Embedding derivatives of weighted Hardy spaces into Lebesgue spaces

1994 ◽  
Vol 116 (1) ◽  
pp. 151-166 ◽  
Author(s):  
Daniel Girela ◽  
María Lorente ◽  
María Dolores Sarrión
Keyword(s):  
Hardy Space ◽  
Sufficient Conditions ◽  
Studia Math ◽  
Necessary Condition ◽  
Weighted Hardy Spaces ◽  
Borel Measures ◽  
Negative Function ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Derivatives Of

AbstractLet 1 ≤ p < ∞ and let ω be a non-negative function defined on the unit circle T which satisfies the Ap condition of Muckenhoupt. The weighted Hardy space Hp(ω) consists of those functions f in the classical Hardy space H1 whose boundary values belong to Lp(ω). Recently McPhail (Studia Math. 96, 1990) has characterized those positive Borel measures μ on the unit disc Δ for which Hp(ω) is continuously contained in Lp(dμ). In this paper we study the question of finding necessary and sufficient conditions on a positive Borel measure μ on Δ for the differentiation operator D defined by Df = f′ to map Hp(ω) continuously into Lp(dμ). We prove that a necessary condition is that there exists a positive constant C such thatwhere for any interval I ⊂ T,We prove that this condition is also sufficient in some cases, namely for 2 ≤ p < ∞ and ω(et0) = |θ|α, (|θ| ≤ π), – 1 < α < p – 1, but not in general. In the general case we prove the sufficiency of a condition which is slightly stronger than (A).

scholarly journals VARIATION PROBLEM CONTAINING SECOND DERIVATIVES OF UNKNOWN FUNCTIONS

2021 ◽  
Vol 6 (1(34)) ◽  
pp. 30-42
Author(s):  
Misraddin Allahverdi oglu Sadigov
Keyword(s):  
Variational Problem ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary Condition ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Variation Problem ◽  
Second Derivatives ◽  
Necessary And Sufficient ◽  
Derivatives Of

The property subdifferential of an integral and terminal functional in a space of the type of absolutely continuous functions is studied. Necessary and sufficient conditions for an extremum for a variational problem containing the second derivatives of unknown functions are obtained. With the help of the subdifferential introduced by the author, a nonconvex generalized variational problem containing the second derivatives of unknown functions is considered, and the necessary condition for an extremum is obtained.

Some generalized characters associated to a transitive permutation group

2020 ◽  
Vol 23 (3) ◽  
pp. 393-397
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Frobenius Group ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Transitive Permutation Group ◽  
Point Stabilizer ◽  
Necessary And Sufficient ◽  
Regular Character ◽  
Permutation Character

AbstractLet G be a finite transitive permutation group of degree n, with point stabilizer {H\neq 1} and permutation character π. For every positive integer t, we consider the generalized character {\psi_{t}=\rho_{G}-t(\pi-1_{G})}, where {\rho_{G}} is the regular character of G and {1_{G}} the 1-character. We give necessary and sufficient conditions on t (and G) which guarantee that {\psi_{t}} is a character of G. A necessary condition is that {t\leq\min\{n-1,\lvert H\rvert\}}, and it turns out that {\psi_{t}} is a character of G for {t=n-1} resp. {t=\lvert H\rvert} precisely when G is 2-transitive resp. a Frobenius group.

scholarly journals On circulant codes with prescribed distances

1977 ◽  
Vol 16 (3) ◽  
pp. 361-369
Author(s):  
M. Deza ◽  
Peter Eades
Keyword(s):  
Sufficient Conditions ◽  
Difference Sets ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Weighing Matrices ◽  
Cyclic Difference Sets ◽  
The Matrix ◽  
Circulant Codes ◽  
Necessary And Sufficient ◽  
Square Matrix

Necessary and sufficient conditions are given for a square matrix to te the matrix of distances of a circulant code. These conditions are used to obtain some inequalities for cyclic difference sets, and a necessary condition for the existence of circulant weighing matrices.

Separability criteria based on the Bloch representation of density matrices

2007 ◽  
Vol 7 (7) ◽  
pp. 624-638
Author(s):  
J. de Vicente
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Pure Product ◽  
Separability Problem ◽  
Arbitrary Dimensions ◽  
Necessary And Sufficient ◽  
Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

scholarly journals Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms

2020 ◽  
Vol 13 (1) ◽  
pp. 53-74 ◽  
Author(s):  
Adisak Seesanea ◽  
Igor E. Verbitsky
Keyword(s):  
Sufficient Conditions ◽  
Nonlinear Elliptic Equations ◽  
Classical Case ◽  
Finite Energy ◽  
Natural Growth ◽  
Nonlinear Elliptic ◽  
Borel Measures ◽  
Inhomogeneous Problems ◽  
Necessary And Sufficient ◽  
Quasilinear Elliptic

AbstractWe obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation-\Delta_{p}u=\sigma u^{q}+\mu\quad\text{on }\mathbb{R}^{n}in the sub-natural growth case {0<q<p-1}, where {\Delta_{p}} ({1<p<\infty}) is the p-Laplacian, and σ, μ are positive Borel measures on {\mathbb{R}^{n}}. Uniqueness of such a solution is established as well. Similar inhomogeneous problems in the sublinear case {0<q<1} are treated for the fractional Laplace operator {(-\Delta)^{\alpha}} in place of {-\Delta_{p}}, on {\mathbb{R}^{n}} for {0<\alpha<\frac{n}{2}}, and on an arbitrary domain {\Omega\subset\mathbb{R}^{n}} with positive Green’s function in the classical case {\alpha=1}.

Multivariate Poisson flows on Markov step processes

1982 ◽  
Vol 19 (2) ◽  
pp. 289-300 ◽  
Author(s):  
Frederick J. Beutler ◽  
Benjamin Melamed
Keyword(s):  
Counting Process ◽  
Sufficient Conditions ◽  
Infinitesimal Operator ◽  
Negative Function ◽  
Rate Of Increase ◽  
Weakened Version ◽  
Target Set ◽  
Empty Target ◽  
Necessary And Sufficient ◽  
Mutually Independent

A Markov step process Z equipped with a possibly non-denumerable state space X can model a variety of queueing, communication and computer networks. The analysis of such networks can be facilitated if certain traffic flows consist of mutually independent Poisson processes with respective deterministic intensities λi (t). Accordingly, we define the multivariate counting process N = (N1, N2, · ·· Nc) induced by Z; a count in Ni occurs whenever Z jumps from x (χ into a (possibly empty) target set . We study N through the infinitesimal operator à of the augmented Markov process W = (Z, N), and the integral relationship connecting à with the transition operator Tt of W. It is then shown that Ni depends on a non-negative function ri defined on χ; ri (x) may be interpreted as the expected rate of increase in Ni, given that Z is in state x.A multivariate N is Poisson (i.e., composed of mutually independent Poisson streams Ni) if and only if simultaneous jumps are impossible in a certain sense, and if the conditional expectation E[ri (Z(t) | 𝒩i] = E[ri (Z(t))] for i = 1, 2, ···, c and each t ≧ 0, where 𝒩i is the σ-algebra σ{N(s), s ≦ t}. Necessary and sufficient conditions are also specified that, for each s ≦ t, the variates [Ni(t)-Ni(s)] are mutually independent Poisson distributed; this involves a weakened version of E[ri(Z(v)) | N(v) – N(u)] = E [ri(Z(v))] for i = 1, 2, ···, c and all 0 ≦ u ≦ v.It is shown that the above criteria are automatically met by the more stringent classical requirement that N(t) and Z(t) be independent for each t ≧ 0.

scholarly journals Topological entropy in totally disconnected locally compact groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2163-2186 ◽  
Author(s):  
ANNA GIORDANO BRUNO ◽  
SIMONE VIRILI
Keyword(s):  
Topological Entropy ◽  
Sufficient Conditions ◽  
Invariant Subgroup ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Totally Disconnected ◽  
Dynamical Interpretation

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$. We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$. As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$. Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.

Moving Weighted Averages

1993 ◽  
Vol 45 (3) ◽  
pp. 449-469 ◽  
Author(s):  
M. A. Akcoglu ◽  
Y. Déniel
Keyword(s):  
Weight Function ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Fixed Weight ◽  
Finite Measure Space ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averagesof a function f X —> ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in whichwhere φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk > 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

scholarly journals Sufficient conditions for matchings

1972 ◽  
Vol 18 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Ian Anderson
Keyword(s):  
Simple Proof ◽  
Perfect Matching ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Number Of Components ◽  
Image Position ◽  
Necessary And Sufficient

A graph G is said to possess a perfect matching if there is a subgraph of G consisting of disjoint edges which together cover all the vertices of G. Clearly G must then have an even number of vertices. A necessary and sufficient condition for G to possess a perfect matching was obtained by Tutte (3). If S is any set of vertices of G, let p(S) denote the number of components of the graph G – S with an odd number of vertices. Then the conditionis both necessary and sufficient for the existence of a perfect matching. A simple proof of this result is given in (1).

A NOTE ON INEQUALITY CONSTRAINTS IN THE GARCH MODEL

2008 ◽  
Vol 24 (3) ◽  
pp. 823-828 ◽  
Author(s):  
Henghsiu Tsai ◽  
Kung-Sik Chan
Keyword(s):  
Generating Function ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Conditional Variance ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Economic Statistics ◽  
Absolute Monotonicity ◽  
Necessary And Sufficient ◽  
The Garch Model

We consider the parameter restrictions that need to be imposed to ensure that the conditional variance process of a GARCH(p,q) model remains nonnegative. Previously, Nelson and Cao (1992, Journal of Business ’ Economic Statistics 10, 229–235) provided a set of necessary and sufficient conditions for the aforementioned nonnegativity property for GARCH(p,q) models with p ≤ 2 and derived a sufficient condition for the general case of GARCH(p,q) models with p ≥ 3. In this paper, we show that the sufficient condition of Nelson and Cao (1992) for p ≥ 3 actually is also a necessary condition. In addition, we point out the linkage between the absolute monotonicity of the generalized autoregressive conditional heteroskedastic (GARCH) generating function and the nonnegativity of the GARCH kernel, and we use it to provide examples of sufficient conditions for this nonnegativity property to hold.

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