Boundedness of commutators for Marcinkiewicz integrals on weighted Herz-type Hardy spaces

In this paper,the authors study the boundedness of the operator μ b Ω,the commutator generated by a function b ∈Lip β (R n)(0 < β < 1) and the Marcinkiewicz integral μΩ on weighted Herz-type Hardy spaces.     

Continuity of Marcinkiewicz integrals on homogeneous Morrey-Herz spaces

Let μmΩ,b be the higher order commutator generated by Marcinkiewicz integral μΩ and a BMO(Rn) function b(x). In this paper, we will study the continuity ofμΩ and μmΩ,b on homogeneous Morrey-Herz spaces.     

Decompositions of the Hardy space Hz^1（Ω）

We give a decomposition of the Hardy space Hz^1（Ω） into ＂div-curl＂ quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1（Ω） into Jacobians det Du, u ∈ W0^1,2 （Ω,R^2） for Ω in R^2. This partially answers a well-known open problem.     

On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.     

Boundedness of commutators related to Marcinkiewicz integrals on hardy type spaces

In this paper, the authors study the boundedness of the operator [μΩ, b], the commutator generated by a function b ∈ Lipβ(Rn)(0 ＜β≤ 1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω∈ Lipα(Sn-1)(0 ＜α≤ 1).     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

Potential space estimates in local Hardy spaces for Green potentials in convex domains

Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)＜p≤1.     

Estimates for parametric Marcinkiewicz integrals in BMO and Campanato spaces

In this paper, the authors consider the behaviors of a class of parametric Marcinkiewicz integrals μ _Ω ^ρ , μ _Ω,λ ^*,ρ and μ _Ω,S ^ρ on BMO(ℝ ⁿ ) and Campanato spaces with complex parameter ρ and the kernel Ω in Llog⁺ L(S ⁿ⁻¹). Here μ _Ω,λ ^*,ρ and μ _Ω,S ^ρ are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g _λ *-function and the Lusin area function S, respectively. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO(ℝ ⁿ ) or to a certain Campanato space, then [μ _Ω,λ ^*,ρ (f)]², [μ _Ω,S ^ρ (f)]² and [μ _Ω ^ρ (f)]² are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established.     

The boundedness of Toeplitz-type Operators on vanishing-morrey spaces

In this note,we prove that the Toeplitz-type Operator Θ b α generated by the generalized fractional integral,Calderón-Zygmund operator and VMO funtion is bounded from L p,λ (R n) to L q,μ (R n).We also show that under some conditions Θαb f ∈ V L q,μ (B R),the vanishing-Morrey space.     

Variational functionals defined on spaces ofBV functions and their dependence on boundary data

Summary We extend in a suitable way a class of functionals defined on W ^1,1 (Ω) to the space BV_b(Ω) ⊗ (C(∂Ω))*. Optimization theorems for the extended functional are given. As an application we obtain a known result about minimal surface problems. Entrata in Redazione il 18 luglio 1978. This work was supported in part by Laboratorio per la Matematica Applicata del C.N.R., Italy.     

Degenerate triply nonlinear problems with nonhomogeneous boundary conditions

The paper addresses the existence and uniqueness of entropy solutions for the degenerate triply nonlinear problem: b(v) _t − div α(v, ▽g(v)) = f on Q:= (0, T) × Ω with the initial condition b(v(0, ·)) = b(v ₀) on Ω and the nonhomogeneous boundary condition “v = u” on some part of the boundary (0, T) × ∂Ω”. The function g is continuous locally Lipschitz continuous and has a flat region [A ₁, A ₂,] with A ₁ ≤ 0 ≤ A ₂ so that the problem is of parabolic-hyperbolic type.     

Analytic Semigroups on H 1 (Ω) Generated by Degenerate Elliptic Operators

One proves that Ay=aΔ y+b⋅ \nabla y , D(A)={y∈ H ¹ (Ω ); aΔ y+b⋅ \nabla y∈ H ₀ ¹ (Ω ), \sqrt a Δ y∈ L ² (Ω )} generates, under suitable conditions on a and b , a C ₀ -analytic semigroup on H ¹ (Ω) . September 5, 1999     

Irreducibility and Mixed Boundary Conditions

In this paper we consider positive semigroups on L^p(Ω) generated by elliptic operators A subject to mixed Dirichlet-Neumann boundary conditions on non-smooth domains Ω. We show in particular that these semigroups as well as those generated by multiplicative perturbations bA of A are irreducible, provided b ∈ L^∞(Ω) is real and satisfies b ≥ δ for some δ > 0. In memoriam Helmut H. Schaefer     

Conjugation of injections by permutations

Let Ω be a countably infinite set, Inj(Ω) the monoid of all injective endomaps of Ω, and Sym(Ω) the group of all permutations of Ω. Also, let f,g,h∈Inj(Ω) be any three maps, each having at least one infinite cycle. (For instance, this holds if f,g,h∈Inj(Ω)∖Sym(Ω).) We show that there are permutations a,b∈Sym(Ω) such that h=afa ⁻¹ bgb ⁻¹ if and only if |Ω∖(Ω)f|+|Ω∖(Ω)g|=|Ω∖(Ω)h|. We also prove a generalization of this statement that holds for infinite sets Ω that are not necessarily countable.     

Compact embeddings of besov spaces involving only slowly varying smoothness

We characterize compact embeddings of Besov spaces B _p,r ^0,b (ℝ ⁿ ) involving the zero classical smoothness and a slowly varying smoothness b into Lorentz-Karamata spaces L<sub>p,q;[`(b)]</sub> {L_{p,q;\overline b }}(Ω), where is a bounded domain in ℝ <sup>n</sup> and [`(b)]\overline b is another slowly varying function.     

Boundedness of parabolic singular integrals and Marcinkiewicz integrals on Triebel-Lizorkin spaces

In this paper,we obtain the boundedness of the parabolic singular integral operator T with kernel in L(logL)1/γ(Sn-1) on Triebel-Lizorkin spaces.Moreover,we prove the boundedness of a class of Marcinkiewicz integrals μΩ,q(f) from ∥f∥ F˙p0,q(Rn) into Lp(Rn).     

Commutators of integral operators with variable kernels on Hardy spaces

LetT _Ω,α (0 ≤ α< n) be the singular and fractional integrals with variable kernel Ω(x, z), and [b, T_Ω,α] be the commutator generated by T_Ω,α and a Lipschitz functionb. In this paper, the authors study the boundedness of [b, T_Ω,α] on the Hardy spaces, under some assumptions such as theL ^r -Dini condition. Similar results and the weak type estimates at the end-point cases are also given for the homogeneous convolution operators . The smoothness conditions imposed on are weaker than the corresponding known results.     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

On the behavior of solutions of a semilinear parabolic or elliptic equation satisfying a nonlinear boundary condition in a cylindrical domain

One considers a semilinear parabolic equation u _t = Lu − a(x)f(u) or an elliptic equation u _tt + Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ₊ with the nonlinear boundary condition , where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝⁿ; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems for t → ∞. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007.     

N-widths of Ω r ′ in L p

Let Ω _ϕ ^r ={f:f ^(r-1) abs. cont. on [0,1], ‖q_r(D)f‖_p≤1, f^(2K+σ) (0)=f^(2K+σ)=0, (k)=0,...,l-1}. where , and I is an identical operator. Denote Kolmogorov, linear, Geelfand and Bernstein n-widths of Ω _ϕ ^r in L^p byd _n (Ω _ϕ ^r ;L ^p ),δ _n (Ω _ϕ ^r ;L ^p ),d ⁿ (Ω _p ^r ;L ^p ) andb _n (Ω _p ^r ;L ^p ), respectively. In this paper, we find a method to get an exact estimation of these n-widths. Related optimal subspaces and an optimal linear operator are given. For another subset , similar results are also derrived.