Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality

It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,Z Rn Z Rn f(x)g(y)|x||x.y||y|dxdy6 B(p,q,,,,n)kfkLp(Rn)kgkLq(Rn).The main purpose of this paper is to give the sharp constants B(p,q,,,,n)for the above inequality for three cases:(i)p=1 and q=1;(ii)p=1 and 1q 6∞,or 1p 6∞and q=1;(iii)1p,q∞and 1p+1q=1.In addition,the explicit bounds can be obtained for the case 1p,q∞and 1p+1q1.     

Approximation by M. Riesz Kernels in L p for p<1

Let α > 0. We consider the linear span $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right)$ of scalar Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^\alpha }}} \right\}_{a \in \mathbb{R}^n }$ and the linear span $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right)$ of vector Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^{\alpha + 1} }}\left( {x - a} \right)} \right\}_{a \in \mathbb{R}^n }$ . We study the following problems. (1) When is the intersection $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n } \right)$ dense in L^p(?ⁿ)? (2) When is the intersection $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n ,\mathbb{R}^n } \right)$ dense in L^p(?ⁿ, ?ⁿ)? Bibliography: 15 titles.     

Boundedness of the fractional maximal operator in generalized Morrey space on the Heisenberg group

In this paper we study the fractional maximal operator M _α , 0 ≤ α < Q on the Heisenberg group ? _n in the generalized Morrey spaces M _{p, ?}(? _n ), where Q = 2n + 2 is the homogeneous dimension of ? _n . We find the conditions on the pair (? ₁, ? ₂) which ensures the boundedness of the operator M _α from one generalized Morrey space M _{p, ?1}(? _n ) to another M _{q, ?2}(? _n ), 1 < p < q < ∞, 1/p?1/q = α/Q, and from the space M _{1, ?1}(? _n ) to the weak space WM _{q, ?2}(? _n ), 1 < q < ∞, 1 ? 1/q = α/Q. We also find conditions on the φ which ensure the Adams type boundedness of M _α from $M_{p,\phi ^{\tfrac{1} {p}} } \left( {\mathbb{H}_n } \right)$ to $M_{q,\phi ^{\tfrac{1} {q}} } \left( {\mathbb{H}_n } \right)$ for 1 < p < q < ∞ and from M _{1, ?}(? _n ) to $WM_{q,\phi ^{\tfrac{1} {q}} } \left( {\mathbb{H}_n } \right)$ for 1 < q < ∞. As applications we establish the boundedness of some Schrödinger type operators on generalized Morrey spaces related to certain nonnegative potentials V belonging to the reverse Hölder class B _∞(” _n ).     

Weighted inequalities for some integral operators with rough kernels

In this paper we study integral operators with kernels $$K(x,y) = k_1 (x - A_1 y) \cdots k_m \left( {x - A_m y} \right),$$ $k_i \left( x \right) = {{\Omega _i \left( x \right)} \mathord{\left/ {\vphantom {{\Omega _i \left( x \right)} {\left| x \right|}}} \right. \kern-0em} {\left| x \right|}}^{{n \mathord{\left/ {\vphantom {n {q_i }}} \right. \kern-0em} {q_i }}}$ where Ω _i : ? ⁿ → ? are homogeneous functions of degree zero, satisfying a size and a Dini condition, A _i are certain invertible matrices, and n/q ₁ +…+n/q _m = n?α, 0 ≤ α < n. We obtain the appropriate weighted L ^p -L ^q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.     

The Boundedness for Commutator of Fractional Integral Operator With Rough Variable Kernel

For b?∈?BMO(? ⁿ ) and 0?<?α?≤?1/2, the commutator of the fractional integral operator T _Ω,α with rough variable kernel is defined by $$ [b, T_{\Omega, \alpha}]f(x)= \int_{\mathbb{R}^n} \frac{\Omega(x,x-y)}{|x-y|^{n-\alpha}}(b(x)-b(y))f(y)dy. $$ In this paper the authors prove that the commutator [b, T _Ω,α ] is a bounded operator from $L^{\frac{2n}{n+2\alpha}}(\mathbb{R}^n)$ to L ²(? ⁿ ). The result obtained in this paper is substantial improvement and extension of some known results.     

On the representation theory of Möbius groups inR n*

We will solve several fundamental problems of Möbius groupsM(R ⁿ) which have been matters of interest such as the conjugate classification, the establishment of a standard form without finding the fixed points and a simple discrimination method. Let \(g = \left[ {\begin{array}{*{20}c} a &; b \\ c &; d \\ \end{array} } \right]\) be a Clifford matrix of dimensionn, c ≠ 0. We give a complete conjugate classification and prove the following necessary and sufficient conditions:g is f.p.f. (fixed points free) iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; 0 \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α|<1 and |E?AE ¹| ≠ 0;g is elliptic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; \beta \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α| <1 and |E?AE ¹|=0;g is parabolic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; 0 \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α|=1; andg is loxodromic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; \beta \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α| >1 or rank (E?AE ¹) ≠ rank (E?AE ¹,ac ^?1+c ^?1 d), where α is represented by the solutions of certain linear algebraic equations and satisfies $\left| {c^{ - 1} \alpha '} \right| = \left| {\left( {E - AE^1 } \right)^{ - 1} \left( {\alpha c^{ - 1} + c^{ - 1} \alpha '} \right)} \right|.$      

Convergence of the Vallée-Poussin means for Fourier-Jacobi sums

Letf εC[?1, 1], ?1<α,β≤0, let $f \in C[ - 1, 1], - 1< \alpha , \beta \leqslant 0$ , letS _n ^{α, β} (f, x) be a partial Fourier-Jacobi sum of ordern, and let $$\nu _{m, n}^{\alpha , \beta } = \nu _{m, n}^{\alpha , \beta } (f) = \nu _{m, n}^{\alpha , \beta } (f,x) = \frac{1}{{n + 1}}[S_m^{\alpha ,\beta } (f,x) + ... + S_{m + n}^{\alpha ,\beta } (f,x)]$$ be the Vallée-Poussin means for Fourier-Jacobi sums. It was proved that if 0<a≤m/n≤b, then there exists a constantc=c(α, β, a, b) such that ‖ν _{m, n} ^{α, β} ‖ ≤c, where ‖ν _{m, n} ^{α, β} ‖ is the norm of the operator ν _{m, n} ^{α, β} inC[?1,1].     

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.     

Rational approximation to 19-0119-0119-01on [0, 1]

В стАтьЕ пОлУЧЕНО УсИ лЕНИЕ НЕскОлькИх РЕж УльтАтОВ О РАцИОНАльНОИ АппРОк сИМАцИИx ^α НА [0,1]. ДОкАжАНО, ЧтО НАИ лУЧшИЕ пРИБлИжЕНИьr _n (x ^α) ФУНкцИИx ^α НА [0,1] РАцИОНА льНыМИ ДРОБьМИ пОРьДкАn Дль лУБОгО НЕцЕлОгО пОлО жИтЕльНОгО А УДОВлЕтВОРьУт сООт НОшЕНИУ $$\mathop {\lim }\limits_{n \to \infty } r_n^{1/\sqrt n } (x^\alpha ) = \exp ( - 2\pi \sqrt \alpha ).$$ гИпОтЕжА О спРАВЕДлИ ВОстИ ЁтОИ ОцЕНкИ Был А ВыскАжАНА А. А. гОНЧАРОМ В 1974 г. кРОМЕ тОгО, тОЧНАь ОцЕ НкА, пОлУЧЕННАь Н. с. Вь ЧЕслАВОВНы Дль слУЧАь А=1/2 РАс-пРОс тРАНьЕтсь В РАБОтЕ НА пРОИжВОль НыЕ пОлОжИтЕльНыЕ РА цИОНАльНыЕ ЧИслА α: $$b_\alpha |\sin \pi \alpha |< r_n (x^\alpha )\exp (2\pi \sqrt {\alpha n} )< B_{p,q} ,$$ жДЕсь α=p/q.     

Positive solutions to m-point boundary value problem of fractional differential equation

In this paper, we study the multiplicity of positive solutions to the following m-point boundary value problem of nonlinear fractional differential equations: Dqu(t) + f(t, u(t)) = 0, 0 < t < 1, u(0) = 0, u(1) =sum (μiDpu(t)|t = ξi ) from i =1 to ∞ m-2, where q ∈R , 1 相似文献   

Fractional BVPs with strong time singularities and the limit properties of their solutions

In the first part, we investigate the singular BVP \(\tfrac{d} {{dt}}^c D^\alpha u + (a/t)^c D^\alpha u = \mathcal{H}u\) , u(0) = A, u(1) = B, ^c D ^α u(t)| _t=0 = 0, where \(\mathcal{H}\) is a continuous operator, α ∈ (0, 1) and a < 0. Here, ^c D denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems \(\tfrac{d} {{dt}}^c D^{\alpha _n } u + (a/t)^c D^{\alpha _n } u = f(t,u,^c D^{\beta _n } u)\) , u(0) = A, u(1) = B, \(\left. {^c D^{\alpha _n } u(t)} \right|_{t = 0} = 0\) where a < 0, 0 < β _n ≤ α _n < 1, lim _n→∞ β _n = 1, and solutions of u″+(a/t)u′ = f(t, u, u′) satisfying the boundary conditions u(0) = A, u(1) = B, u′(0) = 0.     

Optimal Cubature Formulas in Weighted Besov Spaces with A ∞ Weights on Multivariate Domains

For the unit ball $\mathit{UB}_{\tau}^{\alpha}(L_{p,w})$ of the weighted Besov space $B_{\tau}^{\alpha}(L_{p,w})$ with an A _∞ weight w on the domain Ω, which denotes either the unit sphere, or the unit ball, or the standard simplex of the Euclidean space ? ^d , the sharp asymptotic order of the quantity $$\mathop{\inf}_{{\lambda}_1, \ldots, {\lambda}_n \in \mathbb{R}\atop{\xi_1,\ldots, \xi_n \in\varOmega }} \sup_{f\in \mathit{UB}_\tau^\alpha(L_{p,w})} \biggl|\int _{\varOmega } f(x) w(x)\, dx-\sum_{j=1}^n{\lambda}_j f(\xi_j) \biggr|$$ is obtained as n→∞. A similar result is also established on unweighted spherical caps.     

On a linear combination of some expressions in the theory of the univalent functions

LetH(α) denote the class of regular functionsf(z) normalized so thatf(0)=0 andf′(0)=1 and satisfying in the unit discE the condition $$\operatorname{Re} \left\{ {(1 - \alpha )f'(z) + \alpha (1 + zf''(z)/f'(z))} \right\} > 0$$ for fixed α. It is known thatH(0) is a particular class NW of close-to-convex univalent functions. The authors show the following results:Theorem 1. Letf(z)∈H(α). Thenf(z)∈NW if α≤0 andz∈E.Theorem 2. Letf(z)∈NW. Thenf(z)∈H(α) in |z|=r<r _α where i) \(r_\alpha = (1 + \sqrt {2\alpha } )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}\) , α≥0 and ii) \(r_\alpha = \sqrt {\frac{{1 - \alpha - \sqrt {\alpha (\alpha - 1)} }}{{1 - \alpha }}}\) , α<0. All results are sharp.Theorem 3. Iff(z)=z+a ₂ z ²+a ₃ z ³+... is inH(α) and if μ is an arbitrary complex number, then $$\left| {1 + \alpha } \right|\left| {a_3 - \mu a_2^2 } \right| \leqslant ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})\max \left[ {1,\left| {1 + 2\alpha - {3 \mathord{\left/ {\vphantom {3 {2\mu }}} \right. \kern-\nulldelimiterspace} {2\mu }}(1 + \alpha )} \right|} \right].$$ .     

On the properties of the strong Cesàro summability and strong convergence of series

Говорят, что ряд \(\mathop \sum \limits_{k = 0}^\infty a_k \) сумм ируется к s в смысле (С, gа), gа >?1, если $$\sigma _n^{(k)} - s = o(1),n \to \infty ,$$ в смысле [C,α] _λ , α<0, λ>0, если $$\frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n \left| {\sigma _k^{(\alpha - 1)} - s} \right|^\lambda = o(1),n \to \infty ,$$ и в смысле [C,0] _λ , λ>0, если $$\frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n \left| {(k + 1)(s_k - 1) - k(s_{k - 1} - 1)} \right|^\lambda = o(1),n \to \infty ,$$ где σ _n ^(α) обозначаетn-ое ч езаровское среднее р яда. Суммируемость [C,α] _λ , α>?1, λ ≧1 о значает, что $$\mathop \sum \limits_{k = 0}^\infty k^{\lambda - 1} \left| {\sigma _k^{(\alpha )} - \sigma _{k - 1}^{(\alpha )} } \right|^\lambda< \infty .$$ В данной статье содер жится продолжение ис следований свойств [C,α] _λ -суммиру емо сти, которые начали Винн, Х ислоп, Флетт, Танович-М иллер и автор, в частности свя зей между указанными методами суммирования. Наконец, даны некотор ые простые приложени я к вопросам суммируемости ортог ональных рядов.     

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

Commutators of Marcinkiewicz integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator [b, ??_??,?? ] from the homogeneous Sobolev space $\dot L_\gamma ^p \left( {\mathbb{R}^n } \right)$ to the Lebesgue space L ^p (? ⁿ ) for 1 < p < ??, where ??_??,?? denotes the Marcinkiewicz integral with rough hypersingular kernel defined by $\mu _{\Omega ,\gamma } f\left( x \right) = \left( {\int_0^\infty {\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega \left( {x - y} \right)}} {{\left| {x - y} \right|^{n - 1} }}f\left( y \right)dy} } \right|^2 \frac{{dt}} {{t^{3 + 2\gamma } }}} } \right)^{\frac{1} {2}} ,$ , with ?? ?? L ¹(S ^n?1) for $0 < \gamma < min\left\{ {\frac{n} {2},\frac{n} {p}} \right\}$ or ?? ?? L(log⁺ L) ^?? (S ^n?1) for $\left| {1 - \frac{2} {p}} \right| < \beta < 1\left( {0 < \gamma < \frac{n} {2}} \right)$ , respectively.     

Bergman type operators on a class of weakly pseudoconvex domains

A necessary and sufficient condition for the boundedness of the operator: $(T_{s,u,u} f)(\xi ) = h^{u + \tfrac{v}{a}} (\xi )\smallint _{\Omega _a } h^s (\xi ')K_{s,u,v} (\xi ,\xi ')f(\xi ')dv(\xi ') on L^p (\Omega _a ,dv_\lambda ),1< p< \infty $ , is obtained, where $\Omega _a = \left\{ {\xi = (z,w) \in \mathbb{C}^{n + m} :z \in \mathbb{C}^n ,w \in \mathbb{C}^m ,|z|^2 + |w|^{2/a}< 1} \right\},h(\xi ) = (1 - |z|^2 )^a - |w|^2 $ andK _x,u,v (ξ,ξ′).This generalizes the works in literature from the unit ball or unit disc to the weakly pseudoconvex domain ω _a . As an appli cation, it is proved thatf?L _H ^p (ω _a ,dv _λ) implies $h\tfrac{{|a|}}{a} + |\beta |(\xi )D_2^a D_z^\beta f \in L^p (\Omega _a ,dv_\lambda ),1 \leqslant p< \infty $ , for any multi-indexa=(α₁,?,α _n and ß = (ß₁, —ß). An interesting question is whether the converse holds.     

Asymptotic properties of least squares estimators for discrete compound distributions

Let (X, Λ) be a pair of random variables, where Λ is an Ω (a compact subset of the real line) valued random variable with the density functiong(Θ: α) andX is a real-valued random variable whose conditional probability function given Λ=Θ is P {X=x|Θ} withx=x ₀, x₁, …. Based onn independent observations ofX, x ⁽ⁿ⁾, we are to estimate the true (unknown) parameter vectorα=(α ₁, α₂, ...,α_m) of the probability function ofX, P_α(X=∫_ΩP{X=x|Θ}g(Θ:α)dΘ. A least squares estimator of α is any vector \(\hat \alpha \left( {X^{\left( n \right)} } \right)\) which minimizes $$n^{ - 1} \sum\limits_{i = 1}^n {\left( {P_\alpha \left( {x_i } \right) - fn\left( {x_i } \right)} \right)^2 } $$ wherex ⁽ⁿ⁾=(x₁, x₂,…,x _n) is a random sample ofX andf _n(x_i)=[number ofx _i inx ⁽ⁿ⁾]/n. It is shown that the least squares estimators exist as a unique solution of the normal equations for all sufficiently large sample size (n) and the Gauss-Newton iteration method of obtaining the estimator is numerically stable. The least squares estimators converge to the true values at the rate of \(O\left( {\sqrt {2\log \left( {{{\log n} \mathord{\left/ {\vphantom {{\log n} n}} \right. \kern-0em} n}} \right)} } \right)\) with probability one, and has the asymptotically normal distribution.     

Some remarks on the moments of \left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right| in short intervals

Some new results on power moments of the integral $$ J_k (t,G) = \frac{1} {{\sqrt {\pi G} }}\int_{ - \infty }^\infty { \left| {\varsigma \left( {\tfrac{1} {2} + it + iu} \right)} \right|^{2k} } e^{ - (u/G)^2 } du $$ (t ? T, T ^? ≦ G ? T, κ ∈ N) are obtained when κ = 1. These results can be used to derive bounds for moments of $ \left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right| $ .     

Freud's conjecture and approximation of reciprocals of weights by polynomials

LetW(x) be a function that is nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ² (x) are finite. Let {p _n (W ²;x)} ₀ ^∞ denote the sequence of orthonormal polynomials with respect to the weightW ², and let {α _n } ₁ ^∞ and {β _n } ₁ ^∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = \alpha _{n + 1} p_{n + 1} (W^2 ,x) + \beta _n p_n (W^2 ,x) + \alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW ^?1 by reciprocals of polynomials, for $$\mathop {\lim }\limits_{n \to \infty } {{\alpha _n } \mathord{\left/ {\vphantom {{\alpha _n } {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{\beta _n } \mathord{\left/ {\vphantom {{\beta _n } {c_{n + 1} }}} \right. \kern-\nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec _{n 1} ^∞ is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR.