Essential norms of some singular integral operators

Let a\alpha and b\beta be bounded measurable functions on the unit circle T. The singular integral operator S_a, bS_{\alpha ,\,\beta } is defined by S_a, b f = aPf + bQf(f ? L² (T))S_{\alpha ,\,\beta } f = \alpha Pf + \beta Qf(f \in L^2 (T)) where P is an analytic projection and Q is a co-analytic projection. In the previous paper, the norm of S_a, bS_{\alpha ,\,\beta } was calculated in general, using a,b\alpha ,\beta and a[`(b)] + H<sup>￥</sup>\alpha \bar {\beta } + H^\infty where H<sup>￥</sup>H^\infty is a Hardy space in L<sup>￥</sup> (T).L^\infty (T). In this paper, the essential norm ||S<sub>a, b</sub> ||<sub>e</sub>\Vert S_{\alpha ,\,\beta } \Vert _e of S<sub>a, b</sub>S_{\alpha ,\,\beta } is calculated in general, using a[`(b)] + H^￥ + C\alpha \bar {\beta } + H^\infty + C where C is a set of all continuous functions on T. Hence if a[`(b)]\alpha \bar {\beta } is in H^￥ + CH^\infty + C then ||S_a, b ||_e = max(||a||_￥ , ||b||_￥ ).\Vert S_{\alpha ,\,\beta } \Vert _e = \max (\Vert \alpha \Vert _\infty , \Vert \beta \Vert _\infty ). This gives a known result when a, b\alpha , \beta are in C.     

Operator-Lipschitz functions in Schatten–von Neumann classes

This paper resolves a number of problems in the perturbation theory of linear operators, linked with the 45-year-old conjecure of M. G. Kreĭn. In particular, we prove that every Lipschitz function is operator-Lipschitz in the Schatten–von Neumann ideals S ^α , 1 < α < ∞. Alternatively, for every 1 < α < ∞, there is a constant c _α > 0 such that

Fourier Series with the Continuous Primitive Integral

Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted A_c(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}) and is a Banach space under the Alexiewicz norm, ||f||_\mathbbT=sup_{|I| ￡ 2p}|ò_I f|\|f\|_{\mathbb{T}}=\sup_{|I|\leq2\pi}|\int_{I} f|, the supremum being taken over intervals of length not exceeding 2π. It contains the periodic functions integrable in the sense of Lebesgue and Henstock–Kurzweil. Many of the properties of L ¹ Fourier series continue to hold for this larger space, with the L ¹ norm replaced by the Alexiewicz norm. The Riemann–Lebesgue lemma takes the form [^(f)](n)=o(n)\hat{f}(n)=o(n) as |n|→∞. The convolution is defined for f ? A_c(\mathbbT)f\in{\mathcal{A}}_{c}(\mathbb{T}) and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate ||f*g||_￥ ￡ ||f||_\mathbbT ||g||_BV\|f\ast g\|_{\infty}\leq\|f\|_{\mathbb{T}} \|g\|_{\mathcal{BV}}. For g ? L¹(\mathbbT)g\in L^{1}(\mathbb{T}), ||f*g||_\mathbbT ￡ ||f||_{\mathbb T} ||g||₁\|f\ast g\|_{\mathbb{T}}\leq\|f\|_{\mathbb {T}} \|g\|_{1}. As well, [^(f*g)](n)=[^(f)](n) [^(g)](n)\widehat{f\ast g}(n)=\hat{f}(n) \hat{g}(n). There are versions of the Salem–Zygmund–Rudin–Cohen factorization theorem, Fejér’s lemma and the Parseval equality. The trigonometric polynomials are dense in A_c(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}). The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let D _n be the Dirichlet kernel and let f ? L¹(\mathbbT)f\in L^{1}(\mathbb{T}). Then ||D_n*f-f||_\mathbbT?0\|D_{n}\ast f-f\|_{\mathbb{T}}\to0 as n→∞. Fourier coefficients of functions of bounded variation are characterized. The Appendix contains a type of Fubini theorem.     

On approximation of periodic functions by Fourier sums

Let L _p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm || f ||_p = ( ò _{- p}^p | f |^p )^{1 \mathord}

Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem

We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l ¹(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||_A(\mathbbT) = || [^(f)] ||_{l¹ (\mathbbZ)}\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || e^inf ||_A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || e^inf ||_A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.     

Uniqueness of a solution to the problem of atmosphere dynamics

We consider the model of atmosphere dynamics and prove the uniqueness of a solution in a bounded domain W ì \mathbbR³ \Omega \subset {\mathbb{R}^3} in the space V(Q) of weak solutions equipped with the finite norm

Minimal rotational hypersurfaces in Minkowski (<Emphasis Type="Italic">α</Emphasis>, <Emphasis Type="Italic">β</Emphasis>)-space

In Finsler geometry, minimal surfaces with respect to the Busemann-Hausdorff measure and the Holmes-Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. In this paper, we give the explicit expressions of BH-minimal and HT-minimal rotational hypersurfaces generated by plane curves rotating around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski (α, β)-space (\mathbbVⁿ⁺¹,[(F_b)\tilde]){(\mathbb{V}^{n+1},\tilde{F_b})} , where \mathbbVⁿ⁺¹{\mathbb{V}^{n+1}} is an (n+1)-dimensional real vector space, [(F_b)\tilde]=[(a)\tilde]f([(b)\tilde]/[(a)\tilde]), [(a)\tilde]{\tilde{F_b}=\tilde{\alpha}\phi(\tilde{\beta}/\tilde{\alpha}), \tilde{\alpha}} is the Euclidean metric, [(b)\tilde]{\tilde{\beta}} is a one form of constant length b:=||[(b)\tilde]||_[(a)\tilde], [(b)\tilde]^\sharp{b:=\|\tilde{\beta}\|_{\tilde{\alpha}}, \tilde{\beta}^{\sharp}} is the dual vector of [(b)\tilde]{\tilde{\beta}} with respect to [(a)\tilde]{\tilde{\alpha}} . As an application, we first give the explicit expressions of the forward complete BH-minimal rotational surfaces generated around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski Randers 3-space (\mathbbV³,[(a)\tilde]+[(b)\tilde]){(\mathbb{V}^{3},\tilde{\alpha}+\tilde{\beta})} .     

White noise eliminates instability

Let x₁,..., x_n be points in the d-dimensional Euclidean space E^d with || x_i-x_j|| ￡ 1\| x_{i}-x_{j}\| \le 1 for all 1 \leqq i,j \leqq n1 \leqq i,j \leqq n, where || .||\| .\| denotes the Euclidean norm. We ask for the maximum M(d,n) of \mathop?_i, j=1ⁿ|| x_i-x_j|| ²\textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| ^{2} (see [4]). This paper deals with the case d = 2. We calculate M(2, n) and show that the value M(2, n) is attained if and only if the points are distributed as evenly as possible among the vertices of a regular triangle of edge-length 1. Moreover we give an upper bound for the value \mathop?_i, j=1ⁿ|| x_i-x_j|| \textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| , where the points x₁,...,x_n are chosen under the same constraints as above.     

Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye–Hückel system

In this paper we establish some regularizing and decay rate estimates for mild solutions of the Debye–Hückel system. We prove that if the initial data belong to the critical Lebesgue space L^\fracn2(\mathbbRⁿ){L^{\frac{n}{2}}(\mathbb{R}^{n})} , then the L ^q -norm ( \fracn2 ￡ q ￡ ￥{\frac{n}{2} \leq q \leq \infty}) of the βth order spatial derivative of mild solutions are majorized by K₁(K₂|b|)^|b|t^{-\frac|b|2-1+\fracn2q}{K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2}-1+\frac{n}{2q}}} for some constants K ₁ and K ₂. These estimates particularly imply that mild solutions are analytic in the space variable, and provide decay estimates in the time variable for higher-order derivatives of mild solutions. We also prove that similar estimates also hold for mild solutions whose initial data belong to the critical homogeneous Besov space [(B)\dot]^-2+\fracnp_p,￥(\mathbbRⁿ){\dot{B}^{-2+\frac{n}{p}}_{p,\infty}(\mathbb{R}^n)} ( \fracn2 < p < n{\frac{n}{2} < p < n}).     

The Jacobi-Dunkl transform on ℝ and the convolution product on new spaces of distributions

In this work, we consider the Jacobi-Dunkl operator Λ _α,β , a 3 b 3 \frac-12\alpha\geq\beta\geq\frac{-1}{2} , a 1 \frac-12\alpha\neq\frac{-1}{2} , on ℝ. The eigenfunction Y_l^a,b\Psi_{\lambda}^{\alpha,\beta} of this operator permits to define the Jacobi-Dunkl transform. The main idea in this paper is to introduce and study the Jacobi-Dunkl transform and the Jacobi-Dunkl convolution product on new spaces of distributions     

Best constant in a three-parameter Poincaré inequality

We study the problem of finding the best constant in the generalized Poincaré inequality

Compact Toeplitz Operators for Weighted Bergman Spaces on Bounded Symmetric Domains

Let W ì \mathbb C^d{\Omega \subset{\mathbb C}^{d}} be an irreducible bounded symmetric domain of type (r, a, b) in its Harish–Chandra realization. We study Toeplitz operators Tⁿ_g{T^{\nu}_{g}} with symbol g acting on the standard weighted Bergman space H_n²{H_\nu^2} over Ω with weight ν. Under some conditions on the weights ν and ν ₀ we show that there exists C(ν, ν ₀) > 0, such that the Berezin transform [(g)\tilde]_n0{\tilde{g}_{\nu_{0}}} of g with respect to H²_n0{H^2_{\nu_0}} satisfies:

Numerical verification of condition for approximately midconvex functions

Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR₊ ? \mathbbR₊{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if

The Łojasiewicz exponent of an analytic function at an isolated zero

Let f be a real analytic function defined in a neighborhood of 0 ? \Bbb Rⁿ 0 \in {\Bbb R}^n such that f^-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|^a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|^b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|^q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|^(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.     

Recurrence in β-expansion over formal Laurent series

In this paper, we study the quantitative recurrence and hitting sets of β-transformation T _β on the unit disk I of formal Laurent series field

Dirichlet and Bergman Spaces of Holomorphic Functions on the Unit Ball of C n

Let B denote the unit ball in [(?)\tilde] \widetilde{\nabla\hskip-4pt}\hskip4pt denote the volume measure and gradient with respect to the Bergman metric on B. In the paper we consider the weighted Dirichlet spaces D_g{{\cal D}_{\gamma}} , $\gamma > (n-1)$\gamma > (n-1) , and weighted Bergman spaces A^p_a{A^p_{\alpha}} , 0 < p < ￥0 < p < \infty , $\alpha > n$\alpha > n , of holomorphic functions f on B for which D_g( f)D_{\gamma}(\,f) and || f||_A^pa\Vert\, f\Vert_{A^p_{\alpha}} respectively are finite, where D_g( f)=ò_B (1-|z|²)^g|[(?)\tilde]  f(z)|²dt(z),D_{\gamma}(\,f)=\int_B (1-\vert z\vert^2)^{\gamma}\vert\widetilde{\nabla\hskip-4pt}\hskip4pt f(z)\vert^2d\tau(z), and || f||^p_A^pa=ò_B(1-|z|²)^a| f(z)|^pdt(z).\Vert\, f\Vert^p_{A^p_{\alpha}}=\int_B(1-\vert z\vert^2)^{\alpha}\vert\, f(z)\vert^pd\tau(z). The main result of the paper is the following theorem.Theorem 1. Let f be holomorphic on B and $\alpha > n$\alpha > n .     

Maps Between Uniform Algebras Preserving Norms of Rational Functions

Let A, B be uniform algebras. Suppose that A ₀, B ₀ are subgroups of A ⁻¹, B ⁻¹ that contain exp A, exp B respectively. Let α be a non-zero complex number. Suppose that m, n are non-zero integers and d is the greatest common divisor of m and n. If T : A ₀ → B ₀ is a surjection with ||T(f)^mT(g)ⁿ - a||_￥ = ||f^mgⁿ - a||_￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f,g ? A₀{f,g \in A_0}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)^d = (T(f)/T(1))^d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? A₀{f \in A_0}. This result leads to the following assertion: Suppose that S _A , S _B are subsets of A, B that contain A ⁻¹, B ⁻¹ respectively. If m, n > 0 and a surjection T : S _A → S _B satisfies ||T(f)^mT(g)ⁿ - a||_￥ = ||f^mgⁿ - a||_￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f, g ? S_A{f, g \in S_A}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)^d = (T(f)/T(1))^d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? S_A{f \in S_A}. Note that in these results and elsewhere in this paper we do not assume that T(exp A) = exp B.     

A convexity theorem for multiplicative functions

We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function g : \mathbbR^m ? [0, ￥]{g : \mathbb{R}^m \rightarrow [{0}, {\infty}]}, the function ${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}, is convex if β ≥ 0 and α ≥ β ₁ + ··· + β _n . We also provide further generalization to functions of the form (x,y₁, . . . , y_n)? g(x)^1+af₁(y₁)^-b₁ ···f_n(y_n)^-b_n{({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} } with the f _k concave, positively homogeneous and nonnegative on their domains.     

Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms

We introduce new potential type operators J^a_b = (E+(-D)^b/2)^-a/bJ^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}, (α > 0, β > 0), and bi-parametric scale of function spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta , p}({\mathbb{R}}^n) associated with J^α_β. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta, p}({\mathbb{R}}^n) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1).