Об аппроксимации свермками и баэисах иэ сдвигов функции

For 2π-periodic functions f ∈ L ^p ( $ \mathbb{T} $ ), 1 ≤ p < ∞, σ ∈ V ( $ \mathbb{T} $ ) and g ∈ L( $ \mathbb{T} $ ), we consider the convolutions $$ (f*d\sigma )_T (x) = \int_0^{2\pi } {f(x - t)d\sigma (t), } (f*g)_T (x) = \int_0^{2\pi } {f(x - t)g(t)dt.} $$ For fixed functions σ ∈ V ( $ \mathbb{T} $ ) and g ∈ L( $ \mathbb{T} $ ), necessary and sufficient conditions are obtained for the density of the ranges of these operators in L ^p . Similar result is proved for the dyadic convolution $$ (f*g)_2 (x) = \int_0^1 {f(x \oplus t)g(t)dt,} $$ where ⊕ is the operation of dyadic addition on [0, 1). Moreover, it is proved that in the spaces L ^p ( $ \mathbb{T} $ ), 1 ≤ p ∞, and C( $ \mathbb{T} $ ) there exist no bases of shifts of a function. Similar results are obtained for the spaces L ^p [0, 1]*, 1 ≤ p < ∞, and C[0, 1]* relative to dyadic shifts, where [0, 1]* is the modified segment [0, 1]. It is also proved that in the space L(?₊) there exists no basis of dyadic shifts of a function.     

Almost periodic solutions for nonlinear delay evolutions with nonlocal initial conditions

We consider the nonlinear delay differential evolution equation $$\left\{\begin{array}{ll} u'(t) \in Au(t) + f(t, u_t), \quad \quad t \in \mathbb{R}_+,\\ u(t) = g(u)(t),\qquad \qquad \quad t \in [-\tau, 0], \end{array} \right.$$ u ′ ( t ) ∈ A u ( t ) + f ( t , u t ) , t ∈ R + , u ( t ) = g ( u ) ( t ) , t ∈ [ - τ , 0 ] , where τ ≥ 0, X is a real Banach space, A is the infinitesimal generator of a nonlinear semigroup of contractions whose Lipschitz seminorm decays exponentially as ${t \mapsto {\rm{e}}^{-\omega t}}$ t ? e - ω t when ${t \to + \infty}$ t → + ∞ and ${f : {\mathbb{R}}_+ \times C([-\tau, 0]; \overline{D(A)}) \to X}$ f : R + × C ( [ - τ , 0 ] ; D ( A ) ¯ ) → X is jointly continuous. We prove that if f Lipschitz with respect to its second argument and its Lipschitz constant ? satisfies the condition ${\ell{\rm{e}}^{\omega\tau} < \omega, g : C_b([-\tau, +\infty); \overline{D(A)}) \to C([-\tau, 0]; \overline{D(A)})}$ ? e ω τ < ω , g : C b ( [ - τ , + ∞ ) ; D ( A ) ¯ ) → C ( [ - τ , 0 ] ; D ( A ) ¯ ) is nonexpansive and (I – A)^?1 is compact, then the unique C ⁰-solution of the problem above is almost periodic.     

Invariant Translative Mappings and a Functional Equation

Let \({K,M,N : \mathbb{R}^{2} \rightarrow \mathbb{R}}\) be translative functions. Then K is invariant with respect to the mapping \({(M,N) : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}}\) if and only if the functions \({h = K(\cdot, 0), f = M(\cdot, 0), g = N(\cdot, 0)}\) satisfy the functional equation $$h(x) = h(f(x) - g(x)) + g(x),\,\, x\in \mathbb{R}.$$ If K, M, N are means, then h(0) =  f(0) =  g(0) = 0. The formal power solutions and analytic solutions of this functional equation, satisfying these initial conditions, are considered.     

An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

We consider the weighted space W ₁ ⁽²⁾ (?,q) of Sobolev type $$W_1^{(2)} (\mathbb{R},q) = \left\{ {y \in A_{loc}^{(1)} (\mathbb{R}):\left\| {y''} \right\|_{L_1 (\mathbb{R})} + \left\| {qy} \right\|_{L_1 (\mathbb{R})} < \infty } \right\} $$ and the equation $$ - y''(x) + q(x)y(x) = f(x),x \in \mathbb{R} $$ Here f ε L ₁(?) and 0 ? q ∈ L ₁ ^loc (?). We prove the following:

1. The problems of embedding W ₁ ⁽²⁾ (?q) ? L ₁(?) and of correct solvability of (1) in L ₁(?) are equivalent
2. an embedding W ₁ ⁽²⁾ (?,q) ? L ₁(?) exists if and only if $$\exists a > 0:\mathop {\inf }\limits_{x \in R} \int_{x - a}^{x + a} {q(t)dt > 0} $$

     

Commutators of the fractional maximal function on variable exponent Lebesgue spaces

Let \({M_\beta }\) be the fractional maximal function. The commutator generated by \({M_\beta }\) and a suitable function b is defined by \([{M_\beta },b]f = {M_\beta }(bf) - b{M_\beta }(f)\) . Denote by P(? ⁿ ) the set of all measurable functions p(·): ? ⁿ → [1,∞) such that $1 < p_ - : = \mathop {es\sin fp(x)}\limits_{x \in \mathbb{R}^n } andp_ + : = \mathop {es\operatorname{s} \sup p(x) < \infty }\limits_{x \in \mathbb{R}^n } ,$ and by B(? ⁿ ) the set of all p(·) ∈ P(? ⁿ ) such that the Hardy-Littlewood maximal function M is bounded on L ^p(·)(? ⁿ ). In this paper, the authors give some characterizations of b for which \([{M_\beta },b]\) is bounded from L ^p(·)(? ⁿ ) into L ^q(·)(? ⁿ ), when p(·) ∈ P(? ⁿ ), 0 < β < n/p ₊ and 1/q(·) = 1/p(·) ? β/n with q(·)(n ? β)/n ∈ B(? ⁿ ).     

On mappings preserving orthogonality of non-singular vectors

Letq be a regular quadratic form on a vector space (V,F) and letf be the bilinear form associated withq. Then, \(\dot V: = \{ z \in V|q(z) \ne 0\} \) is the set of non-singular vectors ofV, and forx, y ∈ \(\dot V\) , ?(x, y) ?f(x, y) ²/(q(x) · q(y)) is theq-measure of (x, y), where ?(x,y)=0 means thatx, y are orthogonal. For an arbitrary mapping \(\sigma :\dot V \to \dot V\) we consider the functional equations $$\begin{gathered} (I)\sphericalangle (x,y) = 0 \Leftrightarrow \sphericalangle (x^\sigma ,y^\sigma ) = 0\forall x,y \in \dot V, \hfill \\ (II)\sphericalangle (x,y) = \sphericalangle (x^\sigma ,y^\sigma )\forall x,y \in \dot V, \hfill \\ (III)f(x,y)^2 = f(x^\sigma ,y^\sigma )^2 \forall x,y \in \dot V, \hfill \\ \end{gathered} $$ and we state conditions on (V,F,q) such thatσ is induced by a mapping of a well-known type. In case of dimV ∈N?{0, 1, 2} ∧ ∣F∣ > 3, each of the assumptions (I), (II), (III) implies that there exist aρ-linear injectionξ :V →V and a fixed λ ∈F?{0} such thatF x ^σ =F x ^ξ ?x ∈ \(\dot V\) andf(x ^ξ,y ^ξ)=λ · (f(x, y))^ρ ?x, y ∈V. Moreover, (II) implies ρ =id _F ∧q(x ^ξ) = λ ·q(x) ?x ∈ \(\dot V\) , and (III) implies ρ=id _F ∧ λ ∈ {1,?1} ∧x ^σ ∈ {x ^ξ, ?x ^ξ} ?x ∈ \(\dot V\) . Other results obtained in this paper include the cases dimV = 2 resp. dimV ?N resp. ∣F∣ = 3.     

The completeness of systems of functions of the Mittag-Leffler type for weighted uniform approximation in a complex domain

For a givenρ(1/2 <ρ < + ∞) let us set L _ρ = {z: |arg z| = π/(2ρ)} and assume that a real valued measurable function ?(t) such that ?(t) ≥ 1(t ∈ L _ρ ) and \(\mathop {\lim }\limits_{|t| \to + \infty } \varphi (t) = + \infty (t \in L_\rho )\) is defined on L _ρ . Let C _? (L _ρ ) denote the space of continuous functionsf(t) on L _ρ such that \(\lim \tfrac{{f(t)}}{{\varphi (t)}} = 0\) , where the norm of an elementf is defined as: \(\parallel f\parallel = \mathop {\sup }\limits_{t \in L_\rho } \tfrac{{|f(t)|}}{{\varphi (t)}}\) . In this note we pose the question about the completeness of the system of functions of the Mittag-Leffler type {Eρ(ut; μ)} (μ ≥ 1, 0 ≤ u ≤a) or, what is the same thing, of the system of functions \(p(t) = \int_0^a {E_\rho (ut;\mu )d\sigma (u)} \) in C _? (L _ρ ). The following theorem is proved: The system of functions of the Mittag-Leffler type is complete in C _? (L _ρ ) if and only if sup |p(z)| ≡ +∞, z ∈ L _ρ , where the supremum is taken over the set of functions p(t) such that ∥p(t) (t + 1)^?1 ∥ ≤ 1.     

A sharp weightedL 2-estimate for the solution to the time-dependent Schrödinger equation

For Ξ∈R ⁿ ,t∈R andf∈S(R ⁿ ) define $\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$ . We determine the optimal regularitys ₀ such that $\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$ holds whereC is independent off∈S(R ⁿ ) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11]. Our theorems can be generalized to the case where the exp(it|ξ|²) is replaced by exp(it|ξ|^a),a≠2. The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL ²(R ⁿ ) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity.     

On a result of Hardy and Littlewood concerning Fourier series

Let f ∈ L ¹( $ \mathbb{T} $ ) and assume that $$ f\left( t \right) \sim \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos kt + b_k \sin kt} \right)} $$ Hardy and Littlewood [1] proved that the series $ \sum\limits_{k = 1}^\infty {\frac{{a_k }} {k}} $ converges if and only if the improper Riemann integral $$ \mathop {\lim }\limits_{\delta \to 0^ + } \int_\delta ^\pi {\frac{1} {x}} \left\{ {\int_{ - x}^x {f(t)dt} } \right\}dx $$ exists. In this paper we prove a refinement of this result.     

Nonlinear elliptic equations of order 2m and subdifferentials

LetA be an operator of the calculus of variations of order 2m onW ^m,p (Ω) andj a normal convex integrand. Forf ∈L ^p (Ω), the equation $$\mathcal{A}u + \partial j(x,u) \ni f, in \Omega , u - \phi \in W_0^{m,p} (\Omega ),$$ may have no strong solutions whenm>1, even ifj is independent ofx and φ=0. However, we obtain existence results whenj is everywhere finite and $$\int_\Omega {j(x,\phi ) dx< + \infty ,} $$ by the study of the subdifferential of the function $$\upsilon \mapsto \int_\Omega {j(x,\upsilon + \phi ) dx on W_0^{m,p} (\Omega ).} $$      

Bounds for singular integrals associated with a class of hypersurfaces

For the hypersurface Γ=(y,γ(y)), the singular integral operator along Γ is defined by. $$Tf(x,x_n ) = P.V.\int_{\mathbb{R}^n } {, f(x - y,x_n ) - } \gamma (y))_{\left| y \right|^{n - 1} }^{\Omega (v)} dy$$ where Σ is homogeneous of order 0, $ \int_{\Sigma _{n \lambda } } {\Omega (y')dy'} = 0 $ . For a certain class of hypersurfaces, T is shown to be bounded on L^p(Rⁿ) provided Ω∈L _α ¹ (Σ _n?2),P>1.     

On certain properties of linear summation methods of Fourier series on the classesC(ɛ)

Для линейных методов суммирования рядов Ф урье (1) $$L_n (f;x) = \frac{1}{\pi }\mathop \smallint \limits_{ - \pi }^\pi f(x + t)\left( {\frac{1}{2} + \sum\limits_{k = 1}^n {\lambda _{k,n} } \cos kt} \right)dt$$ на классах $$C(\varepsilon ) = \{ f:E_n (f) \leqq \varepsilon _n ;\forall n \geqq 0\} ,\varepsilon = \{ \varepsilon _n \} _{n = 0.}^\infty \varepsilon _n \downarrow 0,$$ доказываются:

1. оценки для порядка р оста норм ∥{L_n∥, если из вестен порядок приближения операторами (1) некоторого классаС (?) (при этом, если опера торы (1) приближают класс С(е) с наилучшим порядком, то находится точная а симптотика возрастания норм {∥ L_n∥);
2. сравнительные оцен ки порядков приближе ния классовС(?) операторами (1), если известен порядок при ближения ими некотор ого более узкого класса С(?*).

В том случае, когда опе раторы (1) приближают кл асс С(?*) с наилучшим порядком, получаются точные по рядковые оценки для л юбого более широкого класса С(?).     

An abstract functional differential equation and a related nonlinear Volterra equation

We study the existence, uniqueness, regularity and dependence upon data of solutions of the abstract functional differential equation 1 $$\frac{{du}}{{dt}} + Au \ni G(u) (0 \leqq t \leqq T), u(0) = x,$$ , whereT>0 is arbitrary,A is a givenm-accretive operator in a real Banach spaceX, and \(G:C([0,T]; \overline {D(A)} ) \to L^1 (0, T; X)\) is a given mapping. This study provides simple proofs of generalizations of results by several authors concerning the nonlinear Volterra equation 2 $$u(t) + b * Au(t) \ni F(t) (0 \leqq t \leqq T),$$ , for the case in which X is a real Hilbert space. In (2) the kernelb is real, absolutely continuous on [0,T],b*g(t)=∫ ₀ ¹ (t?s)g(s)ds, andf∈W ^1,1(0,T;X).     

On the homogenization of noncoercive variational problems

The paper deals with variational problems of the form $$\mathop {\inf }\limits_{u \in W^{1,p} (\Omega )} \int\limits_\Omega {a(\varepsilon ^{ - 1} x)(\left| {\nabla u} \right|^p + \left| {u - g} \right|^p )} dx,$$ where Ω is a bounded Lipschitzian domain in ? ^N , g∈L^p(Ω). The function a(x) is assumed to satisfy the following conditions:

1. a(x) is periodic and lower semicontinuous;
2. 0≤a(x)≤1 and the set {∈? ^N , a(x)>0} is connected in ? ^N Under these conditions, basic properties of homogenization (convergence of energies and generalized solutions) and properties of Г-convergence type are proved. Bibliography: 3 titles.

     

An extremum problem on a class of differentiable functions of several variables

On the multidimensional classW ₀ ^r H _ω ⁽ⁿ⁾ of continuous periodic functionsF with therth derivativeD ^r F from $$H_\omega ^{(n)} = \left\{ {f \in C| |f(x) - f(y)| \leqslant \sum\limits_{i = 1}^n {\omega _i } (|x_i - y_i |)\forall x, y \in \mathbb{R}^n } \right\}$$ (where the ω _i (x _i ) are the convex moduli of continuity) and zero mean with respect to each variable, we obtain the exact value of $$M_r (\omega ) = \mathop {\sup }\limits_{F \in W_0^r H_\omega ^{(n)} } \left\| F \right\|c$$ .     

A priori estimates for a class of degenerate elliptic equations

In this paper we investigate the regularity of solutions for the following degenerate partial differential equation $$\left \{\begin{array}{ll} -\Delta_p u + u = f \qquad {\rm in} \,\Omega,\\ \frac{\partial u}{\partial \nu} = 0 \qquad \qquad \,\,\,\,\,\,\,\,\,\, {\rm on} \,\partial \Omega, \end{array}\right.$$ when ${f \in L^q(\Omega), p > 2}$ and q ≥ 2. If u is a weak solution in ${W^{1, p}(\Omega)}$ , we obtain estimates for u in the Nikolskii space ${\mathcal{N}^{1+2/r,r}(\Omega)}$ , where r = q(p ? 2) + 2, in terms of the L ^q norm of f. In particular, due to imbedding theorems of Nikolskii spaces into Sobolev spaces, we conclude that ${\|u\|^r_{W^{1 + 2/r - \epsilon, r}(\Omega)} \leq C(\|f\|_{L^q(\Omega)}^q + \| f\|^{r}_{L^q(\Omega)} + \|f\|^{2r/p}_{L^q(\Omega)})}$ for every ${\epsilon > 0}$ sufficiently small. Moreover, we prove that the resolvent operator is continuous and compact in ${W^{1,r}(\Omega)}$ .     

On a functional equation of Bruce Ebanks

In the paper Brillouët-Belluot and Ebanks (Aequationes Math 60:233–242, 2000), the authors found all continuous functions f: [0, 1] → [0, + ∞) which verify f(0) = f(1) = 0 and the functional equation $$f(xy +c f(x) f(y)) = x f(y) + y f(x) +d \, f(x) f(y)$$ where c and d are given real numbers with c ≠ 0. In the present paper we obtain all continuous solutions ${f: \mathbb{R} \rightarrow \mathbb{R}}$ of the functional equation (1).     

On the continuity of the sharp constant in the Jackson-Stechkin inequality in the space L 2

This paper deals with the continuity of the sharp constant K(T,X) with respect to the set T in the Jackson-Stechkin inequality $E(f,L) \leqslant K(T,X)\omega (f,T,X),$ , where E(f,L) is the best approximation of the function f ∈ X by elements of the subspace L ? X, and ω is a modulus of continuity, in the case where the space L ²( $\mathbb{T}^d $ , ?) is taken for X and the subspace of functions g ∈ L ²( $\mathbb{T}^d $ , ?), for L. In particular, it is proved that the sharp constant in the Jackson-Stechkin inequality is continuous in the case where L is the space of trigonometric polynomials of nth order and the modulus of continuity ω is the classical modulus of continuity of rth order.     

Estimates of maximal functions measuring local smoothness

Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(I ⁿ ) (I≡[0,1]. Set $${\mathcal{N}}_\eta f(x) = \sup \frac{1}{{\left| Q \right|\eta (\left| Q \right|^{1/n} )}} \smallint _Q \left| {f(t) - f(x)} \right|dt,$$ , where the supremum is taken over all cubes containing the pointx. Forη=t ^α (0<α≤1) this definition was given by A.Calderón. In the paper we prove estimates of the maximal functions ${\mathcal{N}}_\eta f$ , along with some embedding theorems. In particular, we prove the following Sobolev type inequality: if $$1 \leqslant p< q< \infty , \theta \equiv n(1/p - 1/q)< 1, and \eta (t) \leqslant t^\theta \sigma (t),$$ , then $$\parallel {\mathcal{N}}_\sigma {f} {\parallel_{q,p}} \leqslant c \parallel {\mathcal{N}}_\eta {f} {\parallel_p} .$$ . Furthermore, we obtain estimates of ${\mathcal{N}}_\eta f$ in terms of theL ^p -modulus of continuity off. We find sharp conditions for ${\mathcal{N}}_\eta f$ to belong toL ^p (I ⁿ ) and the Orlicz class?(L), too.     

Multiplicity and concentration of positive solutions for the Schr?dinger�CPoisson equations

This paper is concerned with the multiplicity and concentration of positive solutions for the nonlinear Schr?dinger?CPoisson equations $$ \left\{ \begin{array}{l@{\quad}l} -\varepsilon^2\triangle u+V(x)u+\phi(x) u=f(u)& {\rm in}\,{\mathbb R}^3, \\ -\varepsilon^2\triangle \phi=u^2 & {\rm in}\,{\mathbb R}^3, \\ u\in H^1({\mathbb R}^3), u(x) > 0,& \forall x\in{\mathbb R}^3, \\ \end{array} \right. $$ where ???>?0 is a parameter, ${V: {\mathbb R}^3\rightarrow{\mathbb R}}$ is a continuous function and ${f: {\mathbb R}\rightarrow {\mathbb R}}$ is a C ¹ function having subcritical growth. The proof of the main result is based on minimax theorems and the Ljusternik?CSchnirelmann theory.