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

Product structures and fractional integration along curves in the space

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 $).
Citation: Valentina Casarino, Paolo Ciatti, Silvia Secco. Product structures and fractional integration along curves in the space. Discrete and 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.

[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.

[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.

[4]

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

[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.

[6]

M. Kashiwara, B-functions and holonomic systems, Invent. Math., 38 (1976/77), 33-53.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[6]

M. Kashiwara, B-functions and holonomic systems, Invent. Math., 38 (1976/77), 33-53.

[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.

[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.

[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.

[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.

[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.

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