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 ＜∞.     

On the tensor products of operators in lebesgue spaces

The following question concerning the computation of the norms of the tensor products of operators in the Lebesgue spaces is studied: Is it true that the norm of the tensor product A?B: L^p(μ?μ)→L^q(ν?ν) of operators A: L^p(μ)→L^q(ν) and B: L^p(μ)→L^q(ν) coincides with the product ‖A‖ ‖B‖ of their norms? An answer is positive if and only if 1≤p≤q≤+∞. Bibliography: 26 titles.     

乘子算子及其交换子在Morrey空间上的有界性

证明了乘子算子(M_p~q(R~n),Lip(β-n/q))的有界性和(M_p~q(R~n),BMO(R~n))的有界性.还得到乘子算子及其交换子在广义Morrey空问Lp,L_(p,φ)(R~n)上的有界性.     

满足H(m)条件的奇异积分算子的交换子的有界性

主要讨论了满足H(m)条件的奇异积分算子与Lipschitz函数的交换子在L~p和Hardy空间的有界性,并把这个结果应用于与薛定谔算子相关的Riesz变换.     

On a Hardy Type General Weighted Inequality in Spaces L p(·)

A Hardy type two-weighted inequality is investigated for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces L ^p(·). Equivalent necessary and sufficient conditions are found for the ${L^{p(\cdot)} \longrightarrow L^{q(\cdot)}}A Hardy type two-weighted inequality is investigated for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces L ^p(·). Equivalent necessary and sufficient conditions are found for the L^p(·) ? L^q(·){L^{p(\cdot)} \longrightarrow L^{q(\cdot)}} boundedness of the Hardy operator when exponents q(0) < p(0), q(∞) < p(∞). It is proved that the condition for such an inequality to hold coincides with the condition for the validity of two-weighted Hardy inequalities with constant exponents if we require of the exponents to be regular near zero and at infinity.     

Hardy-Littlewood Maximal Operator,Singular Integral Operators and the Riesz Potentials on Generalized Morrey Spaces

Any continuous linear operator T: L^p → L^q has a natural vector-valued extension T: L^p(l) → L^q(l) which is automatically continuous. Relations between the norms of these operators in the cases of p = q and r = 2 were considered by Marcinkiewicz -Zygmund [28], Herz [14] and Krivine [19] - [21]. In this paper we study systematically these relations and given some applications. It turns out that some known results can be proved in a simple way as a consequence of these developments.     

Two remarks concerning the equation Πp(Χ,·)=Ip(Χ,·)

It is proved that the analog of Grothendieck's theorem is valid for a diskalgebra “up to a logarithmic factor.” Namely, if Tε? (C_A, L¹) and then π₂(T)?C (1+logn) ¦T¦. The question of whether the logarithmic factor is actually necessary remains open. It is also established that C _A ^* is a space of cotype q for any q, q > 2. The proofs are based on a theorem of Mityagin-Pelchinskii: π_p(T)?C·p·i_p(T), p?2 for any operator T acting from a disk-algebra to an arbitrary Banach space.     

Dirichlet operators and the positive maximum principle

Let (A, D(A)) denote the infinitesimal generator of some strongly continuous sub-Markovian contraction semigroup onL ^p (m), p1 andm not necessarily -finite. We show under mild regularity conditions thatA is a Dirichlet operator in all spacesL ^q (m), qp. It turns out that, in the limitq,A satisfies the positive maximum principle. If the test functionsC _c D(A), then the positive maximum principle implies thatA is a pseudo-differential operator associated with a negative definite symbol, i.e., a Lévy-type operator. Conversely, we provide sufficient criteria for an operator (A, D(A)) onL ^p(m) satisfying the positive maximum principle to be a Dirichlet operator. If, in particular,A onL ² (m) is a symmetric integro-differential operator associated with a negative definite symbol, thenA extends to a generator of a regular (symmetric) Dirichlet form onL ² (m) with explicitly given Beurling-Deny formula.     

Two-point inequalities,the Hermite semigroup and the Gauss-Weierstrass semigroup

Let e^?zH, Re z ? 0, be the Hermite semigroup on R with Gauss measure μ. Necessary and sufficient conditions for e^?zH to be a bounded map from L^p(μ) into L^q(μ), 1 ? p, q ? ∞, are found and in many cases it is proved that e^?zH: L^p(μ) → L^q(μ) is in fact a contraction. Furthermore, these results and a formula relating the Hermite semigroup with the Gauss-Weierstrass semigroup e^zΔ enable one to calculate the precise norm of e^zΔ:L^p(dx) → L^q(dx) in a large number of cases.     

Fixed point iterations for certain classes of nonlinear mappings

Let X = L_p or L_p, 2≤p<∞, and let K be a nonempty closed convex bounded subset of X. It is proved that for some classes of nonlinear mappings T:K → K (more precisely, for T P₂ or C in the terminology of F.E. Browder and W.V. Pretryshyn; and B.E. Rhoades), the iteration process: x₁ ?K,X_n+1 = (1-C_n)x_n+C_n Tx_n, n ≥1,under suitable conditions on K and the real sequence {C_n}_n=1 ^∞ converges strongly to a fixed point of T. While our thorems generalize serveral known results, our method is also of independent interest     

FUNCTIONAL CALCULI FOR HILBERT SPACE OPERATORS

Abstract

Let T be a bounded operator on a Hilbert space H with Von Neumann spectral set X. If there exists no non-zero reducing subspace of H restricted to which T is a normal operator with spectrum contained in the boundary of X and if the uniform algebra R(X) is pointwise boundedly dense in H^∞ (X°), then there exists a functional calculus f → f(T) for f ε H^∞ (X°). A similar result for the two-variable case is also proved.     

Asymmetry of twisted convolution operators

Let K be a distribution on $\text{R}$². We denote by λ(K) the twisted convolution operator f → K × f defined by the formula K × f(x, y) = ∝∝ dudvK(x ? u, y ? v) f(u, v) exp(ixv ? iyu). We show that there exists K such that the operator λ(K) is bounded on L^p(R)² for every p in (1, 2¦, but is unbounded on L^q(R)² for every q > 2.     

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 < ∞.     

AN EIGENSPACE CHARACTERIZATION OF LORENTZ-IMPROVING MEASURES ON COMPACT GROUPS

Abstract

A measure μ on a compact group is called Lorentz-improving if for some 1 > p > ∞ and 1 → q ₁ > q ₂ ∞ μ *L (p, q ₂) ? L(p, q ₁). Let T _μ denote the operator on L ² defined by T _μ(f) = μ * f. Lorentz-improving measures are characterized in terms of the eigenspaces of T _μ, if T _μ is a normal operator, and in terms of the eigenspaces of |T _μ| otherwise. This result generalizes our recent characterization of Lorentz-improving measures on compact abelian groups and is modelled after Hare's characterization of L ^p -improving measures on compact groups.     

Bernstein widths of embeddings of Lebesgue spaces

Let (K, μ) be a measurable space with μ(K)=1. Let Ip,q: L^p (K, μ)→L^q (K, μ) be the embedding operator. The Bernstein widths of I_{p, q} are considered. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 166–169. Translated by S. V. Kislyakov.     

FACTORIZATION OF UNBOUNDED THIN AND COTHIN OPERATORS

Abstract

Let X and Y be normed spaces and T: D(T) ? X → Y a linear operator. Following R.D. Neidingcr [N1] we recall the Davis, Figiel, Johnson, Pelczynski factorization of T corresponding to a parameter p (1 ≤ p ≤ ∞) and apply the corresponding factorization result in [N1] to unbounded thin operators. Properties equivalent to ubiquitous thinness arc derived. Defining an operator T to be cothin if its adjoint is thin, a dual factorization result for cothin operators is obtained, where for each 1 < p < ∞, the intermediate space in the factorization is cohereditarily l_p. This result is shown to hold more generally for the cases when T is either partially continuous or closable; in particular, such operators are strictly cosingular. A condition for a closable weakly compact operator to be strictly cosingular follows as a corollary.     

The annihilator of Bergman space

LetD be a bounded plane domain (with some smoothness requirements on its boundary). LetB _p(D), 1≤p<∞, be the Bergmanp-space ofD. In a previous paper we showed that the “natural projection”P, involving the Bergman kernel forD, is a bounded projection fromL _p(D) ontoB _p(D), 1<p<∞. With this we have the decompositionL _p(D)=B _p(D)⊕B _q ^⊥ (D,p ^–1+q ^–=1, 1<p< ∞. Here, we show that the annihilatorB _q ^⊥ (D) is the space of allL _p-complex derivatives of functions belonging to Sobolev space and which vanish on the boundary ofD. This extends a result of Schiffer for the casep=2. We also study certain operators onL _p(D). Especially, we show that , whereI is the identity operator and ? is an operator involving the adjoint of the Bergman kernel. Other relationships relevant toB _q ^⊥ (D) are studied.     

BOUNDED LINEAR OPERATORS THAT COMMUTE WITH SHIFTS ARE SCALED IDENTITY

We prove that every bounded linear operator on L^p?L^p(R'),s≥1 and 1≤p<∞, that commutes with both the spatial shift operator and the phase shift operator must be a constant multiple of the identity. We also apply this result to identify analysis-synthesis dual L^q-L^p paris and to characterize bi-orthogonal unconditional bases of L^q-L^p, where p^?1+q^?1=1.     

Some new Triebel-Lizorkin spaces on spaces of homogeneous type and their frame characterizations

Let(X,p,μ)d,θ be a space of homogeneous type,(?) ∈(0,θ],|s|＜(?) andmax{d/(d＋(?)),d/(d＋s＋(?))}＜q≤∞.The author introduces the new Triebel-Lizorkin spaces (?)_∞q~s(X) and establishes the framecharacterizations of these spaces by first establishing a Plancherel-P(?)lya-type inequalityrelated to the norm of the spaces (?)_∞q~s(X).The frame characterizations of the Besovspace (?)_pq~s(X) with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p≤∞ and 0＜q≤∞and the Triebel-Lizorkin space (?)_pq~s(X)with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p＜∞ and max{d/(d＋(?)),d/(d＋s＋(?))}＜q≤∞ are also presented.Moreover,the au-thor introduces the new TriebeI-Lizorkin spaces b(?)_∞q~s(X) and H(?)_∞q~s(X) associated to agiven para-accretive function b.The relation between the space b(?)_∞q~s(X) and the spaceH(?)_∞q~s(X) is also presented.The author further proves that if s＝0 and q＝2,thenH(?)_∞q~s(X)＝(?)_∞q~s(X),which also gives a new characterization of the space BMO(X),since (?)_∞q~s(X)＝BMO(X).     

Compact and strictly singular operators on Orlicz spaces

If 0 < p < 1 andT: L_p(0,1) →E is a continuous linear operator into a topological vector space, there is an infinite-dimensional subspaceX ofL _p on whichT is an isomorphism; thus there are no compact operators onL _p . Results of this type are proved for general non-locally convex Orlicz spaces.