An example of an almost greedy uniformly bounded orthonormal basis for

We construct a uniformly bounded orthonormal almost greedy basis for L_p(0,1), 1<p<∞. The example shows that it is not possible to extend Orlicz's theorem, stating that there are no uniformly bounded orthonormal unconditional bases for L_p(0,1), p≠2, to the class of almost greedy bases.     

Finite semigroups with slowly growing pn-sequences

The p _n -sequence of a semigroup S is said to be polynomially bounded, if there exist a positive constant c and a positive integer r such that the inequality p _n (S) ≤cn ^r holds for all n≥ 1. In this paper, we fully describe all finite semigroups having polynomially bounded p _n -sequences. First we give a characterization in terms of identities satisfied by these semigroups. In the sequel, this result will allow an insight into the structure of such semigroups. We are going to deal with certain ideals and the construction of ideal extension of semigroups. In addition, we supply an effective procedure for deciding whether a finite semigroup has polynomially bounded p _n -sequence and give some examples. Received March 5, 1999; accepted in final form November 1, 1999.     

Noncommutative Figà-Talamanca–Herz Algebras for Schur Multipliers

In this work, we introduce a noncommutative analogue of the Figà-Talamanca–Herz algebra A _p (G) on the natural predual of the operator space \frakM_p,cb{\frak{M}_{p,cb}} of completely bounded Schur multipliers on the Schatten space S _p . We determine the isometric Schur multipliers and prove that the space \frakM_p{\frak{M}_{p}} of bounded Schur multipliers on the Schatten space S _p is the closure in the weak operator topology of the span of isometric multipliers.     

Local regularity for nonlocal equations with variable exponents

In this paper, we study local regularity properties of minimizers of nonlocal variational functionals with variable exponents and weak solutions to the corresponding Euler–Lagrange equations. We show that weak solutions are locally bounded when the variable exponent p is only assumed to be continuous and bounded. Furthermore, we prove that bounded weak solutions are locally Hölder continuous under some additional assumptions on p. On the one hand, the class of admissible exponents is assumed to satisfy a log-Hölder–type condition inside the domain, which is essential even in the case of local equations. On the other hand, since we are concerned with nonlocal problems, we need an additional assumption on p outside the domain.     

BMO Solvability and the <Emphasis Type="Italic">A</Emphasis><Subscript>∞</Subscript> Condition for Elliptic Operators

We establish a connection between the absolute continuity of elliptic measure associated with a second order divergence form operator with bounded measurable coefficients with the solvability of an end-point BMO Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the BMO Dirichlet problem implies L ^p solvability for all p>p ₀.     

Some Estimates for Radial Fourier Multiplier Operators with Slowly Decaying Kernels

We consider the restriction to radial functions of a class of radial Fourier multiplier operators containing the Bochner-Riesz multiplier operator. The convolution kernel K(x) of an operator in this class decays too slowly at infinity to be integrable, but has enough oscillation to achieve L^p -boundedness for p inside a suitable interval (a, b). We prove boundedness results for the maximal operator Kf(x) = sup_{r>0 r}ⁿ∣K(r) * f(x)∣ associated with such a kernel. The maximal operator is shown to be weak type bounded at the lower critical index a, restricted weak type bounded at the upper critical index b, and strong type bounded between. This together with our assumptions on K(x) leads to the pointwise convergence result lim_γ→ γⁿ K(γ·) * f(x) = cf(x) a. e. for radial f ? L^P(?ⁿ), a ≥ p > b.     

Non Integral Regularizing Operators on Lp- Spaces

We present bounded positivity preserving operators from L_p(?) to L_q (?), for 1 < p < ∞, 1/p-1/q < 1/2, which are not integral operators.     

Approximate identities on some homogeneous Banach spaces

We study convergence of approximate identities on some complete semi-normed or normed spaces of locally L ^p functions where translations are isometries, namely Marcinkiewicz spaces M^p{\mathcal{M}^{p}} and Stepanoff spaces S^p{\mathcal{S}^p}, 1 ≤ p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M ^p ) or even unbounded (M^p₀{M^{p}_{0}}). We construct a function f that belongs to these spaces and has the property that all approximate identities f_e * f{\phi_\varepsilon * f} converge to f pointwise but they never converge in norm.     

On the boundedness of some potential‐type operators with oscillating kernels

We consider a class of multidimensional potential‐type operators with kernels that have singularities at the origin and on the unit sphere and that are oscillating at infinity. We describe some convex sets in the (1/p, 1/q)‐plane for which these operators are bounded from L_p into L_q and indicate domains where they are not bounded. We also reveal some effects which show that oscillation and singularities of the kernels may strongly influence on the picture of boundedness of the operators under consideration. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Harmonic maps from a Riemannian manifold with a pole into an Hadamard manifold with negative sectional curvatures

In this paper we show a nonexistence result for harmonic maps with a rotational nondegeneracy condition from a Riemannian manifoldM with polep ₀ to a negatively curved Hadamard manifold under the condition that the metric tensor ofM is bounded and that the sectional curvature ofM at a pointp is bounded from below by −c dist(p ₀,p)⁻² (c: a positive constant) as dist(p ₀,p)→∞. Partly supported by Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan     

On the Uniform Poincaré Inequality

We give a proof of the Poincaré inequality in W ^{1, p} (Ω) with a constant that is independent of Ω ? , where  is a set of uniformly bounded and uniformly Lipschitz domains in ? ⁿ . As a byproduct, we obtain the following: The first non vanishing eigenvalues λ₂(Ω) of the standard Neumann (variational) boundary value problem on Ω for the Laplace operator are bounded below by a positive constant if the domains Ω vary and remain uniformly bounded and uniformly Lipschitz regular.     

Vector lattices of almost polynomial sequences

We study the order structure, various duals and orthomorphisms on vector lattices of sequences that differ from a polynomial (either of bounded or unbounded degree) by a sequence in ℓ _p .     

Toeplitz Operators on Analytic Besov Spaces

We characterize complex measures μ on the unit disk for which the Toeplitz operator T _μ is bounded or compact on the analytic Besov spaces B _p with 1 ≤ p < ∞. Research supported in part by NSF grant, DMS 0200587 (first author); and by a NSERC grant (third author).     

p-harmonic functions on graphs and manifolds

We show that the LiouvilleD _p -property is invariant under rough isometries between a Riemannian manifold of bounded geometry and a graph of bounded degree. The first author was supported partly by the EU HCM contract No. CHRX-CT92-0071. This article was processed by the author using the Springer-Verlag TEX P Jourlg macro package 1991.     

Algebra descent spectrum of operators

We say that a Banach space X satisfies the “descent spectrum equality” (in short, DSE) whenever, for every bounded linear operator T on X, the descent spectrum of T as an operator coincides with the descent spectrum of T as an element of the algebra of all bounded linear operators on X. We prove that the DSE is fulfilled by ℓ₁, all Hilbert spaces, and all Banach spaces which are not isomorphic to any of their proper quotients (so, in particular, by the hereditarily indecomposable Banach spaces [8]), but not by ℓ _p , for 1 < p ≤ ∞ with p ≠ 2. Actually, a Banach space is not isomorphic to any of its proper quotients if and only if it is not isomorphic to any of its proper complemented subspaces and satisfies the DSE.     

On Iterations of Non–Negative Operators and Their Applications to Elliptic Systems

Let H₀, H₁ be Hilbert spaces and L : H₀ → H₁ be a linear bounded operator with ∥L∥ ≤ 1. Then L*L is a bounded linear self–adjoint non–negative operator in the Hilbert space H₀ and one can use the Neumann series Σ^∞_v=0(I — L*L)^v L*f in order to stud solvabilit of the operator equation Lu = f. In particular, applying this method to the ill–posed Cauch problem for solutions to an elliptic system Pu = 0 of linear PDE's of order p with smoothcoefficients we obtain solvabilit conditions and representation formulae for solutions of the problem in Hardy spaces whenever these solutions exist. For the Cauch–Riemann system in ℂ the summands of the Neumann series are iterations of the Cauch type integral.     

The best approximation of Laplace operator by linear bounded operators in the space <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We obtain close two-sided estimates for the best approximation of Laplace operator by linear bounded operators on the class of functions for which the square of the Laplace operator belongs to the L _p -space. We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of values of the Laplace operator on functions from this class defined with an error. In a particular case (p = 2) we solve all three problems exactly.     

Boundedness in nonlinear oscillations

In this paper, the boundedness of all solutions of the nonlinear differential equation (φ_p(x′))′ + αφ_p(x⁺) – βφ_p(x^–) + f(x) = e(t) is studied, where φ_p(u) = |u|^p–2 u, p ≥ 2, α, β are positive constants such that = 2w^–1 with w ∈ ?⁺\?, f is a bounded C⁵ function, e(t) ∈ C⁶ is 2π_p‐periodic, x⁺ = max{x, 0}, x^– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral approximation by L 2 discretization of the Laplace operator on manifolds of bounded geometry

The author considers a discretization of the p-form Laplacian on open complete Riemannian manifolds of bounded geometry. Following Dodziuk and Patodi [8], the eigenvalues below the essential spectrum together with their eigenforms are approximated by eigenvalues and eigencochains of a semicombinatorical Laplacian acting on L ₂-cochains. We obtain a similar result for a ray which is contained in the essential spectrum. An example of a manifold of bounded geometry which admits eigenvalues below the essential spectrum is constructed.     

Test elements for endomorphisms of free groups and algebras

LetE be a bounded Borel subset of ℝⁿ,n≥2, of positive Lebesgue measure andP _E the corresponding ‘Pompeiu transform”. We prove thatP _E is injective onL ^p(ℝⁿ) if 1≤p≤2n/(n-1). We explore the connection between this problem and a Wiener-Tauberian type theorem for theM(n) action onL ^q(ℝⁿ) for various values ofq. We also take up the question of whenP _E is injective in caseE is of finite, positive measure, but is not necessarily a bounded set. Finally, we briefly look at these questions in the contexts of symmetric spaces of compact and non-compact type.