# 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 and 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. Kim, R. 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. Kim, R. 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.
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