# American Institute of Mathematical Sciences

June  2018, 11(3): 493-509. doi: 10.3934/dcdss.2018027

## Some remarks on boundary operators of Bessel extensions

 1 Department of Statistics, University of Auckland, Private Bag 92019, Victoria Street West, Auckland 1142, New Zealand 2 Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 3 National Center for Theoretical Sciences, National Taiwan University, No. 1 Sec. 4 Roosevelt Rd, Taipei, 106, Taiwan

* Corresponding author: dspector@math.nctu.edu.tw.

Received  May 2017 Revised  August 2017 Published  October 2017

Fund Project: The first author is supported in part by the Marsden Fund Council from New Zealand Government funding, managed by the Royal Society of New Zealand. The second author is supported in part by the Taiwan Ministry of Science and Technology under research grants 103-2115-M-009-016-MY2 and 105-2115-M-009-004-MY2.

In this paper we study some boundary operators of a class of Bessel-type Littlewood-Paley extensions whose prototype is
 \begin{align*}Δ_x u(x, y) +\frac{1-2s}{y} \frac{\partial u}{\partial y}(x, y)+\frac{\partial^2 u}{\partial y^2}(x, y)&=0 &&\text{for }x∈\mathbb{R}^d, y>0, \\ u(x, 0)&=f(x) &&\text{for }x∈\mathbb{R}^d.\end{align*}
In particular, we show that with a logarithmic scaling one can capture the failure of analyticity of these extensions in the limiting cases
 $s=k ∈ \mathbb{N}$
.
Citation: Jesse Goodman, Daniel Spector. Some remarks on boundary operators of Bessel extensions. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 493-509. doi: 10.3934/dcdss.2018027
##### References:
 [1] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar [2] S.-Y. A. Chang and M. d. M. González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016. Google Scholar [3] D. DeBlassie, The first exit time of a two-dimensional symmetric stable process from a wedge, Ann. Probab., 18 (1990), 1034-1070. doi: 10.1214/aop/1176990735. Google Scholar [4] I. S. Gradshteyn and M. Ryzhik, Table of Integrals, Series and Products 7$^{th}$ edition, Academic Press, 2007.Google Scholar [5] P. Kim, R. Song and Z. Vondraček, On harmonic functions for trace processes, Math. Nachr., 284 (2011), 1889-1902. doi: 10.1002/mana.200910008. Google Scholar [6] M. Marias, Littlewood-Paley-Stein theory and Bessel diffusions, Bull. Sci. Math. (2), 111 (1987), 313-331. Google Scholar [7] S. A. Molčanov and E. Ostrovskiǐ, Symmetric stable processes as traces of degenerate diffusion processes, Teor. Verojatnost. i Primenen., 14 (1969), 127-130. Google Scholar [8] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations Texts in Applied Mathematics, Springer-Verlag, New York, 2004. Google Scholar [9] L. Roncal and P. R. Stinga, Fractional Laplacian on the torus Commun. Contemp. Math. 18 (2016), 1550033, 26pp. doi: 10.1142/S0219199715500339. Google Scholar [10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J. , 1970. Google Scholar [11] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680. Google Scholar [12] R. Yang, On higher order extensions for the fractional Laplacian, preprint, arXiv: 1302.4413.Google Scholar [13] K. Yoshida, Functional Analysis Classics in Mathematics, Reprint of the 6$^{th}$ edition, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-61859-8. Google Scholar

show all references

##### References:
 [1] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar [2] S.-Y. A. Chang and M. d. M. González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016. Google Scholar [3] D. DeBlassie, The first exit time of a two-dimensional symmetric stable process from a wedge, Ann. Probab., 18 (1990), 1034-1070. doi: 10.1214/aop/1176990735. Google Scholar [4] I. S. Gradshteyn and M. Ryzhik, Table of Integrals, Series and Products 7$^{th}$ edition, Academic Press, 2007.Google Scholar [5] P. Kim, R. Song and Z. Vondraček, On harmonic functions for trace processes, Math. Nachr., 284 (2011), 1889-1902. doi: 10.1002/mana.200910008. Google Scholar [6] M. Marias, Littlewood-Paley-Stein theory and Bessel diffusions, Bull. Sci. Math. (2), 111 (1987), 313-331. Google Scholar [7] S. A. Molčanov and E. Ostrovskiǐ, Symmetric stable processes as traces of degenerate diffusion processes, Teor. Verojatnost. i Primenen., 14 (1969), 127-130. Google Scholar [8] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations Texts in Applied Mathematics, Springer-Verlag, New York, 2004. Google Scholar [9] L. Roncal and P. R. Stinga, Fractional Laplacian on the torus Commun. Contemp. Math. 18 (2016), 1550033, 26pp. doi: 10.1142/S0219199715500339. Google Scholar [10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J. , 1970. Google Scholar [11] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680. Google Scholar [12] R. Yang, On higher order extensions for the fractional Laplacian, preprint, arXiv: 1302.4413.Google Scholar [13] K. Yoshida, Functional Analysis Classics in Mathematics, Reprint of the 6$^{th}$ edition, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-61859-8. Google Scholar
 [1] Radjesvarane Alexandre, Mouhamad Elsafadi. Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations II. Non cutoff case and non Maxwellian molecules. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 1-11. doi: 10.3934/dcds.2009.24.1 [2] Vitali Milman, Liran Rotem. $\alpha$-concave functions and a functional extension of mixed volumes. Electronic Research Announcements, 2013, 20: 1-11. doi: 10.3934/era.2013.20.1 [3] M.T. Boudjelkha. Extended Riemann Bessel functions. Conference Publications, 2005, 2005 (Special) : 121-130. doi: 10.3934/proc.2005.2005.121 [4] Fabrizio Colombo, Graziano Gentili, Irene Sabadini and Daniele C. Struppa. A functional calculus in a noncommutative setting. Electronic Research Announcements, 2007, 14: 60-68. doi: 10.3934/era.2007.14.60 [5] Vladimir V. Kisil. Mobius transformations and monogenic functional calculus. Electronic Research Announcements, 1996, 2: 26-33. [6] Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905 [7] Hassan Emamirad, Arnaud Rougirel. A functional calculus approach for the rational approximation with nonuniform partitions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 955-972. doi: 10.3934/dcds.2008.22.955 [8] Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447 [9] Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 [10] Nicola Abatangelo. Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5555-5607. doi: 10.3934/dcds.2015.35.5555 [11] Nicolas Lerner, Yoshinori Morimoto, Karel Pravda-Starov, Chao-Jiang Xu. Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators. Kinetic & Related Models, 2013, 6 (3) : 625-648. doi: 10.3934/krm.2013.6.625 [12] Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760 [13] Mehar Chand, Jyotindra C. Prajapati, Ebenezer Bonyah, Jatinder Kumar Bansal. Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 539-560. doi: 10.3934/dcdss.2020030 [14] Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157 [15] Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171 [16] Matthias Geissert, Horst Heck, Christof Trunk. $H^{\infty}$-calculus for a system of Laplace operators with mixed order boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1259-1275. doi: 10.3934/dcdss.2013.6.1259 [17] Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48 [18] Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101 [19] Marta García-Huidobro, Raul Manásevich. A three point boundary value problem containing the operator. Conference Publications, 2003, 2003 (Special) : 313-319. doi: 10.3934/proc.2003.2003.313 [20] Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, Jari P. Kaipio, Erkki Somersalo. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map. Inverse Problems & Imaging, 2015, 9 (3) : 767-789. doi: 10.3934/ipi.2015.9.767

2018 Impact Factor: 0.545