We investigate the analytic properties of a new class of special functions that appear in the kernels of a class of integral operators underlying the dynamics of matter relaxation processes in attractive fields. These functions, recently introduced by the second author, generate the kernels of the principal parts of these operators and play an important role in understanding their spectral characteristics

This paper is devoted to studying the boundedness and compactness of commutators for Marcinkiewicz integrals with rough kernels in weighted Lebesgue spaces. The characterized theorems on the boundedness and compactness for such commutators are established, respectively. We improve and extend the previous results. Meanwhile, the quantitative weighted bounds for the commutator of Marcinkiewicz integral

We prove strict necessary density conditions for coherent frames and Riesz sequences on homogeneous groups. Let N be a connected, simply connected nilpotent Lie group with a dilation structure (a homogeneous group) and let (π,Hπ) be an irreducible, square-integrable representation modulo the center Z(N) of N on a Hilbert space Hπ of formal dimension dπ. If g ∈ Hπ is an integrable vector and the set

Sendov’s conjecture states that if all the zeroes of a complex polynomial P(z) of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of P(z). In a paper that appeared in 2014, Dégot proved that, for each a ∈ (0, 1), there exists an integer N such that for any polynomial P(z) with degree greater than N, if P(a) = 0 and all zeroes lie inside the unit

We show that, for each compact subset of the real line of infinite cardinality with an isolated point, the space of Whitney jets on the set does not possess a basis consisting only of polynomials. On the other hand, polynomials are dense in any Whitney space. Thus, there are no general results about stability of bases in Fréchet spaces.

In this paper, we study the problem of approximation to reconstruct the value of a function of several variables from the approximate values of the Fourier coefficients of a function at a given fixed point. A theorem on the reconstruction of the value of a function with several variables at any given fixed point with a small error is given.

Let f(z) be a non-constant meromorphic (entire) function of hyper-order strictly less than 1, n ≥ 3 (n ≥ 2) be an integer. It is shown that if fn(z) and fn(z + c) share a(≠ 0) ∈ ℂ and ∞ CM, then f(z) = t1f(z + c) or f(z) = t2f(z + 2c), where t1 and t2 satisfy \(t_i^n=1\), (i = 1, 2). Some examples are provided to show the sharpness of our results. In addition, we mainly obtain two uniqueness results

We show the following result: Let \(A, B \in \mathbb{B}(\mathcal{H})\) be two strictly positive operators such that A ≤ B and \(m1_{\mathcal{H}}\leq{B}\leq{M1_{\mathcal{H}}}\) for some scalars 0 < m < M. Then $$B^p\leq{\rm{exp}}(\frac{M1_{\mathcal{H}}-B}{M-m}{\rm{ln}} \;m^p+\frac{B-m1_{\mathcal{H}}}{M-m}{\rm{ln}} \;M^p)\leq{K(m, M, p, q)}A^q$$ for p ≤ 0, −1 ≤ q ≤ 0 where K(m, M, p, q) is the generalized

In this article we study the probability that the maximum over a symmetric interval [−a, a] of a univariate polynomial of degree at most n is attained at an endpoint. We give explicit formulas for the degree n = 1, 2,3 cases. The formula for the degree 3 case leads to a lower bound for the true probability. Numerical experiments indicate that this lower bound is actually quite a good approximation

We carry out complete membership to Kaplan classes of polynomials with all zeros on unit circle. In this way we extend Sheil-Small’s and Jahangiri’s theorems.

Using a local analogue of the Wiener-Levi theorem, we investigate the class of measures on Euclidean space with discrete support and spectrum. Also, we find a new sufficient conditions for a discrete set in Euclidean space to be a coherent set of frequencies.

We continue the study in [1] in the setting of pluripotential theory arising from polynomials associated to a convex body C in (ℝ+)d. Here we discuss C-Robin functions and their applications. In the particular case where C is a triangle in (ℝ+)2 with vertices (0, 0), (b, 0), (0, a), a, b > 0, we generalize results of T. Bloom to construct families of polynomials which recover the C-extremal function

We continue the study of several related extremal problems for functions analytic in a simply connected domain G with a rectifiable Jordan boundary Γ. In particular, the problem of optimal recovery of a derivative at a point z0 ∈ G from limit boundary values given with an error on a measurable part γ1 of the boundary Γ for the class Q of functions with limit boundary values bounded by 1 on γ0 = Γ γ1

In this paper, we study a Trudinger-Moser inequality with a vanishing weight in the unit disk. Precisely, let \(B\) be the unit disk and β ≥ 0 be a real number. Denote \({\cal S} = \left\{{u \in W_0^{1,2}\left(B\right);u} \right.\) is a radially symmetric function and ∥∇u∥2≤ 1. Suppose a function h(x) is radially symmetric, nonnegative, continuous on \(\overline{\mathbb{B}}\) and satisfies h(x) > 0

We discuss the problem of the best uniform approximation on the axis, i.e., in the space C(−∞,∞), of the differentiation operator of order k on the class of functions with bounded derivative of order n, 0 < k < n, by bounded linear operators in the space Lr,1 ≤ r < ∞. We give an exact solution of the problem for odd n ≥ 3 and all k, 0 < k < n. For even n, a result close to the best one is obtained

The sum of a sine series \(g\left({b,x} \right) = \sum\nolimits_{k = 1}^\infty {}\)bk sin kx with coefficients forming a convex sequence b is known to be positive on the interval (0,π). To estimate its values near zero Telyakovskiĭ used the piecewise-continuous function \(\sigma \left({{\bf{b}},x} \right) = \left({1/m\left(x \right)} \right)\sum\nolimits_{k = 1}^{m\left(x \right) - 1} {{k^2}\left({{b_k}

The well-known stationary phase formula gives us a way to find asymptotics of oscillating integrals so long as the symbol is regular enough (in comparison to the large parameter controlling the oscillation). However in a number of applications we find ourselves with symbols that are not suitably regular. In this paper we obtain decay bounds for such oscillatory integrals.

In this paper, a new characterization is provided for the boundedness, compactness and essential norm of the difference of two weighted composition operators on weighted-type spaces in the unit ball of ℂn.

In this paper we study analogues of the perfect splines for weighted Sobolev classes of functions defined on the half-line. Maximally oscillating splines play important role in the solution of certain extremal problems. In particular, using these splines, we characterize the modulus of continuity of the differential operator.

Let ℋ and \({\cal K}\) be complex infinite dimensional separable Hilbert spaces. We denote by \({M_C} = \left( {\begin{array}{*{20}{c}} A&C \\ 0&B \end{array}} \right)\) a 2 × 2 upper triangular operator matrix acting on \({\cal H} \oplus {\cal K}\), where \(A \in {\cal B}\left({\cal H} \right),\,B \in {\cal B}\left({\cal K} \right)\) and \(C \in {\cal B}\left({{\cal K},{\cal H}} \right)\). In this

This paper is a follow up of a recent work of the author and Y. Sawano [6], in 2019. We investigate the boundedness of the bilinear fractional integral operator introduced by Grafakos in [3], 1992. When the local integrability index s falls 1 with weights and t exceeds 1, He and Yan obtained some results on this operator was worked on Morrey spaces earlier in [7] 2018. Later in the paper [6] we considered

Steinmetz [16] considered the first order non-linear differential equations$$C(z, f)(f^\prime)^2+B(z, f)f^\prime+A(z, f)=0,$$where A(z, f), B(z, f), C(z, f) are polynomials in f with rational coefficients in z and pointed out that the above equation must reduce into some certain types when it admits transcendental meromorphic solutions. In this paper, we will consider its difference version$$C(z, f)f(z+c)^2+B(z

We consider the Hankel operator with symbols of forms and study the relationship between the compactness of this operator and the compactness of ∂̄-Neumann operator.

In this paper, B-Fredholm linear relations in Hilbert spaces are introduced and some of their properties are given. In particular, it is shown that they are completely characterized in terms of an algebraic decomposition with a Fredholm linear relation and a bounded nilpotent operator. The behaviour of a polynomial in them is also investigated.

We provide sufficient conditions on the wavelets on a local field K for the wavelet system to form an unconditional basis for the Hardy space H1(K) and the Lebesgue spaces Lp(K), 1 < p < ∞.

Let \(\mathcal{B}(\mathcal{H})\) be the algebra of all bounded linear operators on a separable complex Hilbert space \(\mathcal{H}\). We introduce the J-decomposition property for projections in \(\mathcal{B}(\mathcal{H})\), and prove that the projection E in \(\mathcal{B}(\mathcal{H})\) has J-decomposition property with respect to a particular space decomposition, which is related to Hal-mos’ two

We give an exposition of some simple but applicable cases of worst-case growth of a polynomial in terms of its uniform norm on a given compact set K ⊂ ℂd. Included is a direct verification of the formula for the pluripotential extremal function for a real simplex. Throughout we attempt to make the exposition as accessible to a general (analytic) audience as possible, avoiding wherever possible the

We prove a Hille–Yosida type theorem for relatively uniformly continuous positive semigroups on vector lattices. We introduce the notions of relatively uniformly continuous, differentiable, and integrable functions on ℝ+. These notions allow us to study the generators of relatively uniformly continuous semigroups. Our main result provides sufficient and necessary conditions for an operator to be the

In this paper we first introduce the Heron and Heinz means of two convex functionals. Afterwards, some inequalities involving these functional means are investigated. The operator versions of our theoretical functional results are immediately deduced. We also obtain new refinements of some known operator inequalities via our functional approach in a fast and nice way.

The paper addresses questions on existence and structure of universal functions for spaces L1(E), E ⊂ [0, 1)2, with respect to the doubleWalsh system in the sense of signs of Fourier coefficients. It is shown that for each ε > 0 one can find a measurable set Eε ⊂ [0, 1)2 with measure |Eε| > 1−ε, such that by a proper modification of any integrable function f ∈ L1[0, 1)2 outside Eε one can get an integrable

For a locally compact group G, let \(\mathcal{L}^{\Phi}\)(G) and \(\mathcal{L}_\omega^{\Phi}\)(G) be Orlicz and weighted Orlicz spaces, respectively, where Φ is a Young function and ω is a weight on G. We study the harmonic and convolution operators on Orlicz and weighted Orlicz spaces. We prove that under some conditions the harmonic operators on \(\mathcal{L}^{\Phi}\)(G) and \(\mathcal{L}_\omega^{\Phi}\)(G)

In this paper, we study the weight log canonical thresholds of holomorphic functions. We prove the ascending chain condition for certain weight in dimension two.

We study generalized inner functions on a large family of Reproducing Kernel Hilbert Spaces. We show that the only inner functions which are entire are the normalized monomials.

Let λ2 ∈ ℕ, and in dimensions d ≥ 5, let Aλf(x) denote the average of f: ℤd → ℝ over the lattice points on the sphere of radius λ centered at x. We prove ℓp improving properties of Aλ:$$\begin{array}{*{20}{c}} {{{\left\| {{A_\lambda }} \right\|}_{{\ell ^{p \to }}{\ell ^{p'}}}} \leqslant {C_{d,p,\omega ({\lambda ^2})}}{\lambda ^{(1 - \tfrac{2}{p})}},}&{\frac{{d - 1}}{{d + 1}} < p \leqslant \frac{d}{{d

The aim of this paper is to establish the boundedness of the commutator [b1, b2, Tθ], which generated by the bilinear θ-type Calderón–Zygmund operators Tθ and the functions \(b_1, b_2 \in \widetilde {RBMO}(\mu)\), on non-homogeneous metric measure space satisfying the so-called geometrically doubling and the upper doubling conditions. Under the assumption that the dominating function λ satisfies the

In this article we obtain a characterization of the class of p-convergent operators between two Banach spaces in terms of p-(V) subsets of the dual space. Also, for 1 ≤ p < q ≤ ∞, by introducing the concepts of Pelczyński's properties (V)p,q and (V*)p,q, we obtain a condition that ensures that q-convergent operators are p-convergent operators. Some characterizations of the p-Schur property of Banach

Let d ∈ {3, 4, 5,...} and p ∈ (0, 1]. We consider the Hermite operator L = −Δ + |x|2 on its maximal domain in L2(ℝd). Let \(H_L^p(\mathbb{R}^d)\) be the completion of \(\left\{ {f \in {L^2}({\mathbb{R}^d}):{\mathcal{M}_L}f \in {L^p}({\mathbb{R}^d})} \right\}\) with respect to the quasi-norm \({\left\| \cdot \right\|_{H_L^p}} = {\left\| {{\mathcal{M}_L} \cdot } \right\|_{{L^p}}}\), where \({\mathcal{M}_L}f(

We define the weighted relative p(.)-capacity and discuss its properties in the space \(W_\vartheta ^{1,p(.)}({\mathbb{R}^n})\). Also, we investigate some properties of the weighted variable Sobolev capacity. It is shown that there is a relation between these two capacities. Moreover, we introduce the notion of thinness related to this newly defined relative capacity and prove an equivalence statement

It is known that the maximal operator σ* ƒ is of type (Hp,Lp) if the Vilenkin group G is bounded and \(p > \tfrac{1}{2}\). We prove a maximal converse inequality which characterizes the space Hp by means of the operator σ†ƒ:= supn |σMnƒ|, for bounded groups.

This paper aims to highlight new properties of the centroid of the zeroes of a polynomial. As an illustration, we apply these techniques to O-classical orthogonal polynomials, where O is the derivative operator D or the q-derivative Hq.

In this paper we show that the maximum of a polynomial P ∈ ℂ[z] of degree d ≥ 2 in the unit disc can be bounded below by the sum of moduli of its two coefficients, say, for zs and zt under certain assumption on the pair s < t. We also show that this assumption on s, t cannot be removed or weakened and give several examples showing when this lower bound is (or is not) attained.

The primary purpose of this paper is to investigate the question of the invertibility of the sum of operators in the bounded and unbounded setting. Some interesting examples and consequences are given. As an illustrative point, we characterize invertibility for the class of normal operators. Also, we give a very short proof of the self-adjointness of a normal operator which has a real spectrum.

We consider a cofinite Fuchsian group of the first kind with finitely many inequivalent parabolic elements and a unitary multiplier system of an arbitrary weight on it. Through the Gallagher–Koyama approach to the prime geodesic theorem on the corresponding noncompact hyperbolic surface, we reduce the exponent in the error term from \(\frac{3}{4}\) to \(\frac{7}{10}\) outside a set of finite logarithmic

We characterize the dual spaces of the generalized Hardy spaces defined by replacing Lebesgue quasi-norms by Wiener amalgam ones. In these generalized Hardy spaces we prove that some classical linear operators such as Calderón–Zygmund, convolution and Riesz potential operators are bounded.

In this paper we present several sharp upper bounds for the numerical radii of the diagonal and off-diagonal parts of the 2×2 block operator matrix \(\left[ {\begin{array}{*{20}{c}} A&B \\ C&D \end{array}} \right]\). Among extensions of some results of Kittaneh et al., it is shown that if \(T = \left[ {\begin{array}{*{20}{c}} A&0 \\ 0&D \end{array}} \right]\), and f and g are non-negative continuous

We study the autocorrelation coefficients of the Rudin–Shapiro polynomials, proving in particular that their maximum on the interval [1, 2n) is bounded from below by C12αn and is bounded from above by C22α’n where α = 0.7302852 ... and α′ = 0.7302867....

In this paper, the calculus of functions with values in L-spaces is developed. We then consider nonlinear integral equations of Fredholm and Volterra types for functions with values in L-spaces. Such class of equations includes set-valued integral equations, fuzzy integral equations, and many others. We prove theorems of existence and uniqueness of the solutions of such equations and investigate data

In this paper we define and study signed deficient topological measures and signed topological measures (which generalize signed measures) on locally compact spaces. We prove that a signed deficient topological measure is τ-smooth on open sets and τ-smooth on compact sets. We show that the family of signed measures that are differences of two Radon measures is properly contained in the family of signed

By means of the linearization method, we investigate a large class of terminating 3F2(\(\frac{4}{3}\))-series with two extra integer parameters. Three analytical formulae are established and twenty closed expressions are deduced as examples.

The Wigner-Ville distribution (WVD) and the quaternion offset linear canonical transform (QOLCT) are useful tools in signal analysis and image processing. The purpose of this paper is to define the Wigner-Ville distribution associated with the quaternionic offset linear canonical transform (WVD-QOLCT). Actually, this transform combines both the results and flexibility of the two transforms WVD and

We prove that there is an absolute constant A > 0 such that$$\begin{array}{l} \begin{array}{*{20}{c}} {\max } \\ { - 1 \le x \le 1} \\ \end{array}|P'(x)| \ge A\sqrt {n \cdot } \begin{array}{*{20}{c}} {\max } \\ { - 1 \le x \le 1} \\ \end{array}\,|P(x)| \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{array}$$for an arbitrary algebraic polynomial P of degree n whose

A criterion for almost everywhere convergence of Franklin series on a set is proved. The criterion is similar to the ones obtained by Y.S. Chow for martingales and F.G. Arutyunyan for Haar series.

We evaluate the suprema of approximations of bivariate functions by triangular partial sums of the double Fourier-Hermite series on the class of functions Lr2(D) in the space L2,ρ(ℝ2), where D is the second-order Hermite operator. Sharp Jackson-Stechkin type inequalities on the sets L2,ρ(ℝ2) are obtained, in which the best approximation is estimated from above both in terms of moduli of continuity

For any dimension n ≥ 2, we consider the maximal directional Hilbert transform \(\mathscr{H}_U\) on ℝn associated with a direction set \(U\subseteq\mathbb{S}^{n-1}\):$${H_U}f\left( x \right): = \frac{1}{\pi }\mathop {\sup }\limits_{v \in U} \left| {p.v.\int {f\left( {x - tv} \right)\frac{{dt}}{t}} } \right|.$$The main result in this article asserts that for any exponent p ∈ (1,∞), there exists a positive

Let G, H be uniquely 2-divisible Abelian groups. We study the solutions f, g: G → H of Pexider type functional equation$$f(x+y)+f(x-y)+g(x+y)=2f(x)+2f(y)+g(x)+g(y),$$((*))resulting from summing up the well known quadratic functional equation and additive Cauchy functional equation side by side. We show that modulo a constant equation (*) forces f to be a quadratic function, and g to be an additive

We give an estimation for the eigenvalues of matrix power functions. In particular, it has been shown that$$\lambda({(A+B)^p})\leq\lambda({2^{p-1}}({A^p}+{B^p}-\gamma I))\;(P \geq2)$$for all positive semi-definite matrices A, B, where γ is a positive constant. This provides a sharper bound for the known estimation for eigenvalues.

We determine the exact constants for simultaneous L2-approximation of Sobolev classes by piecewise Hermite interpolation with equidistant nodes. The general results are applied to obtain sharp Wirtinger–Sobolev type inequalities.

We obtain necessary conditions for the non-linear complex differential-difference equations$$w(z + 1)w(z - 1) + a(z)\frac{{{w'}(z)}}{{w(z)}} = R(z,w(z))$$to admit transcendental meromorphic solutions w(z) such that ρ2(w) < 1, where R(z,w(z)) is rational in w(z) with rational coefficients, a(z) is a rational function and ρ2(w) is the hyper-order of w(z). Our results can be seen as the product versions

For 0 < β <1, 0 < p < 1, and ω ∈ A1(ℝn), a version of the John–Nirenberg inequality suitable for the weighted Campanato spaces L(β, p,ω) is established. Further, we show that L(β, p,ω) are independent of the scale p ∈ (0,∞) in the sense of norm when 0 < β < 1 and ω ∈ A1(ℝn). As an application we characterize these spaces by the boundedness of the commutators [b, Iα]i (i = 1, 2), generated by bilinear

The main purpose of this short note is to correct a technical error appeared in the proof of [2, Theorem 2.3].