The solution of a mildly singular integral equation of the first kind on a ball

We consider the equations $$\int_{\left| y \right| \leqslant 1} {\frac{{F(y)}}{{\left| {x - y} \right|^\lambda }} dy = G(x)(\left| x \right| \leqslant 1)} $$ where x and y ∈ E^p, p ≥ 3 and λ < p. For λ ∈ (p-2,p) we show that this equation has at most one integrable solution which if G is twice differentiable actually exists and is, in fact, given by an explicit formula in terms of integral operators acting on G and its derivatives. When λ ≤ p-2 and λ ≠ ?2j (j=0,1,?) the equation also has at most one integrable solution for which, assuming it to exist and G to be sufficiently smooth, there also is an explicit formula; in this situation, though, the explicit formula does not usually provide an integrable solution, because, in general, such solutions do not exists when λ ≤ p-2 and λ ≠ ?2j (j=0,1,?), no matter how smooth G is. In the remaining case λ = ?2j (j=0,1,?), neither uniqueness nor existence holds for solutions of the equation.     

A Tauberian theorem for ℓ-adic sheaves on \mathbb{A} 1

Let K ∈ L ¹(?) and let f ∈ L ^∞(?) be two functions on ?. The convolution $$ \left( {K*F} \right)\left( x \right) = \int_\mathbb{R} {K\left( {x - y} \right)f\left( y \right)dy} $$ can be considered as an average of f with weight defined by K. Wiener’s Tauberian theorem says that under suitable conditions, if $$ \mathop {\lim }\limits_{x \to \infty } \left( {K*F} \right)\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \int_\mathbb{R} {\left( {K*A} \right)\left( x \right)} $$ for some constant A, then $$ \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = A $$ We prove the following ?-adic analogue of this theorem: Suppose K, F, G are perverse ?-adic sheaves on the affine line $ \mathbb{A} $ over an algebraically closed field of characteristic p (p ≠ ?). Under suitable conditions, if $ \left( {K*F} \right)|_{\eta _\infty } \cong \left( {K*G} \right)|_{\eta _\infty } $ , then $ F|_{\eta _\infty } \cong G|_{\eta _\infty } $ , where η _∞ is the spectrum of the local field of $ \mathbb{A} $ at ∞.     

A new characterization of the Poisson distribution

In this note we show that an infinitely divisible (i.d.) distribution function F is Poisson if and only if it satisfies the conditions F(+0) > 0, for any 0 < ∈ < 1 $$\int_{ - \infty }^{I - E} {\frac{{\left| x \right|}}{{1 + \left| x \right|}}} dF = 0$$ and for any 0 < β < 1 $$\int_0^\infty {e^{\alpha xln(x + 1)} } dF< \infty $$      

On a multilinear singular integral operator

In this paper, we prove that the maximal operatorsatisfiesis homogeneous of degree 0, has vanishing moment up to order M and satisfies Lq-Dini condition for some     

On a problem of L. Leindler concerning strong approximation by Fourier series and Lipschitz classes

Пусть? — возрастающа я непрерывная фцнкци я на [0,π],?(0)=0 и $$\mathop \smallint \limits_0^h \frac{{\varphi \left( t \right)}}{t}dt = O\left( {\varphi \left( h \right)} \right){\text{ }}\left( {h \to 0} \right).$$ Положим $$\psi \left( h \right) = h\mathop \smallint \limits_h^\pi \frac{{\varphi \left( t \right)}}{{t^2 }}dt \left( {h \in (0, \pi ]} \right).$$ Доказывается следую щая теорема.Пусть f∈ С[?π, π], ω(f, δ)=О(?(δ))) и $$\mathop {\lim }\limits_{h \to 0} \frac{1}{{\varphi \left( {\left| h \right|} \right)}}\left| {f\left( {x + h} \right) - f\left( x \right)} \right| = 0$$ для x∈E?[?π, π], ¦E¦>0. Тогда д ля сопряженной функц ии f почти всюду на E выполн яется соотношение $$\mathop {\lim }\limits_{h \to 0} \frac{1}{{\psi \left( {\left| h \right|} \right)}}\left| {\tilde f\left( {x + h} \right) - \tilde f\left( x \right)} \right| = 0.$$ Из этой теоремы вытек ает положительное ре шение одной задачи Л. Лейндлера.     

Global integrability for minimizers of anisotropic functionals

We consider integral functionals in which the density has growth p _i with respect to ${\frac{\partial u}{\partial x_i}}$ , like in $$\int\limits_{\Omega}\left( \left| \frac{\partial u}{\partial x_1}(x) \right|^{p_1} + \left|\frac{\partial u}{\partial x_2}(x)\right|^{p_2} + \cdots + \left|\frac{\partial u}{\partial x_n}(x) \right|^{p_n} \right) dx.$$ We show that higher integrability of the boundary datum forces minimizer to be more integrable.     

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| $ .     

Convergence of polyharmonic splines on semi-regular grids \mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}} for {\boldsymbol{a}\rightarrow\mathbf {0}}

Let p, n ∈ ? with 2p ≥ n + 2, and let I _a be a polyharmonic spline of order p on the grid ? × a? ⁿ which satisfies the interpolating conditions $I_{a}\left( j,am\right) =d_{j}\left( am\right) $ for j ∈ ?, m ∈ ? ⁿ where the functions d _j : ? ⁿ → ? and the parameter a > 0 are given. Let $B_{s}\left( \mathbb{R}^{n}\right) $ be the set of all integrable functions f : ? ⁿ → ? such that the integral $$ \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi $$ is finite. The main result states that for given $\mathbb{\sigma}\geq0$ there exists a constant c>0 such that whenever $d_{j}\in B_{2p}\left( \mathbb{R}^{n}\right) \cap C\left( \mathbb{R}^{n}\right) ,$ j ∈ ?, satisfy $\left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) $ for all j ∈ ? there exists a polyspline S : ? ⁿ⁺¹ → ? of order p on strips such that $$ \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right) $$ for all y ∈ ? ⁿ , t ∈ ? and all 0 < a ≤ 1.     

Asymptotically sharp error bounds for a quadrature rule for Cauchy principal value integrals based on piecewise linear interpolation

For the numerical evaluation of Cauchy principal value integrals of the form f(x)(x - λ) -1dx withλ ∈ (-1,1)and f ∈ C1[-1,1], we investigate the quadrature formula Q(n+1)Spl1[·;λ] obtained by replacing the integrand function f by its piecewise linear interpolant at an equidistant set of nodes as proposed by Rabinoivitz (Math. Comp. , 51:741 - 747,1988). We give upper bounds for the Peano - type error constantsfor s∈ {1,2}. These are the best possible constants in inequalities of the typeFurthermore, we prove that our upper bounds are asymptotically sharp.     

OnL 1-convergence of Walsh-Fourier series I

We prove the convergence in theL ¹(0, 1)-metric of Walsh-Fourier series \(\sum\limits_{k = 0}^\infty {a_k w_k \left( x \right)} \) of an integrable function with coefficients such that lim_n→∞ and the following Tauberian condition of Hardy-Karamata kind is satisfied: $$\mathop {lim}\limits_{\lambda \to 1 + 0} {\text{ }}\mathop {lim}\limits_{n \to \infty } \sum\limits_{k = n}^{\left[ {\lambda n} \right]} {k^{p - 1} \left| {\Delta a_k } \right|^p } = 0,$$ , wherep>1, [·] denotes the integral part, and Δa _k=a_k?a_k+1.     

On the order of the error function of the square-full integers,II

LetL(x) denote the number of square-full integers not exceedingx. It is well-known that $$L\left( x \right) \sim \frac{{\zeta \left( {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}{{\zeta \left( 3 \right)}}x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \frac{{\zeta \left( {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} \right)}}{{\zeta \left( 2 \right)}}x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} ,$$ whereζ(s) denotes the Riemann Zeta function, LetΔ(x) denote the error function in the asymptotic formula forL(x). On the assumption of the Riemann hypothesis (R.H.), it is known that $$\Delta x = O\left( {x^{13/81 + 8} } \right)$$ for everyε > 0. In this paper, we prove on the assumption of R.H. that $$\frac{1}{x}\int\limits_x^1 {\left| {\Delta \left( t \right)} \right|dt = O\left( {x^{1/10 + ^8 } } \right)} .$$ In fact, we prove a more general result. We conjecture that $$\Delta x = O\left( {x^{1/10 + ^8 } } \right)$$ under the assumption of the R.H.     

Weak type (1,1) bounds for a Marcinkiewicz integral

In this paper, the two-dimensional Marcinkewicz integral introduced by Stein μ(f)(x)=(∫_0~x|∫_(|x-y|≤1) _(|x-y|)~(Ω(x-y))f(y)dy|~2t~(-3)dt)~2is shown to be of weak type (1,1) and weighted weak type (1,1) with respect to power weight |x|~" if- 1< α< 0, where Ω is homogeneous of degree 0. has mean value 0 and belongs to Llog~+L(S~1).     

On the asymptotic behaviour of the ideal counting function in quadratic number fields

LetK be a quadratic number field with discriminantD and denote byF(n) the number of integral ideals with norm equal ton. Forr≥1 the following formula is proved $$\sum\limits_{n \leqslant x} {F(n)F(n + r) = M_K (r)x + E_K (x,r).} $$ HereM _k (r) is an explicitly determined function ofr which depends onK, and for every ε>0 the error term is bounded by \(|E_K (x,r)|<< |D|^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2} + \varepsilon } x^{{5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0em} 6} + \varepsilon } \) uniformly for \(r<< |D|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} x^{{5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0em} 6}} \) Moreover,E _k (x,r) is small on average, i.e \(\int_X^{2X} {|E_K (x,r)|^2 dx}<< |D|^{4 + \varepsilon } X^{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2} + \varepsilon } \) uniformly for \(r<< |D|X^{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-0em} 4}} \) .     

On the convolution equation with positive kernel expressed via an alternating measure

We consider the integral convolution equation on the half-line or on a finite interval with kernel $$K(x - t) = \int_a^b {e^{ - \left| {x - t} \right|s} d\sigma (s)} $$ with an alternating measure dσ under the conditions $$K(x) > 0, \int_a^b {\frac{1}{s}\left| {d\sigma (s)} \right| < + \infty } , \int_{ - \infty }^\infty {K(x)dx = 2} \int_a^b {\frac{1}{s}d\sigma (s) \leqslant 1} .$$ The solution of the nonlinear Ambartsumyan equation $$\varphi (s) = 1 + \varphi (s) \int_a^b {\frac{{\varphi (p)}}{{s + p}}d\sigma (p)} ,$$ is constructed; it can be effectively used for solving the original convolution equation.     

Sufficient conditions for absolute asymptotic stability of linear equations in a Banach space

For a linear differential equation of the type (1) $$\frac{{dx}}{{dt}} = A_0 x(t) + A_1 x(t - \Delta _1 ) + ... + A_n x(t - \Delta _n )$$ we establish the followingTHEOREM. If $$\overline {\left| {z_1 } \right| = ...\underline{\underline \cup } \left| z \right|_n = 1\sigma \left( {A_0 + \sum\nolimits_{k = 1}^n {z_k A_k } } \right)} \subset \left\{ {\lambda :\operatorname{Re} \lambda< 0} \right\}$$ then system (1) is absolutely asymptotically stable.     

On the existence and smoothness of a generalized solution of a nonlinear differential equation with degeneration

Let Ω be an arbitrary open set in R _n , and let σ(x) and g _i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W _p ^r (Ω, σ, $ \vec g $ ), where r ∈ N, p ≥ 1, and $ \vec g $ (x) = (g ₁(x), g ₂(x), ..., g _n (x)), with the norm $$ \left\| {u;W_p^r (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\left\| {u;L_{p,r}^r (\Omega ;\sigma ,\vec g)} \right\|^p + \left\| {u;L_{p,r}^0 (\Omega ;\sigma ,\vec g)} \right\|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where $$ \left\| {u;L_{p,r}^m (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\sum\limits_{\left| k \right| = m} {\int\limits_\Omega {(\sigma (x)g_1^{k_1 - r} (x)g_2^{k_2 - r} (x) \cdots g_n^{k_n - r} (x)\left| {u^{(k)} (x)} \right|)^p dx} } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W _p ^r (Ω, σ, $ \vec g $ ) for the functional $$ \Phi (u) = \sum\limits_{\left| k \right| \leqslant r} {\frac{1} {{p_k }}\int\limits_\Omega {a_k (x)} \left| {u^{(k)} (x)} \right|^{p_k } } dx - \left\langle {F,u} \right\rangle , $$ where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation.     

Scattering theory for elliptic operators of arbitrary order

We prove the main theorems of scattering theory for selfadjoint elliptic partial differential operators of arbitrary order. Under various hypotheses we show that the wave operators exist and are complete, that the intertwining relations hold, and that the invariance principle holds. One of our main hypotheses is that each lower order coefficientq(x) satisfies. $$(1 + \left| x \right|)^\alpha \int\limits_{\left| {x - y} \right|< a} {\left| {q(y)} \right|dy \in L^p (E^n )}$$ for some α≥0,a>0 and forp≤∞ such that $$\alpha > 1 - \frac{{2n}}{{(n + 1)p}}$$      

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.