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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

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