A rough hypersingular integral operator with an oscillating factor

We study certain hypersingular integrals TΩ,α,βf defined on all test functions f∈S(Rn), where the kernel of the operator TΩ,α,β has a strong singularity |y|^−n−α(α>0) at the origin, an oscillating factor ei|y|^−β(β>0) and a distribution Ω∈Hr(Sn−1), 0<r<1. We show that TΩ,α,β extends to a bounded linear operator from the Sobolev space to the Lebesgue space Lp for β/(β−α)<p<β/α, if the distribution Ω is in the Hardy space Hr(Sn−1) with 0<r=(n−1)/(n−1+γ)(0<γ?α) and β>2α>0.     

Boundedness of rough singular integral operators on the Triebel-Lizorkin spaces

We consider the singular integral operator T with kernel K(x)=Ω(x)/n|x| and prove its boundedness on the Triebel-Lizorkin spaces provided that Ω satisfies a size condition which contains the case Ω∈Lr(Sn−1), r>1.     

On hyper-singular integral operators with variable kernels

Let n?2, Sn−1 be the unit sphere in Rn. For 0?α<1, m∈N₀, 1<p?2, and Ω∈L^∞(Rn)×Hr(Sn−1) with (where Hr is the Hardy space if r?1 and Hr=Lr if 1<r<∞), we study the singular integral operator, for r?1, defined by

     

A mean value theorem for cubic fields

Let r(n) denote the number of integral ideals of norm n in a cubic extension K of the rationals, and define and Δ(x)=S(x)−αx where α is the residue of the Dedekind zeta function ζ(s,K) at 1. It is shown that the abscissa of convergence of

     

Haj?asz-Sobolev imbedding and extension

The author establishes some geometric criteria for a Haj?asz-Sobolev -extension (resp. -imbedding) domain of Rn with n?2, s∈(0,1] and p∈[n/s,∞] (resp. p∈(n/s,∞]). In particular, the author proves that a bounded finitely connected planar domain Ω is a weak α-cigar domain with α∈(0,1) if and only if for some/all s∈[α,1) and p=(2−α)/(s−α), where denotes the restriction of the Triebel-Lizorkin space on Ω.     

On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

Let a be a semi-almost periodic matrix function with the almost periodic representatives al and ar at −∞ and +∞, respectively. Suppose p:R→(1,∞) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space Lp(⋅)(R). We prove that if the operator aP+Q with P=(I+S)/2 and Q=(I−S)/2 is Fredholm on the variable Lebesgue space , then the operators alP+Q and arP+Q are invertible on standard Lebesgue spaces and with some exponents ql and qr lying in the segments between the lower and the upper limits of p at −∞ and +∞, respectively.     

Nonlinear random ergodic theorems for affine operators

Let (Ω,ß,μ) be a finite measure space and let (S,F,ν) be another probability measure space on which a measure preserving transformation φ is given. We introduce the so-called affine systems and prove a vector-valued nonlinear random ergodic theorem for the random affine system determined by a strongly F-measurable family of affine operators, where B is a reflexive Banach space, is a strongly F-measurable family of linear contractions on L₁(Ω,B) as well as on L_∞(Ω,B) and ξ is a function in (I−T)Lp(S×Ω,B) (1?p<∞) with the operator T defined by Tf(s,ω)=[Tsfφs](ω) which denotes the F⊗ß-measurable version of Tsfφs(ω). Moreover, some variant forms of the nonlinear random ergodic theorem are also obtained with some examples of affine systems for which the nonlinear ergodic theorems fail to hold.     

A surjection theorem and a fixed point theorem for a class of positive operators

This paper is concerned with α-convex operators on ordered Banach spaces. A surjection theorem for 1-convex operators in order intervals is established by means of the properties of cone and monotone iterative technique. It is assumed that 1-convex operator A is increasing and satisfies Ay−Ax?M(y−x) for θ?x?y?v₀, where θ denotes the zero element and v₀ is a constant. Moreover, we prove a fixed point theorem for -convex operators by using fixed point theorem of cone expansion. In the end, we apply the fixed point theorem to certain integral equations.     

Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem

Let Ω be a C1,1-bounded domain in Rn for n?2. In this paper, we are concerned with the asymptotic behavior of the unique positive classical solution to the singular boundary-value problem Δu+a(x)u−σ=0 in Ω, u|∂Ω=0, where σ?0, a is a nonnegative function in , 0<α<1 and there exists c>0 such that . Here λ?2, μk∈R, ω is a positive constant and δ(x)=dist(x,∂Ω).     

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.     

Generalized Haar integral

The aim of the paper is to generalize the notion of the Haar integral. For a compact semigroup S acting continuously on a Hausdorff compact space Ω, the algebra A(S)⊂C(Ω,R) of S-invariant functions and the linear space M(S) of S-invariant (real-valued) finite signed measures are considered. It is shown that if S has a left and right invariant measure, then the dual space of A(S) is isometrically lattice-isomorphic to M(S) and that there exists a unique linear operator (called the Haar integral) such that for each f∈A(S) and for any f∈C(Ω,R) and s∈S, , where .     

An elliptic system with bifurcation parameters on the boundary conditions

In this paper we consider the elliptic system Δu=a(x)upvq, Δv=b(x)urvs in Ω, a smooth bounded domain, with boundary conditions , on ∂Ω. Here λ and μ are regarded as parameters and p,s>1, q,r>0 verify (p−1)(s−1)>qr. We consider the case where a(x)?0 in Ω and a(x) is allowed to vanish in an interior subdomain Ω₀, while b(x)>0 in . Our main results include existence of nonnegative nontrivial solutions in the range 0<λ<λ₁?∞, μ>0, where λ₁ is characterized by means of an eigenvalue problem, and the uniqueness of such solutions. We also study their asymptotic behavior in all possible cases: as both λ,μ→0, as λ→λ₁<∞ for fixed μ (respectively μ→∞ for fixed λ) and when both λ,μ→∞ in case λ₁=∞.     

Compact operators and Toeplitz algebras on multiply-connected domains

If Ω is a smoothly bounded multiply-connected domain in the complex plane and S belongs to the Toeplitz algebra τ of the Bergman space of Ω, we show that S is compact if and only if its Berezin transform vanishes at the boundary of Ω. We also show that every element S in T, the C^?-subalgebra of τ generated by Toeplitz operators with symbols in H^∞(Ω), has a canonical decomposition for some R in the commutator ideal CT; and S is in CT iff the Berezin transform vanishes identically on the set M₁ of trivial Gleason parts.     

Strong convergence theorems for uniformly continuous pseudocontractive maps

Let K be a nonempty closed convex subset of a real Banach space E and let be a uniformly continuous pseudocontraction. Fix any u∈K. Let {xn} be defined by the iterative process: x₀∈K, xn+1:=μn(αnTxn+(1−αn)xn)+(1−μn)u. Let δ(?) denote the modulus of continuity of T with pseudo-inverse ?. If and {xn} are bounded then, under some mild conditions on the sequences n{αn} and n{μn}, the strong convergence of {xn} to a fixed point of T is proved. In the special case where T is Lipschitz, it is shown that the boundedness assumptions on and {xn} can be dispensed with.     

Hypersurfaces with constant mean curvature in a hyperbolic space

Let Mⁿ be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space Hⁿ⁺¹（c） with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that （1） if the multiplicities of the two distinct principal curvatures are greater than 1,then Mⁿ is isometric to the Riemannian product S^k（r）×H^n-k（-1/（r² + ρ²））, where r > 0 and 1 < k < n - 1;（2）if H² > -c and one of the two distinct principal curvatures is simple, then Mⁿ is isometric to the Riemannian product S^n-1（r） × H¹（-1/（r² +ρ²）） or S¹（r） × H^n-1（-1/（r² +ρ²））,r > 0, if one of the following conditions is satisfied （i） S≤（n-1）t²2+c²t^-22 on Mⁿ or （ii）S≥ （n-1）t²1+c²t^-21 on Mⁿ or（iii）（n-1）t²2+c²t^-22≤ S≤（n-1）t²1+c²t^-21 on Mⁿ, where t₁ and t₂ are the positive real roots of （1.5）.     

Functional calculus under the Tadmor-Ritt condition, and free interpolation by polynomials of a given degree

For Banach space operators T satisfying the Tadmor-Ritt condition ||(zI−T)⁻¹||?C|z−1|⁻¹, |z|>1, we prove that the best-possible constant C_T(n) bounding the polynomial calculus for T, ||p(T)||?C_T(n)||p||_∞, deg(p)?n, behaves (in the worst case) as as n→∞. This result is based on a new free (Carleson type) interpolation theorem for polynomials of a given degree.