Discrete Cesaro operator between weighted Banach spaces on homogenous trees

From a unified vision of vector valued solutions in weighted Banach spaces, this paper establishes the existence and uniqueness for space homogeneous Boltzmann bi-linear systems with conservative collisional forms arising in complex gas dynamical structures. This broader vision is directly applied to dilute multi-component gas mixtures composed of both monatomic and polyatomic gases. Such models can be viewed as extensions of scalar Boltzmann binary elastic flows, as much as monatomic gas mixtures with disparate masses and single polyatomic gases, providing a unified approach for vector valued solutions in weighted Banach spaces. Novel aspects of this work include developing the extension of a general ODE theory in vector valued weighted Banach spaces, precise lower bounds for the collision frequency in terms of the weighted Banach norm, energy identities, angular or compact manifold averaging lemmas which provide coerciveness resulting into global in time stability, a new combinatorics estimate for p-binomial forms producing sharper estimates for the k-moments of bi-linear collisional forms. These techniques enable the Cauchy problem improvement that resolves the model with initial data corresponding to strictly positive and bounded initial vector valued mass and total energy, in addition to only a 2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^+$$\\end{document} moment determined by the hard potential rates discrepancy, a result comparable in generality to the classical Cauchy theory of the scalar homogeneous Boltzmann equation.

When the weight $\mu$ is more general than normal, the complete characterizations in terms of the symbol $g$ and weights for the conditions of the boundedness and compactness of $T_g: H^{\infty}_\nu\rightarrow H^{\infty}_\mu$ and $S_g: H^{\infty}_\nu\rightarrow H^{\infty}_\mu$ are still unknown. Smith et al. firstly gave the sufficient and necessary conditions for the boundedness of Volterra type operators on Banach spaces of bounded analytic functions when the symbol functions are univalent. In this paper, continuing their lines of investigations, we give the complete characterizations of the conditions for the boundedness and compactness of Volterra type operators $T_g$ and $S_g$ between Bloch type spaces $\mathcal{B}^\infty_\nu$ and weighted Banach spaces $H^{\infty}_\nu$ with more general weights, which generalize their works.

On the stability of the positive steady states for a nonhomogeneous semilinear Cauchy problem

We study the relation between multiplication operators M g with closed range on L a p and M g with closed range on H ∞ . Let p>0. For 0 < α < 1 , suppose g ∈ B α . Then the following are equivalent. M g : L a p → L a p is bounded below. M g : H ∞ → H ∞ is bounded below. 0 $ ]]> lim inf z → ∂D | g ( z ) | > 0 .

We consider m-th order linear, uniformly elliptic equations Lu=f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {L}u=f$$\\end{document} with non-smooth coefficients in Banach–Sobolev spaces WXwm(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{X_w}^m (\\Omega )$$\\end{document} generated by weighted Banach Function Spaces (BFS) Xw(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_w (\\Omega )$$\\end{document} on a bounded domain Ω⊂Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset {\\mathbb R}^{n}$$\\end{document}. Supposing boundedness of the Hardy–Littlewood Maximal operator and the Calderón–Zygmund singular integrals in Xw(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_w (\\Omega )$$\\end{document} we obtain solvability in the small in WXwm(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{X_w}^m (\\Omega )$$\\end{document} and establish interior Schauder type a priori estimates. These results will be used in order to obtain Fredholmness of the operator L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {L}$$\\end{document} in Xw(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_w (\\Omega )$$\\end{document}.

It is shown that the monomials Λ = ( z n ) n = 0 ∞ \Lambda =(z^n)_{n=0}^{\infty } are a Schauder basis of the Fréchet spaces A + − γ , γ ≥ 0 , A_+^{-\gamma }, \ \gamma \geq 0, that consists of all the analytic functions f f on the unit disc such that ( 1 − | z | ) μ | f ( z ) | (1-|z|)^{\mu }|f(z)| is bounded for all μ &gt; γ \mu &gt; \gamma . Lusky proved that Λ \Lambda is not a Schauder basis for the closure of the polynomials in weighted Banach spaces of analytic functions of type H ∞ H^{\infty } . A sequence space representation of the Fréchet space A + − γ A_+^{-\gamma } is presented. The case of (LB)-spaces A − − γ , γ &gt; 0 , A_{-}^{-\gamma }, \ \gamma &gt; 0, that are defined as unions of weighted Banach spaces is also studied.

We study the incompleteness of exponential system in C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sub> from a harmonic measure point of view, where C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sub> is the weighted Banach space consists of complex continuous functions f defined on the real axis ℝ with vanishing at infinity, in the uniform norm.

It is proved that in Banach algebras generated by two idempotents and, perhaps, by a certain flip operator the standard identity F4 is fulfilled. The maximal ideal space of such algebras is determined and the corresponding symbol is given. By means of local techniques these results are applied to obtain a symbol calculus for singular integral operators with Carleman shift (changing the orientation) in weighted Banach spaces.

In the year 1944 the well-known French mathematician Jean Favard (1902–1965) introduced a discretely defined operator which is a discrete analogue of the familiar Gauss–Weierstrass singular convolution integral. In the present chapter we sketch the history of this approximation operator during the past 70 years by presenting known results on the operator and its various extensions and variants. The first part after the introduction is dedicated to saturation of the classical Favard operator in weighted Banach spaces. Furthermore, we discuss the asymptotic behaviour of a slight generalization \(F_{n,\sigma _{n}}\) of the Favard operator and its Durrmeyer variant \(\tilde{F }_{n,\sigma _{n}}\). In particular, the local rate of convergence when applied to locally smooth functions is considered. The main result of this part consists of the complete asymptotic expansions for the sequences \(\left (F_{n,\sigma _{n}}f\right )\left (x\right )\) and \(\left (\tilde{F } _{n,\sigma _{n}}f\right )\left (x\right )\) as n tends to infinity. Furthermore, these asymptotic expansions are valid also with respect to simultaneous approximation. A further part is devoted to the recent work of several Polish mathematicians on approximation in weighted function spaces. Finally, we define left quasi-interpolants for the Favard operator and its Durrmeyer variant in the sense of Sablonniere.

For a certain class of sequences with repeated terms,

In this article, we deduce the existence of a solution to the weighted fractional differential equation with functional boundary data involving an ω-weighted fractional derivative with Riemann–Liouville settings, D0+α,ψ,ω of order α∈]n−1,n[, on certain weighted Banach spaces when the nonlinear term contains the proportional delay term and fractional derivatives of order (0,1). After carefully defining a few weighted spaces and building a few weighted projection operators, we use Mawhin’s coincidence theory to derive a number of existence results at resonance. Furthermore, our method generalizes some prior results because numerous fractional differential operators are specific instances of the operator D0+α,ψ,ω and represent functional boundary conditions in a highly generic way. Lastly, we illustrate and support our theoretical results with an example.

In this paper, we establish some new characterizations of a weight \(w\) such that discrete Hardy operator \(\mathcal{H}f(n):=\frac{1}{n}\sum\limits_{s=1}^{n}f(s)\) for quasi-nonincreasing sequence \(f(n)\) is bounded in the Banach space \(\ell _{w}^{p}(\mathbb{Z}_{+})\) when \(0&lt;p&lt;\infty .\) In particular, we will prove that \(\mathcal{H}f\) is bounded in \(\ell _{w}^{p}(\mathbb{Z}_{+})\) if and only if \(w\) belongs to the \(\beta\)-discrete Arino and Muckenhoupt class \(\mathcal{B}_{p,\beta }\). We prove that the self-improving property for the class \(\mathcal{B}_{p,\beta }\) holds, that is we prove that if \(w\in \mathcal{B}_{p,\beta }\) then there exists an \(\varepsilon &gt;0\) such that \(w\in \mathcal{B}_{p-\varepsilon ,\beta }.\)

Optimal fractional Luenberger observers for linear fractional-order systems are developed using the fractional Chebyshev collocation (FCC) method. It is shown that the design method has advantages over existing Luenberger design methods for fractional order systems. To accomplish this, the state transition operator for the solution of linear fractional-order systems is defined in a Banach space and discretized using the FCC method. In addition, the discretized state transition operator is obtained by using the FCC method. Next, the optimal observer gains are obtained by minimizing the spectral radius of the state transition operator for the observer,while ensuring that the observer responds faster than the controller. Finally, a numerical example is provided to demonstrate the validity and the efficiency of the proposed method.

Let E be a complex Banach space with open unit ball $$B_E.$$ For analytic self-maps $$\varphi $$ of $$B_E$$ with $$\varphi (0) =0,$$ we investigate the spectra of weighted composition operators $$uC_\varphi $$ acting on a large class of spaces of analytic functions. This class contains, for example, weighted Banach spaces of $$H^\infty $$-type on $$B_E$$, weighted Bergman spaces $$A^p_\alpha ({\mathbb {B}}_N)$$ and Hardy spaces $$H^p({\mathbb {B}}_N).$$ We present a general approach for deducing new information about the spectrum and for estimating the essential spectral radius of $$uC_\varphi .$$

Tractability results for weighted Banach spaces of smooth functions