scholarly journals Operator-valued Fourier multipliers in vector-valued function spaces and application

2020 ◽  
Vol 4 (4) ◽  
pp. 153-174
Author(s):  
Veli B. Shakhmurov
Keyword(s):  
Function Spaces ◽  
Fourier Multipliers ◽  
Vector Valued Function ◽  
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 POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

2011 ◽  
Vol 84 (1) ◽  
pp. 44-48 ◽  
Author(s):  
MICHAEL G. COWLING ◽  
MICHAEL LEINERT
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Semigroup Of Operators ◽  
Almost Everywhere ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

AbstractA submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

scholarly journals Weighted Estimates for Rough Maximal Functions with Applications to Angular Integrability

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Feng Liu ◽  
Fangfang Xu
Keyword(s):  
Function Spaces ◽  
Maximal Functions ◽  
Rough Kernels ◽  
Weighted Estimates ◽  
Polynomial Curves ◽  
Definition Of ◽  
Vector Valued Function ◽  
Vector Valued

In this note we establish certain weighted estimates for a class of maximal functions with rough kernels along “polynomial curves” on Rn. As applications, we obtain the bounds of the above operators on the mixed radial-angular spaces, on the vector-valued mixed radial-angular spaces, and on the vector-valued function spaces. Particularly, the above bounds are independent of the coefficients of the polynomials in the definition of the operators.

scholarly journals Functional decompositions on vector-valued function spaces via operators

2012 ◽  
Vol 389 (2) ◽  
pp. 1173-1190 ◽  
Author(s):  
Titarii Wootijirattikal ◽  
Sing-Cheong Ong ◽  
Jitti Rakbud
Keyword(s):  
Function Spaces ◽  
Vector Valued Function ◽  
Vector Valued

LINEAR SURJECTIVE ISOMETRIES BETWEEN VECTOR-VALUED FUNCTION SPACES

2016 ◽  
Vol 100 (3) ◽  
pp. 349-373 ◽  
Author(s):  
KAZUHIRO KAWAMURA
Keyword(s):  
Continuous Function ◽  
Extreme Point ◽  
Function Spaces ◽  
Point Method ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Stone Type ◽  
Linear Isometries

We prove some Banach–Stone type theorems for linear isometries of vector-valued continuous function spaces, by making use of the extreme point method.

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