scholarly journals Higher-Order Riesz Transforms in the Inverse Gaussian Setting and UMD Banach Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-28
Author(s):  
Jorge J. Betancor ◽  
Lourdes Rodríguez-Mesa
Keyword(s):  
Banach Spaces ◽  
Gaussian Measure ◽  
Singular Integrals ◽  
Higher Order ◽  
Riesz Transforms ◽  
Inverse Gaussian ◽  
Imaginary Powers ◽  
Umd Banach Spaces

In this paper, we study higher-order Riesz transforms associated with the inverse Gaussian measure given by π n / 2 e x 2 d x on ℝ n . We establish L p ℝ n , e x 2 d x -boundedness properties and obtain representations as principal values singular integrals for the higher-order Riesz transforms. New characterizations of the Banach spaces having the UMD property by means of the Riesz transforms and imaginary powers of the operator involved in the inverse Gaussian setting are given.

scholarly journals Characterization of UMD Banach Spaces by Imaginary Powers of Hermite and Laguerre Operators

2011 ◽  
Vol 7 (4) ◽  
pp. 1019-1048 ◽  
Author(s):  
Jorge J. Betancor ◽  
Alejandro J. Castro ◽  
Jezabel Curbelo ◽  
Lourdes Rodríguez-Mesa
Keyword(s):  
Banach Spaces ◽  
Imaginary Powers ◽  
Umd Banach Spaces

scholarly journals Multipliers and imaginary powers of the Schrödinger operators characterizing UMD Banach spaces

2013 ◽  
Vol 38 ◽  
pp. 209-227 ◽  
Author(s):  
Jorge J. Betancor ◽  
Raquel Crescimbeni ◽  
Juan C. Fariña ◽  
Lourdes Rodríguez-Mesa
Keyword(s):  
Banach Spaces ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Imaginary Powers ◽  
Umd Banach Spaces

scholarly journals Variation and oscillation for harmonic operators in the inverse Gaussian setting

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Víctor Almeida ◽  
Jorge J. Betancor
Keyword(s):  
Banach Spaces ◽  
Fractional Derivatives ◽  
Riesz Transforms ◽  
Semigroups Of Operators ◽  
Inverse Gaussian ◽  
The Family ◽  
Derivatives Of ◽  
Martingale Cotype

<p style='text-indent:20px;'>We prove variation and oscillation <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-inequalities associated with fractional derivatives of certain semigroups of operators and with the family of truncations of Riesz transforms in the inverse Gaussian setting. We also study these variational <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-inequalities in a Banach-valued context by considering Banach spaces with the UMD-property and whose martingale cotype is fewer than the variational exponent. We establish <inline-formula><tex-math id="M3">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-boundedness properties for weighted difference involving the semigroups under consideration.</p>

scholarly journals Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaces

2021 ◽  
Vol 11 (1) ◽  
pp. 72-95
Author(s):  
Xiao Zhang ◽  
Feng Liu ◽  
Huiyun Zhang
Keyword(s):  
Hilbert Transform ◽  
Besov Spaces ◽  
Singular Integrals ◽  
Riesz Transform ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Variation Operators

Abstract This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ Lq (S n−1) (q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.

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 Characterization of Banach valued BMO functions and UMD Banach spaces by using Bessel convolutions

Positivity ◽  
2012 ◽  
Vol 17 (3) ◽  
pp. 535-587
Author(s):  
Jorge J. Betancor ◽  
Alejandro J. Castro ◽  
Lourdes Rodríguez-Mesa
Keyword(s):  
Banach Spaces ◽  
Bmo Functions ◽  
Umd Banach Spaces

scholarly journals Stochastic integration in UMD Banach spaces

2007 ◽  
Vol 35 (4) ◽  
pp. 1438-1478 ◽  
Author(s):  
J. M. A. M. van Neerven ◽  
M. C. Veraar ◽  
L. Weis
Keyword(s):  
Banach Spaces ◽  
Stochastic Integration ◽  
Umd Banach Spaces
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