Studia Math Research Articles

A remark on the approximation of p-compact operators by finite-rank operators

The Banach operator ideal K p of p-compact operators was introduced in [P.D. Sinha, A.K. Karn, Compact operators whose adjoints factor through subspaces of ℓ p , Studia Math. 150 (2002) 17–33]. Let p ⩾ 1 and let X be a closed subspace of an L p ( μ ) -space. We show that X has the approximation property if and only if for every Banach space Y, the linear space F ( Y , X ) of finite-rank operators is K p -norm dense in K p ( Y , X ) , i.e., X has the K p -approximation property.

UPPER BOUND OF THE DIRECTIONAL ENTROPY OF A ℤ2-ACTION

In this short paper, we study the measure directional entropy of a ℤ2-action generated by an invertible one-dimensional linear cellular automaton and the shift map acting on the compact metric space [Formula: see text]. We obtain an upper bound of the measure directional entropy of the ℤ2-action with respect to arbitrary Bernoulli measures without making use of the natural extension previously used in the paper [M. Courbage and B. Kaminski, Studia Math. 153(3), 285–295 (2002)].

On maximal ideals of codimension one in m-convex algebras

In Studia Math. 58:291–298, 1976, W. Żelazko gives a characterization of complex commutative complete unital m-convex algebras in which all maximal ideals are of codimension one. In these algebras every element has a bounded spectrum.

Sous-espaces fermés de séries universelles sur un espace de Fréchet

We improve a result of Charpentier [Studia Math. 198 (2010)]. We prove that even on Fréchet spaces with a continuous norm, the existence of only one restrictively universal series implies the existence of a closed infinite-dimensional subspace of restrict

ONλ-STRICT IDEALS IN BANACH SPACES

AbstractWe define and studyλ-strict ideals in Banach spaces, which forλ=1 means strict ideals. Strict u-ideals in their biduals are known to have the unique ideal property; we prove that so also doλ-strict u-ideals in their biduals, at least forλ>1/2. An open question, posed by Godefroyet al. [‘Unconditional ideals in Banach spaces’,Studia Math.104(1993), 13–59] is whether the Banach spaceXis a u-ideal in Ba(X), the Baire-one functions inX**, exactly whenκu(X)=1; we prove that ifκu(X)=1 thenXis a strict u-ideal in Ba (X) , and we establish the converse in the separable case.

Burkholder’s function via Monge–Ampère equation

We will show how to get Burkholder’s function from (Ann. Probab. 12 (1984) 647–702) by using Monge–Ampère equation. This method is quite different from those in the series of Burkholder’s papers (Ann. Probab. 12 (1984) 647–702, An extension of classical martingale inequality (1986) Marcel Dekker, Astérisque 157–158 (1988) 75–94, In Harmonic analysis and partial differential equations (1989) 1–23 Springer, In École d’Ete de Probabilités de Saint-Flour XIX (1991) 1–66 Springer, Ann. Probab. 22 (1994) 995–1025, Studia Math. 91 (1988) 79–83).

Perturbation theorems for local integrated semigroups

We apply the contraction mapping theorem to establish some bounded and unbounded perturbation theorems concerning nondegenerate local $\alpha $-times integrated semigroups. Some unbounded perturbation results of Wang et al. [Studia Math. 170 (2005)] are a

Heat kernel characterisation of Besov-Lipschitz spaces on metric measure spaces

We give a heat-kernel characterisation of the Besov-Lipschitz spaces Lip (α, p, q)(X) on domains which support a Markovian kernel with appropriate exponential bounds. This extends former results of Grigor’yan et al. (Trans Am Math Soc 355:2065–2095, 2008), Hu and Zahle (Studia Math 170:259–281, 2005), Pietruska-Paluba (Stoch Stoch Rep 67:267–285, 1999; 70:153–164, 2000), which were valid for \({\alpha = \frac{d_w}{2}, p = 2, q = \infty}\), where dw is the walk dimension of the space X.

Gradient estimates below the duality exponent

We show sharp local a priori estimates and regularity results for possibly degenerate non-linear elliptic problems, with data not lying in the natural dual space. We provide a precise non-linear potential theoretic analog of classical potential theory results due to Adams (Duke Math J 42:765–778, 1975) and Adams and Lewis (Studia Math 74:169–182, 1982), concerning Morrey spaces imbedding/regularity properties. For this we introduce a technique allowing for a “non-local representation” of solutions via Riesz potentials, in turn yielding optimal local estimates simultaneously in both rearrangement and non-rearrangement invariant function spaces. In fact we also derive sharp estimates in Lorentz spaces, covering borderline cases which remained open for some while.

Wave Equation and Multiplier Estimates on Damek–Ricci Spaces

Let S be a Damek–Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form $${\rm {e}}^{it\sqrt{ L}}\psi\big(\sqrt{ L}/{\lambda}\big)$$ for arbitrary time t and arbitrary λ>0, where ψ is a smooth bump function supported in [−2,2] if λ<1 and supported in [1,2] if λ≥1. This generalizes previous results in Muller and Thiele (Studia Math. 179:117–148, 2007). We also prove pointwise estimates for the gradient of these convolution kernels. As a corollary, we reprove basic multiplier estimates from Hebish and Steger (Math. Z. 245:37–61, 2003) and Vallarino (J. Lie Theory 17:163–189, 2007) and derive Sobolev estimates for the solutions to the wave equation associated to L.

Dyadic-like maximal operators on [formula omitted] functions

We study the following well-known property of the dyadic maximal operator M d on R n (see [E.M. Stein, Note on the class L log L , Studia Math. 32 (1969) 305–310] for the case of the Hardy–Littlewood maximal function): If ϕ is integrable and supported in a dyadic cube Q then M d ϕ is integrable over sets of finite measure if and only if | ϕ | log ( 1 + | ϕ | ) is integrable and the integral of M d ϕ can be estimated both from above and from below in terms of the integral of | ϕ | log ( 1 + | ϕ | ) over Q. Here we define and explicitly evaluate Bellman functions related to this property and the corresponding estimates (both upper and lower) for the integrals thus producing sharp improved versions of the behavior of M d on the local L log L spaces.

A new characterization of Bergman–Schatten spaces and a duality result

Let B 0 ( D , ℓ 2 ) denote the space of all upper triangular matrices A such that lim r → 1 − ( 1 − r 2 ) ‖ ( A ∗ C ( r ) ) ′ ‖ B ( ℓ 2 ) = 0 . We also denote by B 0 , c ( D , ℓ 2 ) the closed Banach subspace of B 0 ( D , ℓ 2 ) consisting of all upper triangular matrices whose diagonals are compact operators. In this paper we give a duality result between B 0 , c ( D , ℓ 2 ) and the Bergman–Schatten spaces L a 1 ( D , ℓ 2 ) . We also give a characterization of the more general Bergman–Schatten spaces L a p ( D , ℓ 2 ) , 1 ⩽ p < ∞ , in terms of Taylor coefficients, which is similar to that of M. Mateljevic and M. Pavlovic [M. Mateljevic, M. Pavlovic, L p -behaviour of the integral means of analytic functions, Studia Math. 77 (1984) 219–237] for classical Bergman spaces.

A simple inequality for the von Neumann–Jordan and James constants of a Banach space

Let C NJ ( X ) and J ( X ) be the von Neumann–Jordan and James constants of a Banach space X , respectively. We shall show that C NJ ( X ) ⩽ J ( X ) , where equality holds if and only if X is not uniformly non-square. This answers affirmatively to the question in a recent paper by Alonso et al. [J. Alonso, P. Martín, P.L. Papini, Wheeling around von Neumann–Jordan constant in Banach spaces, Studia Math. 188 (2008) 135–150]. This inequality looks quite simple and covers all the preceding results. In particular this is much stronger than Maligranda's conjecture: C NJ ( X ) ⩽ J ( X ) 2 4 + 1 .

Duals of random vectors and processes with applications to prediction problems with missing values

Important results in prediction theory dealing with missing values have been obtained traditionally using difficult techniques based on duality in Hilbert spaces of analytic functions [Nakazi, T., 1984. Two problems in prediction theory. Studia Math. 78, 7–14; Miamee, A.G., Pourahmadi, M., 1988. Best approximations in L p ( d μ ) and prediction problems of Szegö, Kolmogorov, Yaglom, and Nakazi. J. London Math. Soc. 38, 133–145]. We obtain and unify these results using a simple finite-dimensional duality lemma which is essentially an abstraction of a regression property of a multivariate normal random vector ( Rao, 1973) or its inverse covariance matrix. The approach reveals the roles of duality and biorthogonality of random vectors in dealing with infinite-dimensional and difficult prediction problems. A novelty of this approach is its reliance on the explicit representation of the prediction error in terms of the data rather than the predictor itself as in the traditional techniques. In particular, we find a new and explicit formula for the dual of the semi-finite process { X t ; t ≤ n } for a fixed n , which does not seem to be possible using the existing techniques.

Gradient regularity for elliptic equations in the Heisenberg group

We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised in [J.J. Manfredi, G. Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group, Math. Ann. 339 (2007) 485–544], where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven in [A. Domokos, J.J. Manfredi, C 1 , α -regularity for p-harmonic functions in the Heisenberg group for p near 2, in: Contemp. Math., vol. 370, 2005, pp. 17–23]. In turn, using some recent techniques of Caffarelli and Peral [L. Caffarelli, I. Peral, On W 1 , p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1–21], the a priori estimates found are shown to imply the suitable local Calderón–Zygmund theory for the related class of non-homogeneous, possibly degenerate equations involving discontinuous coefficients. These last results extend to the sub-elliptic setting a few classical non-linear Euclidean results [T. Iwaniec, Projections onto gradient fields and L p -estimates for degenerated elliptic operators, Studia Math. 75 (1983) 293–312; E. DiBenedetto, J.J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993) 1107–1134], and to the non-linear case estimates of the same nature that were available in the sub-elliptic setting only for solutions to linear equations.

Equivalence of the Mikusiński–Shiraishi–Itano products in 𝒮′ for various classes of delta-sequences

The equivalence of the Mikusiński-Shiraishi-Itano product in 𝒮′ for the classes Δ1, Δ∞, Δ m of delta-sequences is proved by means of the Antosik-Swartz matrix theorem. The result answers a problem posed in [A. Kamiński, Convolution, product and Fourier transform of distributions, Studia Math. 74 (1982), pp. 83–96] and has various applications, e.g. in the theory of distributors.

Remarks on the structure of the fixed-point sets of uniformly lipschitzian mappings in uniformly convex Banach spaces

It is shown, by asymptotic center techniques, that the set of fixed points of any uniformly k -lipschitzian mapping in a uniformly convex Banach space is a retract of the domain when k is less than a constant bigger than the constant from the paper [K. Goebel, W.A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973) 135–140]. Our result improves a recently result presented in [E. Sędłak, A. Wiśnicki, On the structure of fixed-point sets of uniformly lipschitzian mappings, Topol. Methods Nonlinear Anal. 30 (2007) 345–350].

The p-approximation property in terms of density of finite rank operators

We characterize the p-approximation property ( p-AP) introduced by Sinha and Karn [D.P. Sinha, A.K. Karn, Compact operators whose adjoints factor through subspaces of ℓ p , Studia Math. 150 (2002) 17–33] in terms of density of finite rank operators in the spaces of p-compact and of adjoints of p-summable operators. As application, the p-AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi- p-nuclear operators. This relates the p-AP to Saphar's approximation property AP p ′ . As another application, the p-AP is characterized via a trace condition, allowing to define the trace functional on certain subspaces of the space of nuclear operators.

On the uniqueness of uniform norms and C*-norms

We prove that a semisimple, commutative Banach algebra has either exactly one uniform norm or infinitely many uniform norms; this answers a question asked by S. J. Bhatt and H. V. Dedania [Studia Math. 160 (2004)]. A similar result is proved for $C^\ast $

What is a Sobolev space for the Laguerre function systems?

We discuss the concept of Sobolev space associated to the Laguerre operator $ L_\al = - y\,\frac{d^2}{dy^2} - \frac{d}{dy} + \frac{y}{4} + \frac{\al^2}{4y},\quad y\in (0,\infty).$ We show that the natural definition does not fit with the concept of potential space, defined via the potentials $ (L_\al)^{-s}.$ An appropriate Laguerre-Sobolev spaces are defined in order to have the mentioned coincidence. An application is given to the almost everywhere convergence of solutions of the Schr\odinger equation. Other Laguerre operators are also considered. Published: Studia Math. 192 (2009), no. 2, 147--172.