The Besov and Triebel-Lizorkin spaces with high order on the Lipschitz curves

The Besov spacesB _p ^α,μ (Γ) and Triebel-Lizorkin spacesF _p ^α,μ (Γ) with high order α∈R on a Lipschitz curve Γ are defined. when 1≤p≤∞, 1≤q≤∞. To compare to the classical case a difference characterization of such spaces in the case |α|<1 is given also. The author is supported in part by the Foundation of Zhongshan University Advanced Research Centre and NSF of China.     

Spectral invariance for pseudodifferential operators on weighted function spaces

The spectrum of each symmetric ψ DO of the symbol class S⁰ _{1, γ}, 0≤γ<1, acting on B³ _p,q(w(x)) and F³ _p,q(w(x)), is independent of the choice ofs ∈ℝ, 0<p≤∞ (p<∞ in the F-case), 0<q≤∞ and the weight w(x)∈W.     

Generalized Besov spaces and Triebel-Lizorkin spaces

In this paper the classical Besov spaces Bsp.q and Triebel-Lizorkin spaces Fsp.q for s ∈R are generalized in an isotropy way with the smoothness weights {|2j|aln}∞j=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by Bap.q and Fap.q for a ∈Irk and k ∈N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters a, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between Bs,p.q and ∪tsBt,p.q,and between Fsp.q and ∪ts Ftp.q, respectively. Between Bs,p,q and ∪tsBt,p.qq,and between Fsp,qand ∪tsFtp.q,respectively.     

Weighted approximation of random functions

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Atomic decomposition of Besov-type and Triebel-Lizorkin-type spaces

Abstract With the help of the maximal function caracterizations of the Besov-type space Bs,τp,q and the TriebelLizorkin-type space Fs,τp,q,we present the atomic decomposition of these function spaces.Our results cover the results on classical Besov and Triebel-Lizorkin spaces by taking τ=0.     

Sequence spaces and inverse of an infinite matrix

We give here some properties of the sets _α(uΔ) generalizing the space of generalized difference sequencesl ^∞(uΔ). Then we study spaces related to the sets of sequences that are strongly convergent or strongly bounded. Next we define from the sets of spaces that are (N,q) summable or bounded the sets of spaces that are (N,q)α-bounded orr-bounded. Then we give some properties of these spaces using Banach space of the forms _α.     

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W_p^k(ℝ^s) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W_p^k(ℝ^s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L_p (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ∈L_p(ℝ^s) (φ∈C(ℝ^s) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,L_p(ℝ^s)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167) in the univariate setting to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C20, 41A25, 39B12. Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant G121210654.     

On F-Sobolev and Orlicz-Sobolev inequalities

Let F ∈ C([0,∞)) be a positive increasing function such that Φ(s):= |s|F(|s|) is a Young function. In general, the F-Sobolev inequality and the Φ-Orlicz-Sobolev inequality are not equivalent. In this paper, a growth condition on F is presented for these two inequalities to be equivalent. The main result generalizes the corresponding known one for F(s) = log^δ(1 + s) (δ > 0). As an application, some criteria are presented for the F-Sobolev inequality to hold.     

Applications of utility functions defined on quasi-metric spaces

A quasi-metric space (X,d) is called sup-separable if (X,d^s) is a separable metric space, where d^s(x,y)=max{d(x,y),d(y,x)} for all x,y∈X. We characterize those preferences, defined on a sup-separable quasi-metric space, for which there is a semi-Lipschitz utility function. We deduce from our results that several interesting examples of quasi-metric spaces which appear in different fields of theoretical computer science admit semi-Lipschitz utility functions. We also apply our methods to the study of certain kinds of dynamical systems defined on quasi-metric spaces.     

Commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of Homogeneous type of finite measure

Let X be a space of homogeneous type with finite measure. Let T be a singular integral operator which is bounded on Lp (X), 1 ＜ p ＜∞. We give a sufficient condition on the kernel k(x,y) of Tso that when a function b ∈ BMO (X) ,the commutator [b, T] (f) = T (b f) - bT (f) is aounded on spaces Lp for all p, 1 ＜ p ＜∞.     

Computation of the ε entropy of the space of continuous functions with the Hausdorff metric

The number K_p,q, i.e., the number of (p, q) corridors of closed domains which are convex in the vertical direction, consist of elementary squares of the integral lattice, are situated within a rectangle of the size q × p, and completely cover the side of length p of this rectangle under projection is computed. The asymptotic (K_p,q/q²)^1/p → λ, as p, q → ∞, where λ = 0.3644255… is the maximum root of the equation₁F₁(-1/2 − 1/(16λ), 1/2, 1/(4λ)) = 0,₁F₁ being the confluence hypergeometric function, is established. These results allow us to compute the ε entropy of the space of continuous functions with the Hausdorff metric. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 39–50, January, 1977.     

Uniform algebras as Banach spaces

Let A be a closed subalgebra of the complex Banach algebra C(S), containing the constant functions. We assume that one has found a probability measureμ on S and a function F from L^∞(μ) such that: 1)|F|= 1 a.e. relative to μ; 2) F _μ ε A¹; 3) F is a limit point of the unit ball of the algebra A in the topology δ(L^∞(μ), L¹(μ)). One proves in the paper that under these conditions the space A** contains a complement space, isometric to H^∞. The measure μ and the function F, satisfying the conditions l)-3) indeed exist if the maximal ideal space of the algebra A contains a non-one-point part (and it is very likely that such aμ. and F exist whenever the algebra A is not self-adjoint). Thus, the above-formulated result allows us to extend A. Pelczynski's theorem (Ref, Zh. Mat., 1975, 1B894) regarding the space H^∞ to a very broad class of uniform algebras. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 80–89, 1976.     

Anisotropic Triebel-Lizorkin spaces with doubling measures

We introduce and study anisotropic Triebel-Lizorkin spaces associated with general expansive dilations and doubling measures on ℝⁿ with the use of wavelet transforms. This work generalizes the isotropic methods of dyadic ϕ-transforms of Frazier and Jawerth to nonisotropic settings. We extend results involving boundedness of wavelet transforms, almost diagonality, smooth atomic and molecular decompositions to the setting of doubling measures. We also develop localization techniques in the endpoint case of p = ∞, where the usual definition of Triebel-Lizorkin spaces is replaced by its localized version. Finally, we establish nonsmooth atomic decompositions in the range of 0 < p ≤ 1, which is analogous to the usual Hardy space atomic decompositions.     

The density of rational points on non-singular hypersurfaces

LetF(x) =F[x₁,…,x_n]∈ℤ[x₁,…,x_n] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ ⁿ ;F(x)=0, |x|⩽X}, where . It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X ^n−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process. Dedicated to the memory of Professor K G Ramanathan     

On generalized energy equality of the Navier-Stokes equations

We show that for every initial dataa εL ²(Ω) there exists a weak solutionu of the Navier-Stokes equations satisfying the generalized energy inequality introduced by Caffarelli-Kohn-Nirenberg forn=3. We also show that if a weak solutionu εL ^s (0,T;L ^q (Ω)) with 2/q + 2/s ≤ 1 and 3/q + 1/s ≤ 1 forn=3, or 2/q + 2/s ≤ 1 andq ≥ 4 forn ≥ 4, thenu satisfies both the generalized and the usual energy equalities. Moreover we show that the generalized energy equality holds only under the local hypothesis thatu εL ^s (ε, T; L ^q (K)) for all compact setsK ⊂ ⊂ Ω and all 0 <ε <T with the same (q, s) as above when 3 ≤n ≤ 10.     

A new kind of spaces of Triebel-Lizorkin type

In this note, we define a new kind of spaces TF _p ^aq of Triebel-Lizorkin type and characterize it via sequence space l^p (l^q). Also a Fourier multiplier theorem is established in this new spaces TF _p ^aq . The author is partially supported by the National Natural Science of China and the Zhejiang Provincal Science Foundation.     

Weighted composition operators from F（p,q, s） spaces to Bers-type spaces in the unit ball

This paper deals with the boundedness and compactness of the weighted composition operators from the F（p, q, s） spaces, including Hardy space, Bergman space, Qp space, BMOA space, Besov space and α-Bloch space, to Bers-type spaces Hv^∞（ or little Bers-type spaces Hv,o∞ ）, where v is normal.     

Imbedding of power series spaces and spaces of analytic functions

The diametral dimension of a nuclear Fréchet spaceE, which satisfies (DN) and (Ω), is related to power series spaces Λ₁(ε) and Λ_∞(ε) for some exponent sequence ε. It is proved thatE contains a complemented copy of Λ_∞(ε) provided the diametral dimensions ofE and Λ_∞(ε) are equal and ε is stable. Assuming Λ₁(ε) is nuclear, any subspace of Λ₁(ε) which satisfies (DN), can be imbedded intoE. Applications of these results to spaces of analytic functions are given. Support of Turkish Scientific and Technical Research Council is gratefully acknowledged.     

Elementary characterizations of generalized weighted Morrey-Campanato spaces

Let α∈ (0,∞), p, q ∈ [1,∞), s be a nonnegative integer, and ω∈ A1(Rn) (the class of Muckenhoupt's weights). In this paper, we introduce the generalized weighted Morrey-Campanato space L(α, p, q, s, ω; Rn) and obtain its equivalence on different p ∈ [1,β) and integers s ≥ nα (the integer part of nα), where β = (1q - α)-1 when α 1q or β = ∞ when α≥ 1q. We then introduce the generalized weighted Lipschitz space ∧(α, q, ω; Rn) and prove that L(α, p, q, s, ω; Rn)  ∧(α, q, ω; Rn) when α∈ (0,∞), s ≥ nα , and p ∈ [1,β).     

Boundedness of Convolution-Type Operators on Certain Endpoint Triebel-Lizorkin Spaces

In this paper, we are concerned with the boundedness of convolution-type Calderón-Zygmund operators on some endpoint Triebel-Lizorkin spaces. We establish the boundedness on [(F)\dot]₁^0,q\dot{F}_{1}^{0,q} (2<q<∞) under a very weak pointwise regularity condition. The boundedness is established by the Daubechies wavelets and the atomic-molecular approach.