Maximal operator on variable Lebesgue spaces for almost monotone radial exponent

We study general Lebesgue spaces with variable exponent p. It is known that the classes L and N of functions p are such that the Hardy-Littlewood maximal operator is bounded on them provided p∈L∩P. The class L governs local properties of p and N governs the behavior of p at infinity.In this paper we focus on the properties of p near infinity. We extend the class N to a collection D of functions p such that the Hardy-Littlewood maximal operator is bounded on the corresponding variable Lebesgue spaces provided p∈L∩D and the class D is essentially larger than N.Moreover, the condition p∈D is quite easily verifiable in the practice.     

On Singular Integral Operators with Bounded Measurable Coefficients

Necessary and sufficient analytical conditions are determined for a singular integral operator of the form aP + bQ with bounded measurable coefficients to be a ?-operator on L_p(Γ) for all 1 < p < ∞. where Γ is a closed Lyapunov curve.     

Quasisimilarity of power bounded operators and Blum-Hanson property

We construct a power bounded operator on a Hilbert space which is not quasisimilar to a contraction. To this aim, we solve an open problem from operator ergodic theory showing that there are power bounded Hilbert space operators without the Blum-Hanson property. We also find an example of a power bounded operator quasisimilar to a unitary operator which is not similar to a contraction, thus answering negatively open questions raised by Kérchy and Cassier. On the positive side, we prove that contractions on ?p spaces (1?p<∞) possess the Blum-Hanson property.     

Riesz Means on Graphs and Discrete Groups

Suppose that Γ is a weighted graph or a discrete group. Let $m_{\alpha,R}(\lambda )=\big(1-\big|\frac{\lambda}{R}\big|\big)_{+}^{\alpha}$ be the Riesz means and let Δ be the discrete Laplacian on Γ. We prove that if D is the homogeneous dimension of Γ then the operator m _α,R (Δ) is bounded on L ^p , provided that $\alpha>D|\frac{1}{p}-\frac{1}{2}|$ .     

Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

Norms of multiplication operators on Hardy spaces and weighted composition operators from Hardy spaces to weighted-type spaces on bounded symmetric domains

Let D be a bounded symmetric domain. We calculate operator norm of the multiplication operator on the Hardy space H^p(D), as well as of the weighted composition operator from H^p(D) to a weighted-type space.     

An Extended Faber Operator

   Abstract. One of the basic tools in the theory of polynomial approximation in the uniform norm on compact plane sets is the Faber operator. Usually, the Faber operator is viewed as an operator acting on functions in the disk algebra, that is, functions which are holomorphic in the open unit disk D and continuous on D. We consider an extended Faber operator acting on arbitrary functions continuous on ; D.     

Generalized Hilbert operator on the Dirichlet-type space

The boundedness of the generalized Hilbert operator on the Dirichlet-type space S_p when 0<p<1 is investigated.     

Polar boundary sets for superdiffusions and removable lateral singularities for nonlinear parabolic PDEs

Suppose L is a second-order elliptic differential operator in ℝ^d and D is a bounded, smooth domain in ℝ^d. Let 1 < α ≤ 2 and let Γ be a closed subset of ∂D. It is known [13] that the following three properties are equivalent: (α) Γ is ∂-polar; that is, Γ is not hit by the range of the corresponding (L, α)-superdiffusion in D; (β) the Poisson capacity of Γ is equal to 0; that is, the integral is equal to 0 or ∞ for every measure ν, where ρ(x) is the distance to the boundary and k(x, y) is the corresponding Poisson kernel; and (γ) Γ is a removable boundary singularity for the equation Lu = u^α in D; that is, if u ≥ 0 and Lu = u^α in D and if u = 0 on ∂D \ Γ, then u = 0. We investigate a similar problem for a parabolic operator in a smooth cylinder 𝒬 = ℝ₊ × D. Let Γ be a compact set on the lateral boundary of 𝒬. We show that the following three properties are equivalent: (a) Γ is 𝒢-polar; that is, Γ is not hit by the graph of the corresponding (L, α)-superdiffusion in 𝒬; (b) the Poisson capacity of Γ is equal to 0; that is, the integral is equal to 0 or ∞ for every measure ν, where k(r, x; t, y) is the corresponding (parabolic) Poisson kernel; and (c) Γ is a removable lateral singularity for the equation u˙ + Lu = u^α in 𝒬; that is, if u ≥ 0 and u˙ + Lu = u^α in 𝒬 and if u = 0 on ∂𝒬 \ Γ and on {∞} × D, then u = 0. © 1998 John Wiley & Sons, Inc.     

Endpoint estimates for the commutator of pseudo-differential operators

It is well known that the commutator Tb of the Calderón-Zygmund singular integral operator is bounded on L^p(Rⁿ) for 1 < p < +∞ if and only if b ∈ BMO [1]. On the other hand, the commutator T_b is bounded from H¹(Rⁿ) into L¹(Rⁿ) only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H¹. Let T^σ be the operators that its symbol is S⁰_1,δ with 0 ≤ δ < 1, if b ∈ LMO_∞, then, the commutator [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) and from L¹(Rⁿ) into BMO(Rⁿ); If [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) or L¹(Rⁿ) into BMO(Rⁿ), then, b ∈ LMO_loc.     

Θ-summability of Fourier series

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H _p to L _p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L ₁ converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.     

Fonctions lipschitziennes et espaces de Sobolev fractionnaires

We consider a semigroup of Markovian and symmetric operators to which we associate fractional Sobolev spaces D^α_p (0 < α < 1 and 1 < p < ∞) defined as domains of fractional powers (−A_p)^α/2, where A_p is the generator of the semigroup in L^p. We show under rather general assumptions that Lipschitz continuous functions operate by composition on D^α_p if p ≥ 2. This holds in particular in the case of the Ornstein-Uhlenbeck semigroup on an abstract Wiener space.     

The Faber Operator and its Boundedness

Let G be a domain bounded by a Jordan curve Γ, and let A(G) be the Banach space of functions continuous on G and holomorphic in G. The Faber operator T is a linear mapping from A( ) to A(G) mapping wⁿ onto the nth Faber polynomial F_n(z) (n=0, 1, 2, …). We show that T<∞ if Γ is piecewise Dini-smooth, and give an example of a quasicircle Γ for which T=∞.     

On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line

In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the Dunkl-type fractional maximal operator Mβ, and the Dunkl-type fractional integral operator Iβ from the spaces Lp,α(R) to the spaces Lq,α(R), 1<p<q<∞, and from the spaces L1,α(R) to the weak spaces WLq,α(R), 1<q<∞. In the case , we prove that the operator Mβ is bounded from the space Lp,α(R) to the space L∞,α(R), and the Dunkl-type modified fractional integral operator is bounded from the space Lp,α(R) to the Dunkl-type BMO space BMOα(R). By this results we get boundedness of the operators Mβ and Iβ from the Dunkl-type Besov spaces to the spaces , 1<p<q<∞, 1/p−1/q=β/(2α+2), 1?θ?∞ and 0<s<1.     

Finitely strictly singular operators between James spaces

An operator between Banach spaces is said to be finitely strictly singular if for every ε>0 there exists n such that every subspace E⊆X with dimE?n contains a vector x such that ‖Tx‖<ε‖x‖. We show that, for 1?p<q<∞, the formal inclusion operator from Jp to Jq is finitely strictly singular. As a consequence, we obtain that the strictly singular operator with no invariant subspaces constructed by C. Read is actually finitely strictly singular. These results are deduced from the following fact: if k?n then every k-dimensional subspace of Rn contains a vector x with ‖x?_∞‖=1 such that xmi=i(−1) for some m₁<?<mk.     

Weighted and Ergodic Theorems for Series of Differences of Convolutions

We extend some results of C. Demeter about the boundedness of series of difference of convolutions to the setting of one-sided A _p weights. We need to use a different approach. To be precise, we prove that the kernel of the associated operator satisfies a generalized Hörmander condition. The weighted inequalities allow us to transfer the result to the ergodic case, when the operator is induced by a mean bounded, invertible, positive groups.     

Weighted Hardy and singular operators in Morrey spaces

We study the weighted boundedness of the Cauchy singular integral operator SΓ in Morrey spaces Lp,λ(Γ) on curves satisfying the arc-chord condition, for a class of “radial type” almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces Lp,λ(0,?), ?>0. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves.     

The L resolvents of elliptic operators with uniformly continuous coefficients

We consider the strongly elliptic operator A of order 2m in the divergence form with bounded measurable coefficients and assume that the coefficients of top order are uniformly continuous. For 1<p<∞, A is a bounded linear operator from the L^p Sobolev space H^m,p into H^−m,p. We will prove that (A−λ)⁻¹ exists in H^−m,p for some λ and estimate its operator norm.