American Institute of Mathematical Sciences

Advanced
June  2013, 6(3): 619-635. doi: 10.3934/dcdss.2013.6.619

Product structures and fractional integration along curves in the space

Valentina Casarino 1, , Paolo Ciatti 2, and  Silvia Secco 2,

1. 

DTG, Università degli Studi di Padova, Stradella San Nicola 3, 36100 Vicenza, Italy
2. 

DICEA, Università degli Studi di Padova, Via Marzolo 9, 35131 Padova, Italy, Italy

Received  March 2010 Revised  February 2012 Published  December 2012

In this paper we establish $L^p$ boundedness ($1 < p < \infty$) for a double analytic family of fractional integrals $S^{\gamma}_{z}$, $\gamma,z ∈\mathbb{C}$, when $\Re e z=0$. Our proof is based on product-type kernels arguments. More precisely, we prove that the convolution kernel of $S^{\gamma}_{z}$ is a product kernel on $\mathbb{R}^3$, adapted to the polynomial curve $x_1\mapsto (x_1^m,x_1^n)$ (here $m,n∈\mathbb{N},m ≥ 1, n > m $).
Keywords: fractional integration along curves in the space, Product kernels, strong $L^p$ bounds..
Mathematics Subject Classification: Primary: 42B20, 26A33; Secondary: 44A3.
Citation: Valentina Casarino, Paolo Ciatti, Silvia Secco. Product structures and fractional integration along curves in the space. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 619-635. doi: 10.3934/dcdss.2013.6.619
References:
[1]

V. Casarino, P. Ciatti and S. Secco, Product kernels adapted to curves in the space, Revista Matematica Iberoamericana, 27 (2011), 1023-1057. doi: 10.4171/RMI/662.  Google Scholar

[2]

V. Casarino and S. Secco, $L^p-L^q$ boundedness of analytic families of fractional integrals, Studia Mathematica, 184 (2008), 153-174. doi: 10.4064/sm184-2-5.  Google Scholar

[3]

R. Fefferman and E. M. Stein, Singular integrals on product spaces, Adv. in Math., 45 (1982), 117-143. doi: 10.1016/S0001-8708(82)80001-7.  Google Scholar

[4]

G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups," Mathematical Notes, Princeton University Press, 1982.  Google Scholar

[5]

L. Grafakos, Strong type endpoint bounds for analytic families of fractional integrals, Proc. Amer. Math. Soc., 117 (1993), 653-663 doi: 10.2307/2159123.  Google Scholar

[6]

M. Kashiwara, B-functions and holonomic systems,, Invent. Math., 38 (): 33.   Google Scholar

[7]

D. Müller, F. Ricci and E. M. Stein, Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups I., Invent. Math., 119 (1995), 119-233. doi: 10.1007/BF01245180.  Google Scholar

[8]

A. Nagel, F. Ricci and E. M. Stein, Singular integrals with flag kernels and analysis on quadratic CR manifolds, J. Funct. Anal., 181 (2001), 29-118. doi: 10.1006/jfan.2000.3714.  Google Scholar

[9]

A. Nagel and E. M. Stein, On the product theory of singular integrals, Rev. Mat. Iberoamericana, 20 (2004), 531-561. doi: 10.4171/RMI/400.  Google Scholar

[10]

A. Nagel and E. M. Stein, The $\overline{\partial}_b$-complex on decoupled boundaries in $\mathbbC^n$, Ann. of Math. (2), 164 (2006), 649-713. doi: 10.4007/annals.2006.164.649.  Google Scholar

[11]

E. M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc., 84 (1978), 1239-1295. doi: 10.1090/S0002-9904-1978-14554-6.  Google Scholar

show all references

References:
[1]

V. Casarino, P. Ciatti and S. Secco, Product kernels adapted to curves in the space, Revista Matematica Iberoamericana, 27 (2011), 1023-1057. doi: 10.4171/RMI/662.  Google Scholar

[2]

V. Casarino and S. Secco, $L^p-L^q$ boundedness of analytic families of fractional integrals, Studia Mathematica, 184 (2008), 153-174. doi: 10.4064/sm184-2-5.  Google Scholar

[3]

R. Fefferman and E. M. Stein, Singular integrals on product spaces, Adv. in Math., 45 (1982), 117-143. doi: 10.1016/S0001-8708(82)80001-7.  Google Scholar

[4]

G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups," Mathematical Notes, Princeton University Press, 1982.  Google Scholar

[5]

L. Grafakos, Strong type endpoint bounds for analytic families of fractional integrals, Proc. Amer. Math. Soc., 117 (1993), 653-663 doi: 10.2307/2159123.  Google Scholar

[6]

M. Kashiwara, B-functions and holonomic systems,, Invent. Math., 38 (): 33.   Google Scholar

[7]

D. Müller, F. Ricci and E. M. Stein, Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups I., Invent. Math., 119 (1995), 119-233. doi: 10.1007/BF01245180.  Google Scholar

[8]

A. Nagel, F. Ricci and E. M. Stein, Singular integrals with flag kernels and analysis on quadratic CR manifolds, J. Funct. Anal., 181 (2001), 29-118. doi: 10.1006/jfan.2000.3714.  Google Scholar

[9]

A. Nagel and E. M. Stein, On the product theory of singular integrals, Rev. Mat. Iberoamericana, 20 (2004), 531-561. doi: 10.4171/RMI/400.  Google Scholar

[10]

A. Nagel and E. M. Stein, The $\overline{\partial}_b$-complex on decoupled boundaries in $\mathbbC^n$, Ann. of Math. (2), 164 (2006), 649-713. doi: 10.4007/annals.2006.164.649.  Google Scholar

[11]

E. M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc., 84 (1978), 1239-1295. doi: 10.1090/S0002-9904-1978-14554-6.  Google Scholar

[1]

Thabet Abdeljawad. Fractional operators with boundary points dependent kernels and integration by parts. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 351-375. doi: 10.3934/dcdss.2020020
[2]

Zehui Shao, Huiqin Jiang, Aleksander Vesel. L(2, 1)-labeling of the Cartesian and strong product of two directed cycles. Mathematical Foundations of Computing, 2018, 1 (1) : 49-61. doi: 10.3934/mfc.2018003
[3]

Der-Chen Chang, Jie Xiao. $L^q$-Extensions of $L^p$-spaces by fractional diffusion equations. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1905-1920. doi: 10.3934/dcds.2015.35.1905
[4]

Ling Lin, Dong He, Zhiyi Tan. Bounds on delay start LPT algorithm for scheduling on two identical machines in the $l_p$ norm. Journal of Industrial & Management Optimization, 2008, 4 (4) : 817-826. doi: 10.3934/jimo.2008.4.817
[5]

Martin Fraas, David Krejčiřík, Yehuda Pinchover. On some strong ratio limit theorems for heat kernels. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 495-509. doi: 10.3934/dcds.2010.28.495
[6]

Haixia Yu. Hilbert transforms along double variable fractional monomials. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1433-1446. doi: 10.3934/cpaa.2019069
[7]

Seung-Yeal Ha, Ho Lee, Seok Bae Yun. Uniform $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with external forces. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 115-143. doi: 10.3934/dcds.2009.24.115
[8]

Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761
[9]

Hongyong Cui, Peter E. Kloeden, Wenqiang Zhao. Strong $ (L^2,L^\gamma\cap H_0^1) $-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension. Electronic Research Archive, 2020, 28 (3) : 1357-1374. doi: 10.3934/era.2020072
[10]

T. Diogo, N. B. Franco, P. Lima. High order product integration methods for a Volterra integral equation with logarithmic singular kernel. Communications on Pure & Applied Analysis, 2004, 3 (2) : 217-235. doi: 10.3934/cpaa.2004.3.217
[11]

Martins Bruveris. Completeness properties of Sobolev metrics on the space of curves. Journal of Geometric Mechanics, 2015, 7 (2) : 125-150. doi: 10.3934/jgm.2015.7.125
[12]

Xavier Ros-Oton, Joaquim Serra. Local integration by parts and Pohozaev identities for higher order fractional Laplacians. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2131-2150. doi: 10.3934/dcds.2015.35.2131
[13]

Kevin Zumbrun. L∞ resolvent bounds for steady Boltzmann's Equation. Kinetic & Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048
[14]

Ravi Vakil and Aleksey Zinger. A natural smooth compactification of the space of elliptic curves in projective space. Electronic Research Announcements, 2007, 13: 53-59.

[15]

Fabio Cipriani, Gabriele Grillo. On the $l^p$ -agmon's theory. Conference Publications, 1998, 1998 (Special) : 167-176. doi: 10.3934/proc.1998.1998.167
[16]

Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175
[17]

Hua Chen, Hong-Ge Chen. Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021117
[18]

Piotr Kokocki. Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2315-2334. doi: 10.3934/cpaa.2015.14.2315
[19]

Samer Dweik. $ L^{p, q} $ estimates on the transport density. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3001-3009. doi: 10.3934/cpaa.2019134
[20]

Karina Samvelyan, Frol Zapolsky. Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces. Electronic Research Announcements, 2017, 24: 28-37. doi: 10.3934/era.2017.24.004

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences