\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Some remarks on boundary operators of Bessel extensions

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.
Abstract Full Text(HTML) Related Papers Cited by
  • 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 $s=k ∈ \mathbb{N}$ .

    Mathematics Subject Classification: Primary: 35J70; Secondary: 47D03, 33C10.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
  • 加载中
SHARE

Article Metrics

HTML views(1203) PDF downloads(183) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return