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.

[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.

[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.

[4]

I. S. Gradshteyn and M. Ryzhik, Table of Integrals, Series and Products 7$^{th}$ edition, Academic Press, 2007.

[5]

P. KimR. Song and Z. Vondraček, On harmonic functions for trace processes, Math. Nachr., 284 (2011), 1889-1902. doi: 10.1002/mana.200910008.

[6]

M. Marias, Littlewood-Paley-Stein theory and Bessel diffusions, Bull. Sci. Math. (2), 111 (1987), 313-331.

[7]

S. A. Molčanov and E. Ostrovskiǐ, Symmetric stable processes as traces of degenerate diffusion processes, Teor. Verojatnost. i Primenen., 14 (1969), 127-130.

[8]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations Texts in Applied Mathematics, Springer-Verlag, New York, 2004.

[9]

L. Roncal and P. R. Stinga, Fractional Laplacian on the torus Commun. Contemp. Math. 18 (2016), 1550033, 26pp. doi: 10.1142/S0219199715500339.

[10]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J. , 1970.

[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.

[12]

R. Yang, On higher order extensions for the fractional Laplacian, preprint, arXiv: 1302.4413.

[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.

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.

[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.

[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.

[4]

I. S. Gradshteyn and M. Ryzhik, Table of Integrals, Series and Products 7$^{th}$ edition, Academic Press, 2007.

[5]

P. KimR. Song and Z. Vondraček, On harmonic functions for trace processes, Math. Nachr., 284 (2011), 1889-1902. doi: 10.1002/mana.200910008.

[6]

M. Marias, Littlewood-Paley-Stein theory and Bessel diffusions, Bull. Sci. Math. (2), 111 (1987), 313-331.

[7]

S. A. Molčanov and E. Ostrovskiǐ, Symmetric stable processes as traces of degenerate diffusion processes, Teor. Verojatnost. i Primenen., 14 (1969), 127-130.

[8]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations Texts in Applied Mathematics, Springer-Verlag, New York, 2004.

[9]

L. Roncal and P. R. Stinga, Fractional Laplacian on the torus Commun. Contemp. Math. 18 (2016), 1550033, 26pp. doi: 10.1142/S0219199715500339.

[10]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J. , 1970.

[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.

[12]

R. Yang, On higher order extensions for the fractional Laplacian, preprint, arXiv: 1302.4413.

[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.

[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]

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

[6]

Vladimir V. Kisil. Mobius transformations and monogenic functional calculus. Electronic Research Announcements, 1996, 2: 26-33.

[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]

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

[9]

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

[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]

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

[14]

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

[15]

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

[16]

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

[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]

Uri Shapira. On a generalization of Littlewood's conjecture. Journal of Modern Dynamics, 2009, 3 (3) : 457-477. doi: 10.3934/jmd.2009.3.457

[20]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295

2016 Impact Factor: 0.781

Metrics

  • PDF downloads (4)
  • HTML views (11)
  • Cited by (0)

Other articles
by authors

[Back to Top]