Vector-Valued Singular Integrals Revisited–-with Random Dyadic Cubes

2012 ◽  
Vol 60 (3) ◽  
pp. 269-283 ◽  
Author(s):  
Tuomas P. Hytönen
Keyword(s):  
Singular Integrals ◽  
Dyadic Cubes ◽  
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.

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