Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L²(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL whose kernels pt(x,y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp-norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L² estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.     

Littlewood–Paley and Spectral Multipliers on Weighted L p Spaces

Let L be a non-negative self-adjoint operator acting on L ²(X), where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e ^?tL whose kernel p _t (x,y) has a Gaussian upper bound but there is no assumption on the regularity in variables x and y. In this article we study weighted L ^p -norm inequalities for spectral multipliers of L. We show that a weighted Hörmander-type spectral multiplier theorem follows from weighted L ^p -norm inequalities for the Lusin and Littlewood–Paley functions, Gaussian heat kernel bounds, and appropriate L ² estimates of the kernels of the spectral multipliers.     

Maximal commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of homogeneous type

Let X be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, via a new Cotlar type inequality linking commutators and corresponding maximal operators, a weighted Lp(X) estimate with general weights and a weak type endpoint estimate with A₁(X) weights are established for maximal operators corresponding to commutators of BMO(X) functions and singular integral operators with non-smooth kernels.     

Approximation by integral type Meyer-König-Zeller operators on a simplex

Let T={(x₁,x₂):x₁+x₂≤1, x₁,x₂≤0} be a simplex in ?². The integral type Meyer-König-Zeller operators are constructed on the simplex T, and the degree of approximation of these operators for L^p-functions is obtained.     

Classifying Hilbert bundles. II

A Hilbert bundle (p, B, X) is a type of fibre space p: B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X, even when X is connected. We give two “homotopy” type classification theorems for Hilbert bundles having primarily finite dimensional fibres. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle over (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. As a special case, we show that if X is a compact metric space, C₊X the upper cone of the suspension SX, then the isomorphism classes of (m, n)-bundles over (SX, C₊X) are in one-to-one correspondence with the members of [X, V_m($\text{C}$ⁿ)] where V_m($\text{C}$ⁿ) is the Stiefel manifold. The results are all applicable to the classification of separable, continuous trace C¹-algebras, with specific results given to illustrate.     

Schatten classes of integration operators on Dirichlet spaces

We address the problem of determining membership in Schatten-Von Neumann ideals S _p of integration operators (T _g f)(z) = ∫ ₀ ^z = ∫ ₀ ^z f(ξ)g′(ξ)dξ acting on Dirichlet type spaces. We also study this problem for multiplication, Hankel and Toeplitz operators. In particular, we provide an extension of Luecking's result on Toeplitz operators [10, p. 347].     

A pointwise abelian Ergodic theorem for Lp semigroups, 1 ⩽ p < ∞

Let (X, ∑, μ) be a σ-finite measure space and L_p(μ) = L_p(X, ∑, μ), 1 ? p ? ∞, the usual Banach spaces of complex-valued functions. Let {T_t: t ? 0} be a strongly continuous semigroup of positive L_p(μ) operators for some 1 ? p < ∞. Denote by R_λ the resolvent of {T_t}. We show that f?L_p(μ) implies λR_λf(x) → f(x) a.e. as λ → ∞.     

Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators

Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.     

Banach space sequences and projective tensor products

In this paper we use results from the theory of tensor products of Banach spaces to establish the isometry of the space of (1,p)-summing sequences (also known as strongly p-summable sequences) in a Banach space X, the space of nuclear X-valued operators on ?^p and the complete projective tensor product of ?^p with X. Through similar techniques from the theory of tensor products, the isometry of the sequence space L^p〈X〉 (recently introduced in a paper by Bu, Quaestiones Math. (2002), to appear), the space of nuclear X-valued operators on L^p(0,1) (with a suitable equivalent norm) and the complete projective tensor product of L^p(0,1) with X is established. Moreover, we find conditions for the space of (p,q)-summing multipliers to have the GAK-property (generalized AK-property), use multiplier sequences to characterize Banach space valued bounded linear operators on the vector sequence space of absolutely p-summable sequences in a Banach space and present short proofs for results on p-summing multipliers.     

Connection between random curves,changes of time,and regenerative times of stochastic processes

The product of spaces Φ × D is considered, where Φ is the set of all continuous, nondecreasing functions ?:[0,∞)→(0,∞), ?(0)=0, ?(t)→∞(t→∞), and D is the set of all right continuous functions ξ:(0,∞)→X; here X is some metric space. Two mappings are defined: the first is the projection q(?,ξ)=ξ, and the second is the change of time U(?,ξ)=ξº?. The following equivalence relation is defined on D: $$\xi _1 \sim \xi _2 \Leftrightarrow \exists _{\varphi _1 , \varphi _1 } \in \Phi :\xi _1 ^\circ \varphi _1 = \xi _2 ^\circ \varphi _2 $$ . Let? be the set of all equivalence classes, and let L be the mapping ξ₄～ξ₂, Lξ is called the curve corresponding to ξ. The following theorem is proved: two stochastic processes with probability measures P¹ and P² on D possess identical random curves (i.e.,P¹ºL^?1=P²ºL^?1) if and only if there exist two changes of time (i.e., probability measures Q¹ and Q² on ?×D for which P¹=Q¹ºq^?1, P²=Q²ºq^?1 which take these two processes into a process with measure \(\tilde P\) (i.e., Q¹ºu^?1=Q²ºu^?1,=～P) If (P _x ¹ )_x∈X and (P _x ² )_x∈X are two families of probability measures for which P _x ¹ ºL^?1=P _x ² ºL^?1?x∈X then for each x ε X the corresponding measures Q _X ¹ andQ _X ² can be found in the following manner. The set of regenerative times of the family \(\left( {\tilde P_x } \right)_{x \in X} \) contains all stopping times which are simultaneously regenerative times of the families (p _x ¹ )_x∈X and (P _x ² )_x∈X and possess a certain special property of first intersection.     

Rellich type inequalities related to Grushin type operator and Greiner type operator

We prove some sharp Hardy inequality associated with the gradient ? _?? = (? _x ,|x| ^?? ? _y ) by a direct and simple approach. Moreover, similar method is applied to obtain some weighted sharp Rellich inequality related to the Grushin operator in the setting of L ^p . We also get some weighted Hardy and Rellich type inequalities related to a class of Greiner type operators.     

Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained.     

Singular integrals associated to the Laplacian on the affine groupax+b

We consider singular integral operators of the form (a)Z ₁L^?1Z₂, (b)Z ₁Z₂L^?1, and (c)L ^?1Z₁Z₂, whereZ ₁ andZ ₂ are nonzero right-invariant vector fields, andL is theL ²-closure of a canonical Laplacian. The operators (a) are shown to be bounded onL ^p for allp∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type (p, p) for anyp∈[1, ∞).     

Function Spaces as Path Spaces of Feller Processes

Let {X_t}_{t ≥ 0} be a Feller process with infinitesimal generator (A, D(A)). If the test functions are contained in D(A), —A |C^∞_c (ℝⁿ) is a pseudo–differential operator p(x, D) withsymbol p(x, ξ). We investigate local and global regularity properties of the sample paths t ↦ X_t in terms of (weighted) Besov B^s_pq (ℝ, ρ) and Triebel–Lizorkin F^s_pq (ℝ, ρ) spaces. The parameters for these spaces are determined by certain indices that describe the asymptotic behaviour of the symbol p(x, ξ). Our results improve previous papers on Lévy [5, 9] and Feller processes [22].     

Hardy Spaces, Regularized BMO Spaces and the Boundedness of Calderón–Zygmund Operators on Non-homogeneous Spaces

One defines a non-homogeneous space (X,μ) as a metric space equipped with a non-doubling measure μ so that the volume of the ball with center x, radius r has an upper bound of the form r ⁿ for some n>0. The aim of this paper is to study the boundedness of Calderón–Zygmund singular integral operators T on various function spaces on (X,μ) such as the Hardy spaces, the L ^p spaces, and the regularized BMO spaces. This article thus extends the work of X. Tolsa (Math. Ann. 319:89–149, 2011) on the non-homogeneous space (? ⁿ ,μ) to the setting of a general non-homogeneous space (X,μ). Our framework of the non-homogeneous space (X,μ) is similar to that of Hytönen (2011) and we are able to obtain quite a few properties similar to those of Calderón–Zygmund operators on doubling spaces such as the weak type (1,1) estimate, boundedness from Hardy space into L ¹, boundedness from L ^∞ into the regularized BMO, and an interpolation theorem. Furthermore, we prove that the dual space of the Hardy space is the regularized BMO space, obtain a Calderón–Zygmund decomposition on the non-homogeneous space (X,μ), and use this decomposition to show the boundedness of the maximal operators in the form of a Cotlar inequality as well as the boundedness of commutators of Calderón–Zygmund operators and BMO functions.     

Spectral properties of zeroth-order pseudodifferential operators

Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, {E_S}_{S∈$\text{R}$} as a bounded operator on L²(X, dx).Using theorems of Weyl (Rend. Circ. Mat. Palermo, 27 (1909), 373–392) and Kato (“Perturbation Theory for Linear Operators,” Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular Spec_ESSQ = range(q(x, ξ)) where q is the principal symbol of Q. Turning the attention to the spectral family {E_S}_{S∈$\text{R}$}, it is shown that if $\text{dE}\text{ds}$ is considered as a distribution on $\text{R}$×X×X it is in fact a Lagrangian distribution near the set $\text{{σ=0}?}\text{T}{}^1\text{(}\text{R}\text{×X×X)}\text{0}$ where (s, x, y, σ, ξ,η) are coordinates on T¹($\text{R}$×X×X) induced by the coordinates (s, x, y) on $\text{R}$×X×X. This leads to an easy proof that$\text{?(Q)}$ is a pseudodifferential operator if ?∈C^∞($\text{R}$) and to some results on the microlocal character of E_s. Finally, a look at the wavefront set of $\text{dE}\text{ds}$ leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q(x, ξ).     

Classifying Hilbert bundles

A Hilbert bundle (p, B, X) is a type of fibre space p:B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, G_m($\text{C}$ⁿ)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.     

On a sequence of integral operators on weighted L p spaces

Continuing some investigations started in previous papers, we introduce and study a sequence of multidimensional positive integral operators which generalize the Gauss-Weierstrass operators. We show that this sequence is an approximation process in some classes of weighted L ^p spaces on ? ^N , N ≥ 1. Estimates of the rate of convergence are also obtained. Our mean tool is a Korovkin-type theorem which we establish in the context of L ^p (X, µ) spaces, X being a locally compact Hausdorff space and µ a regular positive Borel measure on X. Several examples are explicitly indicated as well.     

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?ⁿ. We prove interpolation theorems for the subadditive operators of weak type (?₀, ?₀) bounded in L _∞(?ⁿ) and subadditive operators of weak types (?₀, ?₀) and (?₁, ?₁) in L _?(? ⁿ ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?₀, ?₀) bounded from L _∞(?ⁿ) to BMO(? ⁿ). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.     

Asymptotic confidence intervals for Poisson regression

Let (X,Y) be a R^d×N₀-valued random vector where the conditional distribution of Y given X=x is a Poisson distribution with mean m(x). We estimate m by a local polynomial kernel estimate defined by maximizing a localized log-likelihood function. We use this estimate of m(x) to estimate the conditional distribution of Y given X=x by a corresponding Poisson distribution and to construct confidence intervals of level α of Y given X=x. Under mild regularity conditions on m(x) and on the distribution of X we show strong convergence of the integrated L₁ distance between Poisson distribution and its estimate. We also demonstrate that the corresponding confidence interval has asymptotically (i.e., for sample size tending to infinity) level α, and that the probability that the length of this confidence interval deviates from the optimal length by more than one converges to zero with the number of samples tending to infinity.