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1.

Old and New Morrey Spaces with Heat Kernel Bounds

**总被引：1，自引：0，他引：1**Given p ∈ [1,∞) and λ ∈ (0, n), we study Morrey space
of all locally integrable complex-valued functions f on
such that for every open Euclidean ball B ⊂
with radius r

and derive old and new, two essentially different cases arising from either choosing
or replacing c by
—where t

_{B}there are numbers C = C(f ) (depending on f ) and c = c(f,B) (relying upon f and B) satisfying_{B}is scaled to r_{B}and p_{t}(·, ·) is the kernel of the infinitesimal generator L of an analytic semigroup on Consequently, we are led to simultaneously characterize the old and new Morrey spaces, but also to show that for a suitable operator L, the new Morrey space is equivalent to the old one. 相似文献2.

Let be the infinitesimal generator of an analytic semigroup on with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space by means of an area integral function associated with the operator . By using a variant of the maximal function associated with the semigroup , a space of functions of BMO type was defined by Duong and Yan and it generalizes the classical BMO space. In this paper, we show that if has a bounded holomorphic functional calculus on , then the dual space of is where is the adjoint operator of . We then obtain a characterization of the space in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces of BMO when is a second-order elliptic operator of divergence form and when is a Schrödinger operator, and study the inclusion between the classical BMO space and spaces associated with operators.

3.

On commutators of fractional integrals

**总被引：1，自引：0，他引：1** Xuan Thinh Duong Li Xin Yan 《Proceedings of the American Mathematical Society》2004,132(12):3549-3557

Let be the infinitesimal generator of an analytic semigroup on with Gaussian kernel bounds, and let be the fractional integrals of for . For a BMO function on , we show boundedness of the commutators from to , where . Our result of this boundedness still holds when is replaced by a Lipschitz domain of with infinite measure. We give applications to large classes of differential operators such as the magnetic Schrödinger operators and second-order elliptic operators of divergence form.

4.

Let be a space of homogeneous type, and be the generator of a semigroup with Gaussian kernel bounds on . We define the Hardy spaces of for a range of , by means of area integral function associated with the Poisson semigroup of , which is proved to coincide with the usual atomic Hardy spaces on spaces of homogeneous type.

5.

Thierry Coulhon Xuan Thinh Duong 《Transactions of the American Mathematical Society》1999,351(3):1151-1169

It has been asked (see R. Strichartz,

*Analysis of the Laplacian*, J. Funct. Anal.**52**(1983), 48-79) whether one could extend to a reasonable class of non-compact Riemannian manifolds the boundedness of the Riesz transforms that holds in . Several partial answers have been given since. In the present paper, we give positive results for under very weak assumptions, namely the doubling volume property and an optimal on-diagonal heat kernel estimate. In particular, we do not make any hypothesis on the space derivatives of the heat kernel. We also prove that the result cannot hold for under the same assumptions. Finally, we prove a similar result for the Riesz transforms on arbitrary domains of .6.

In this paper, we give a new characterization of the Morrey–Campanato spaces by using the convolution

_{t}_{B}**f*(*x*) to replace the minimizing polynomial*P*_{B}*f*of a function*f*in the Morrey-Campanato norm, where is an appropriate Schwartz function.D.G. Deng and L.X. Yan are partially supported by NSF of China and the Foundation of Advanced Research Center, Zhongshan University. X.T. Duong and L.X. Yan are supported by a Discovery grant from Australia Research Council. 相似文献7.

Xuan Thinh Duong 《Journal of Functional Analysis》2002,196(2):443-485

We study general spectral multiplier theorems for self-adjoint positive definite operators on

*L*^{2}(*X*,*μ*), where*X*is any open subset of a space of homogeneous type. We show that the sharp Hörmander-type spectral multiplier theorems follow from the appropriate estimates of the*L*^{2}norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. The sharp Hörmander-type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the Riesz means summability, which we also study here. We discuss several examples, which include sharp spectral multiplier theorems for a class of scattering operators on**R**^{3}and new spectral multiplier theorems for the Laguerre and Hermite expansions. 相似文献8.

Let $A=-(\nabla-i\vec{a})^2+VLet be a magnetic Schr?dinger operator acting on

*L*

^{2}(

**R**

^{ n }),

*n*≥1, where and 0≤

*V*∈

*L*

^{1}

_{loc}. Following [1], we define, by means of the area integral function, a Hardy space

*H*

^{1}

_{ A }associated with

*A*. We show that Riesz transforms (∂/∂

*x*

_{ k }-

*i*

*a*

_{ k })

*A*

^{-1/2}associated with

*A*, , are bounded from the Hardy space

*H*

^{1}

_{ A }into

*L*

^{1}. By interpolation, the Riesz transforms are bounded on

*L*

^{ p }for all 1<

*p*≤2. 相似文献

9.

Consider a second-order elliptic partial differential operator

*L*in divergence form with real, symmetric, bounded measurable coefficients, under Dirichlet or Neumann conditions on the boundary of a strongly Lipschitz domain Ω. Suppose that 1 <*p <*∞ and μ > 0. Then*L*has a bounded H_{∞}functional calculus in L^{p}(Ω), in the sense that ¦¦*f*(*L*+*cI*)u¦¦_{p}≤*C*sup_{¦arλ¦<μ}^{¦f¦}¦‖u¦‖_{p}for some constants*c*and*C*, and all bounded holomorphic functions*f*on the sector ¦ argλ¦ < μ that contains the spectrum of*L*+*cI.*We prove this by showing that the operators*f(L + cI)*are Calderón-Zygmund singular integral operators. 相似文献10.