n} = \tau ^{\prime} $ . Then k????n for every chain $ \tau = {\tau ^{\prime}_0} < {\tau ^{\prime}_1} < \cdots < {\tau ^{\prime}_k} = \tau ^{\prime} $ of topologies from $ \mathfrak{M} $ , and also n?=?k if and only if $ {\tau ^{\prime}_i}{ \prec_\mathfrak{M}}{\tau ^{\prime}_{i + 1}} $ for all 0????i?<?k.     

Subspace Arrangements of Type B n and D n

Let ${\mathcal{D}}_{n,k} $ be the family of linear subspaces of ?ⁿ given by all equations of the form $\varepsilon _1 x_{i_1 } = \varepsilon _2 x_{i_2 } = \cdot \cdot \cdot \varepsilon _k x_{i_k } ,$ for 1 ≤ < ? ? ? < i _k ≤ i and $\left( {\varepsilon _1 ,...,\varepsilon _k } \right)\varepsilon \left\{ { + 1, - 1} \right\}^k $ Also let ${\mathcal{B}}_{n,k,h} $ be ${\mathcal{D}}_{n,k} $ enlarged by the subspaces $x_{j_1 } = x_{j_2 } = \cdot \cdot \cdot x_{j_h } = 0,$ for 1 ≤. The special cases ${\mathcal{B}}_{n,2,1} $ and ${\mathcal{D}}_{n,2} $ are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type B _nand D _n respectively. In this paper we study combinatorial and topological properties of the intersection lattices of these subspace arrangements. Expressions for their Möbius functions and characteristic polynomials are derived. Lexicographic shellability is established in the case of ${\mathcal{B}}_{n,k,h,} 1 \leqslant h < k$ , which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold $\begin{gathered} {\mathcal{D}}_{n,2} \\ M_{n,k,h,} = {\mathbb{R}}^n \backslash \bigcup {{\mathcal{B}}_{n,k,h,} } \\ \end{gathered} $ . For instance, it is shown that $H^d \left( {M_{n,k,k - 1} } \right)$ is torsion-free and is nonzero if and only if d = t(k ? 2) for some $t,0 \leqslant t \leqslant \left[ {{n \mathord{\left/ {\vphantom {n k}} \right. \kern-0em} k}} \right]$ . Torsion-free cohomology follows also for the complement in ?ⁿof the complexification ${\mathcal{B}}_{n,k,h}^C ,1 \leqslant h < k$ .     

Die Ungleichungen von Vietoris

The inequalities $$P_{k,l} = \frac{1}{{B(l + 1,k)}}\int\limits_0^{l/(k + l)} {x^l (1 - x)^{k - 1} dx = I_{l/(k + l)} (l + 1,k)< \frac{1}{2}}$$ and $$\Phi _{k,l,\mu } = \frac{1}{{B(k,k\mu + 1)}}\int\limits_0^1 {x^{k - 1} (1 - x)^{k\mu } I_x (l + 1,l\mu )dx< \frac{1}{2}}$$ are shown to be valid for any positive real numbersk, l, μ.     

The Two-Weighted Inequalities for Sublinear Operators Generated by B Singular Integrals in Weighted Lebesgue Spaces

In this paper, the authors establish several general theorems for the boundedness of sublinear operators (B sublinear operators) satisfies the condition (1.2), generated by B singular integrals on a weighted Lebesgue spaces $L_{p,\omega,\gamma}(\mathbb{R}_{k,+}^{n})$ , where $B=\sum_{i=1}^{k} (\frac{\partial^{2}}{\partial x_{k}^{2}} + \frac{\gamma_{i}}{x_{i}}\frac{\partial}{\partial x_{i}} )$ . The condition (1.2) are satisfied by many important operators in analysis, including B maximal operator and B singular integral operators. Sufficient conditions on weighted functions ω and ω ₁ are given so that B sublinear operators satisfies the condition (1.2) are bounded from $L_{p,\omega,\gamma}(\mathbb{R}_{k,+}^{n})$ to $L_{p,\omega_{1},\gamma}(\mathbb{R}_{k,+}^{n})$ .     

Asymptotic properties for the loglog laws under positive association

Let {X _n : n ?? 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $S_n = \sum\limits_{k = 1}^n {X_k }$ , $Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$ , n ?? 1. Suppose that $0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$ . In this paper, we prove that if E|X ₁|^2+?? < for some ?? ?? (0, 1], and $\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$ for some ?? > 1, then for any b > ?1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x ₊ = max{x, 0}, N is a standard normal random variable, and ??(·) is a Gamma function.     

Metrization of Weighted Graphs

We find a set of necessary and sufficient conditions under which the weight ${w: E \rightarrow \mathbb{R}^{+}}$ on the graph G = (V, E) can be extended to a pseudometric ${d : V \times V \rightarrow \mathbb{R}^{+}}$ . We describe the structure of graphs G for which the set ${\mathfrak{M}_{w}}$ of all such extensions contains a metric whenever w is strictly positive. Ordering ${\mathfrak{M}_{w}}$ by the pointwise order, we have found that the posets $({\mathfrak{M}_{w}, \leqslant)}$ contain the least elements ρ _0,w if and only if G is a complete k-partite graph with ${k \, \geqslant \, 2}$ . In this case the symmetric functions ${f : V \times V \rightarrow \mathbb{R}^{+}}$ , lying between ρ _0,w and the shortest-path pseudometric, belong to ${\mathfrak{M}_{w}}$ for every metrizable w if and only if the cardinality of all parts in the partition of V is at most two.     

Cubic symmetric polynomials yielding variations of the Erdős–Ginzburg–Ziv theorem

Let p be a prime and let $\varphi\in\mathbb{Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ be a symmetric polynomial, where  $\mathbb {Z}_{p}$ is the field of p elements. A sequence T in  $\mathbb {Z}_{p}$ of length p is called a φ-zero sequence if φ(T)=0; a sequence in $\mathbb {Z}_{p}$ is called a φ-zero free sequence if it does not contain any φ-zero subsequence. Motivated by the EGZ theorem for the prime p, we consider symmetric polynomials $\varphi\in \mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ , which satisfy the following two conditions: (i) every sequence in  $\mathbb {Z}_{p}$ of length 2p?1 contains a φ-zero subsequence, and (ii) the φ-zero free sequences in  $\mathbb {Z}_{p}$ of maximal length are all those containing exactly two distinct elements, where each element appears p?1 times. In this paper, we determine all symmetric polynomials in $\mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ of degree not exceeding 3 satisfying the conditions above.     

Nash equilibria on topological semilattices

A function ${u : X \to \mathbb{R}}$ defined on a partially ordered set is quasi-Leontief if, for all ${x \in X}$ , the upper level set ${\{x\prime \in X : u(x\prime) \geq u(x)\}}$ has a smallest element; such an element is an efficient point of u. An abstract game ${u_{i} : \prod^{n}_{j=1} X_j \to \mathbb{R}, i \in \{1, \ldots , n\}}$ , is a quasi-Leontief game if, for all i and all ${(x_{j})_{j \neq i} \in \prod_{j \neq i} X_{j}, u_{i}((x_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ is quasi-Leontief; a Nash equilibrium x* of an abstract game ${u_{i} :\prod^{n}_{j=1} X_{j} \to \mathbb{R}}$ is efficient if, for all ${i, x^{*}_{i}}$ is an efficient point of the partial function ${u_{i}((x^{*}_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ . We establish the existence of efficient Nash equilibria when the strategy spaces X _i are topological semilattices which are Peano continua and Lawson semilattices.     

Phase transitions of iterated Higman-style well-partial-orderings

We elaborate Weiermann-style phase transitions for well-partial-orderings (wpo) determined by iterated finite sequences under Higman-Friedman style embedding with Gordeev’s symmetric gap condition. For every d-times iterated wpo ${\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}\right)}$ in question, d >? 1, we fix a natural extension of Peano Arithmetic, ${T \supseteq \sf{PA}}$ , that proves the corresponding second-order sentence ${\sf{WPO}\left({\rm S}{\textsc{eq}}^{d}, \trianglelefteq _{d}\right) }$ . Having this we consider the following parametrized first-order slow well-partial-ordering sentence ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}, r\right):}$ $$\left( \forall K > 0 \right) \left( \exists M > 0\right) \left( \forall x_{0},\ldots ,x_{M}\in {\rm S}\text{\textsc{eq}}^{d}\right)$$ $$\left( \left( \forall i\leq M\right) \left( \left| x_{i}\right| < K + r \left\lceil \log _{d} \left( i+1\right) \right\rceil \right)\rightarrow \left( \exists i < j \leq M \right) \left(x_{i} \trianglelefteq _{d} x_{j}\right) \right)$$ for a natural additive Seq ^d -norm |·| and r ranging over EFA-provably computable positive reals, where EFA is an abbreviation for IΔ ₀?+?exp. We show that the following basic phase transition clauses hold with respect to ${T = \Pi_{1}^{0}\sf{CA}_{ < \varphi ^{_{\left( d-1\right) }} \left(0\right) }}$ and the threshold point1.

1. If r <? 1 then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$ is provable in T.

1. If ${r > 1}$ then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$ is not provable in T.

Moreover, by the well-known proof theoretic equivalences we can just as well replace T by PA or ACA ₀ and ${\Delta _{1}^{1}\sf{CA}}$ , if d =? 2 and d =? 3, respectively.In the limit case d → ∞ we replaceEFA-provably computable reals r by EFA-provably computable functions ${f: \mathbb{N} \rightarrow \mathbb{R}_{+}}$ and prove analogous theorems. (In the sequel we denote by ${\mathbb{R}_{+}}$ the set of EFA-provably computable positive reals). In the basic case T?=? PA we strengthen the basic phase transition result by adding the following static threshold clause

1. ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1\right)}$ is still provable in T = PA (actually in EFA).

Furthermore we prove the following dynamic threshold clauses which, loosely speaking are obtained by replacing the static threshold t by slowly growing functions 1 _α given by ${1_{\alpha }\left( i\right)\,{:=}\,1+\frac{1}{H_{\alpha }^{-1}\left(i\right) }, H_{\alpha}}$ being the familiar fast growing Hardy function and ${H_{\alpha }^{-1}\left( i\right)\,{:=}\,\rm min \left\{ j \mid H_{\alpha } \left ( j\right) \geq i \right\}}$ the corresponding slowly growing inversion.

1. If ${\alpha < \varepsilon _{0}}$ , then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1_{\alpha}\right)}$ is provable in T = PA.

1. ${\sf{SWP}\left( {\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2},1_{\varepsilon _{0}}\right)}$ is not provable in T = PA.

We conjecture that this pattern is characteristic for all ${T\supseteq \sf{PA}}$ under consideration and their proof-theoretical ordinals o (T ), instead of ${\varepsilon _{0}}$ .     

The Left Hilbert BVP for h-Regular Functions in Clifford Analysis

Consider the real Clifford algebra ${\mathbb{R}_{0,n}}$ generated by e ₁, e ₂, . . . , e _n satisfying ${e_{i}e_{j} + e_{j}e_{i} = -2\delta_{ij} , i, j = 1, 2, . . . , n, e_{0}}$ is the unit element. Let ${\Omega}$ be an open set in ${\mathbb{R}^{n+1}}$ . u(x) is called an h-regular function in ${\Omega}$ if $$D_{x}u(x) + \widehat{u}(x)h = 0, \quad\quad (0.1)$$ where ${D_x = \sum\limits_{i=0}^{n} e_{i}\partial_{xi}}$ is the Dirac operator in ${\mathbb{R}^{n+1}}$ , and ${\widehat{u}(x) = \sum \limits_{A} (-1)^{\#A}u_{A}(x)e_{A}, \#A}$ denotes the cardinality of A and ${h = \sum\limits_{k=0}^{n} h_{k}e_{k}}$ is a constant paravector. In this paper, we mainly consider the Hilbert boundary value problem (BVP) for h-regular functions in ${\mathbb{R}_{+}^{n+1}}$ .     

Mean Value Properties for the Weinstein Equation Using the Hyperbolic Metric

In this paper we consider solutions of the Weinstein equation $$\begin{aligned} \Delta u-\frac{k}{x_{n}}\frac{\partial u}{\partial x_{n}}+\frac{\ell }{ x_{n}^{2}}u=0, \end{aligned}$$ on some open subset $\Omega \subset \mathbb R ^{n}\cap \{x_{n}>0\}$ subject to the conditions $4\ell \le (k+1)^{2}$ . If $l=0$ , the operator $x_{n}^{2k/n-2}\left( \Delta u-\frac{k}{x_{n}}\frac{\partial u}{\partial x_{n}}\right) $ is the Laplace–Beltrami operator with respect to the Riemannian metric $ds^{2}=x_{n}^{-2k/n-2}\left( \sum _{i=1}^{n}dx_{i} ^{2}\right) $ . In case $k=n-2$ the Riemannian metric is the hyperbolic distance of Poincaré upper half space. The Weinstein equation is connected to the axially symmetric potentials. The solutions of of the Weinstein equation form a so-called Brelot harmonic space and therefore it is known they satisfy the mean value properties with respect to the harmonic measure. We present the explicit mean value properties which give a formula for a harmonic measure evaluated in the center point of the hyperbolic ball. The key idea is to transform the solutions to the eigenfunctions of the Laplace–Beltrami operator in the Poincaré upper half-space model.     

Strong approximation of Riesz means at critical index on Hp(T) (0<p≤1)

This paper is a continuation of [3]. Suppose f∈H^p(T), 0σ _r ^σ f,σ=1/p?1. When p=1, it is just the partial Fourier sums S_kf. In this paper we establish the sharp estimations on the degree of approximation: $$\left\{ { - \frac{1}{{logR}}\int\limits_1^R {\left\| {\sigma _r^\delta f - f} \right\|_{H^p (T)}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqq C{\mathbf{ }}{}_p\omega \left( {f,{\mathbf{ }}( - \frac{1}{{logR}})^{1/p} } \right)_{H^p (T)} ,0< p< 1,$$ and \(\frac{1}{{\log L}}\sum\limits_{k - 1}^L {\frac{{\left\| {S_k f - f} \right\|_H 1_{(T)} }}{k} \leqq Cp\omega (f; - \frac{1}{{\log L}})_H 1_{(T)} } \) Where $$\omega (f,{\mathbf{ }}h)_{H^p (T)} \begin{array}{*{20}c} { = Sup} \\ {0 \leqq \left| u \right| \leqq h} \\ \end{array} \left\| {f( \cdot + u) - f( \cdot )} \right\|_{H^p (T).} $$ .     

Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

We prove that if ${U\subset \mathbb {R}^n}$ is an open domain whose closure ${\overline U}$ is compact in the path metric, and F is a Lipschitz function on ?U, then for each ${\beta \in \mathbb {R}}$ there exists a unique viscosity solution to the β-biased infinity Laplacian equation $$\beta |\nabla u| + \Delta_\infty u=0$$ on U that extends F, where ${\Delta_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}}$ . In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased ${\epsilon}$ -game as follows. The starting position is ${x_0 \in U}$ . At the kth step the two players toss a suitably biased coin (in our key example, player I wins with odds of ${\exp(\beta\epsilon)}$ to 1), and the winner chooses x _k with ${d(x_k,x_{k-1}) < \epsilon}$ . The game ends when ${x_k \in \partial U}$ , and player II pays the amount F(x _k ) to player I. We prove that the value ${u^{\epsilon}(x_0)}$ of this game exists, and that ${\|u^\epsilon - u\|_\infty \to 0}$ as ${\epsilon \to 0}$ , where u is the unique extension of F to ${\overline{U}}$ that satisfies comparison with β-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with β-exponential cones if and only if it is a viscosity solution to the β-biased infinity Laplacian equation.     

Exact multiplicity for the perturbed Q-curvature problem in {\mathbb{R}^N,N \geq 5}

Let N ≥ 5 and \({{\mathcal{D}}^{2,2} (\mathbb{R}^N)}\) denote the closure of \({C_0^\infty (\mathbb{R}^N)}\) in the norm \({\|u\|_{{\mathcal{D}}^{2,2} (\mathbb{R}^N)}^2 := \int\nolimits_{\mathbb{R}^N} |\Delta u|^2.}\) Let \({K \in C^2 (\mathbb{R}^N).}\) We consider the following problem for ? ≥ 0: $$(P_\varepsilon) \left\{\begin{array}{llll}{\rm Find} \, u \in {\mathcal{D}}^{2, 2} (\mathbb{R}^N) \, \, {\rm solving} :\\ \left.\begin{array}{lll}\Delta^2 u = (1+ \varepsilon K (x)) u^{\frac{N+4}{N-4}}\\ u > 0 \end{array}\right\}{\rm in} \, \mathbb{R}^N.\end{array}\right.$$ We show an exact multiplicity result for (P _? ) for all small ? > 0.