scholarly journals Vector-valued multiparameter singular integrals and pseudodifferential operators

2008 ◽  
Vol 217 (2) ◽  
pp. 519-536 ◽  
Author(s):  
Tuomas Hytönen ◽  
Pierre Portal
Keyword(s):  
Singular Integrals ◽  
Pseudodifferential Operators ◽  
Vector Valued

scholarly journals A note on singular integrals with angular integrability

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Function Spaces ◽  
Riesz Transforms ◽  
Hilbert Transforms ◽  
Rotation Method ◽  
Vector Valued Function ◽  
Vector Valued ◽  
L Log L

Abstract In this note we study the rough singular integral $$ T_{\varOmega }f(x)=\mathrm{p.v.} \int _{\mathbb{R}^{n}}f(x-y)\frac{\varOmega (y/ \vert y \vert )}{ \vert y \vert ^{n}}\,dy, $$ T Ω f ( x ) = p . v . ∫ R n f ( x − y ) Ω ( y / | y | ) | y | n d y , where $n\geq 2$ n ≥ 2 and Ω is a function in $L\log L(\mathrm{S} ^{n-1})$ L log L ( S n − 1 ) with vanishing integral. We prove that $T_{\varOmega }$ T Ω is bounded on the mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}( \mathbb{R}^{n})$ L | x | p L θ p ˜ ( R n ) , on the vector-valued mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}(\mathbb{R}^{n},\ell ^{\tilde{p}})$ L | x | p L θ p ˜ ( R n , ℓ p ˜ ) and on the vector-valued function spaces $L^{p}(\mathbb{R}^{n}, \ell ^{\tilde{p}})$ L p ( R n , ℓ p ˜ ) if $1<\tilde{p}\leq p<\tilde{p}n/(n-1)$ 1 < p ˜ ≤ p < p ˜ n / ( n − 1 ) or $\tilde{p}n/(\tilde{p}+n-1)< p\leq \tilde{p}<\infty $ p ˜ n / ( p ˜ + n − 1 ) < p ≤ p ˜ < ∞ . The same conclusions hold for the well-known Riesz transforms and directional Hilbert transforms. It should be pointed out that our proof is based on the Calderón–Zygmund’s rotation method.

scholarly journals THE LOCAL NON-HOMOGENEOUS Tb THEOREM FOR VECTOR-VALUED FUNCTIONS

2014 ◽  
Vol 57 (1) ◽  
pp. 17-82 ◽  
Author(s):  
TUOMAS P. HYTÖNEN ◽  
ANTTI V. VÄHÄKANGAS
Keyword(s):  
Singular Integrals ◽  
Local Version ◽  
Square Functions ◽  
Property A ◽  
Martingale Differences ◽  
Umd Spaces ◽  
Function Lattice ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Similar Extension

AbstractWe extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, ‘vector-valued’ means ‘taking values in a function lattice with the UMD (unconditional martingale differences) property’. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local version requires several elaborations of the previous techniques, and raises new questions about the limits of the vector-valued theory.

SINGULAR INTEGRALS SUPPORTED BY SUBVARIETIES FOR VECTOR-VALUED FUNCTIONS

2017 ◽  
Vol 231 ◽  
pp. 101-114
Author(s):  
HONGHAI LIU
Keyword(s):  
Singular Integrals ◽  
Operator Space ◽  
Space Theory ◽  
Umd Space ◽  
Vector Valued Functions ◽  
Vector Valued

In this paper, we show that singular integrals supported by subvarieties are bounded on $L^{p}(\mathbb{R}^{n};\mathbf{X})$ for $1<p<\infty$ and some UMD space $\mathbf{X}$. In the terminology from operator space theory, we prove that singular integrals supported by subvarieties are completely $L^{p}$-bounded.

scholarly journals vector-valued singular integrals and maximal functions on spaces of homogeneous type

2009 ◽  
Vol 104 (2) ◽  
pp. 296 ◽  
Author(s):  
Loukas Grafakos ◽  
Liguang Liu ◽  
Dachun Yang
Keyword(s):  
Singular Integral ◽  
Maximal Function ◽  
Singular Integrals ◽  
Maximal Functions ◽  
Integral Theory ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Vector Valued

The Fefferman-Stein vector-valued maximal function inequality is proved for spaces of homogeneous type. The approach taken here is based on the theory of vector-valued Calderón-Zygmund singular integral theory in this context, which is appropriately developed.

Vector-Valued Modular Forms of Weight Two Associated With Jacobi-Like Forms

2006 ◽  
Vol 49 (3) ◽  
pp. 428-437 ◽  
Author(s):  
Min Ho Lee
Keyword(s):  
Modular Forms ◽  
Pseudodifferential Operators ◽  
Theta Functions ◽  
Symmetric Tensor ◽  
Tensor Representation ◽  
Vector Valued

AbstractWe construct vector-valued modular forms of weight 2 associated to Jacobi-like forms with respect to a symmetric tensor representation of Γ by using the method of Kuga and Shimura as well as the correspondence between Jacobi-like forms and sequences of modular forms. As an application, we obtain vector-valued modular forms determined by theta functions and by pseudodifferential operators.

Multi-parameter Singular Integrals I: Examples

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Pseudodifferential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Special Cases ◽  
Stratified Lie Groups ◽  
Left And Right ◽  
Flag Kernels

This chapter discusses a few special cases where a theory of multi-parameter singular integral operators has already been developed. These include the product theory of singular integrals, convolution with flag kernels on graded groups, convolution with both the left and right invariant Calderón–Zygmund singular integral operators on stratified Lie groups, and composition of standard pseudodifferential operators with certain singular integrals corresponding to non-Euclidean geometries. The chapter outlines these examples and their applications and relates them to the trichotomy discussed in Chapter 1.

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