## Mateusz Kwaśnicki

Department of Pure Mathematics
Wrocław University of Science and Technology
ul. Wybrzeże Wyspiańskiego 27
50-370 Wrocław, Poland

e-mail:
skype: mateusz.kwasnicki
ORCID: 0000-0003-3896-8124

# Research articles

For an author's copy of any of the following papers, please send me an email at .

## Preprint articles

• Boundary traces of shift-invariant diffusions in half-plane [arXiv]
• Characterisation of the class of bell-shaped functions with Thomas Simon [arXiv]
• Harmonic extension technique for non-symmetric operators with completely monotone kernels [arXiv]
• Random walks are determined by their trace on the positive half-line, Ann. Henri Lebesgue, in press [arXiv]

## Publications

1. A new class of bell-shaped functions, Trans. Amer. Math. Soc. 373(4) (2020): 2255–2280 [online]
2. Minimal Hermite-type eigenbasis of the discrete Fourier transform with Alexey Kuznetsov, J. Fourier Anal. Appl. 25(3) (2019): 1053–1079 [online|MR|zb]
3. Fluctuation theory for Lévy processes with completely monotone jumps, Electron. J. Probab. 24 (2019), no. 40 [online|MR|zb]
Note: (1) In the last lines of equations (5.8) and (5.10), $f^-(-i \xi_2)$ should read $f^-(i \xi_2)$. In the expression for $I$ in line 7 on page 26, $ds$ is missing. (2) In the displayed equation in line 14 on page 31, $\lambda(r)$ should read $\lambda_f(r)$. (3) The assumption that $f(\xi)$ is non-constant is missing in Lemma 5.4, Theorem 5.5 and Corollary 5.6. In Theorem 5.7 and Lemma 6.1, $f(\xi)$ should be assumed to be non-constant rather than non-zero. Finally, in the proof of Theorem 1.1 the trivial case of constant Rogers functions should be treated separately.
(supersedes: Rogers functions and fluctuation theory [arXiv])
4. On the $\ell^p$ norm of the discrete Hilbert transform with Rodrigo Bañuelos, Duke Math. J. 168(3) (2019): 471–504 [online|MR|zb]
5. Pólya’s conjecture fails for the fractional Laplacian with Richard S. Laugesen and Bartłomiej Siudeja, J. Spectral Theory 9(1) 2019: 127–135 [online|MR|zb]
6. Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter Reference, De Gruyter, Berlin, 2019 [online|MR|google]
7. Extension technique for complete Bernstein functions of the Laplace operator with Jacek Mucha, J. Evol. Equ. 18(3) (2018): 1341–1379 [online|MR|zb]
Note: (1) In Theorem A.2, when the string $A(ds)$ contains an atom of mass $b$ at $0$, the expression for $L f$ describes the operator $L = \psi(-\Delta) - b \Delta$, not $L = \psi(-\Delta)$. In order to get the latter one, it is necessary to artificially add $b \Delta$ in the expressions for $L f$ (and assume that $f$ is in the domain of $\Delta$). (2) In the last display on page 1376, as well as in the second line on page 1377, $(\psi(\lambda))^{-1}$ should read $\psi(\lambda)$; the correct definition is $\varphi_\lambda(s) = f_N(s) - \psi(\lambda) f_D(s)$. (3) In the penultimate line on page 1376, $\varphi_\lambda(0) = 0$ should read $\varphi_\lambda(0) = 1$. Thanks to Sigurd Assing for pointing out these errors!
8. Spectral analysis of stable processes on the positive half-line with Alexey Kuznetsov, Electron. J. Probab. 23 (2018), no. 10 [online|MR|zb]
9. Contractivity and ground state domination properties for non-local Schrödinger operators with Kamil Kaleta and József Lőrinczi, J. Spectr. Theory 8 (2018): 165–189 [online|MR|zb]
10. Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes with Tomasz Grzywny, Stoch. Proc. Appl. 128(1) (2018): 1–38 [online|MR|zb]
11. Martin kernels for Markov processes with jumps with Tomasz Juszczyszyn, Potential Anal. 47(3) (2017): 313–335 [online|MR|zb]
12. Eigenvalues of the fractional Laplace operator in the unit ball with Bartłomiej Dyda and Alexey Kuznetsov, J. London Math. Soc. 95 (2017): 500–518 [online|MR|zb]
13. Fractional Laplace operator and Meijer G-function with Bartłomiej Dyda and Alexey Kuznetsov, Constructive Approx. 45(3) (2017): 427–448 [online|MR|zb]
Note: In the list of explicit examples a reference to the result of Port is missing: he gave an explicit expression for the Poisson kernel of the complement of a sphere (or the complement of a hyperplane), see [S. C. Port, The First Hitting Distribution of a Sphere for Symmetric Stable Processes, Tran. Amer. Math. Soc. 133 (1969): 115–125]. In Proposition 1, reference should be given to Proposition 7.1, not 7.2.
14. Ten equivalent definitions of the fractional Laplace operator, Frac. Calc. Appl. Anal. 20(1) (2017): 7–51 [online|MR|zb]
Note: $p \in (1, \infty)$ should read $p \in (1, \tfrac{d}{\alpha})$ in page 22, line 20.
15. The shape of the fundamental sloshing mode in axisymmetric containers with Tadeusz Kulczycki and Bartłomiej Siudeja, J. Eng. Math. 99(1) (2015): 157–183 [online|MR|zb]
16. Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval with Kamil Kaleta and Jacek Małecki, J. Math. Anal. Appl. 439(2) (2016): 896–924 [online|MR|zb]
17. Hitting times of points for symmetric Lévy processes with completely monotone jumps with Tomasz Juszczyszyn, Electron. J. Probab. 20 (2015), no. 48 [online|MR|zb]
18. Boundary Harnack inequality for Markov processes with jumps with Krzysztof Bogdan and Takashi Kumagai, Trans. Amer. Math. Soc. 367(1) (2015): 477–517 [online|MR|zb]
19. New families of subordinators with explicit transition probability semigroup with James Burridge, Alexey Kuznetsov and Andreas Kyprianou, Stoch. Proc. Appl. 124(10) (2014): 3480–3495 [online|MR|zb]
20. The Legacy of Vladimir Andreevich Steklov with Nikolay Kuznetsov, Tadeusz Kulczycki, Alexander Nazarov, Sergey Poborchi, Iosif Polterovich and Bartłomiej Siudeja, Notices Amer. Math. Soc. 61(1) (2014): 9–23 [online|MR|zb]
21. One-dimensional quasi-relativistic particle in the box with Kamil Kaleta and Jacek Małecki, Rev. Math. Phys. 25(8) (2013) 1350014 [online|MR|zb]
Note: In the first displayed formula in Lemma~4.2 the norm in the left-hand side should not be squared.
22. Suprema of Lévy processes with Jacek Małecki and Michał Ryznar, Ann. Probab. 41(3B) (2013): 2047–2065 [online|MR|zb]
23. First passage times for subordinate Brownian motions with Jacek Małecki and Michał Ryznar, Stoch. Proc. Appl. 123 (2013): 1820–1850 [online|MR|zb]
Note: In Prop. 4.5 the last displayed formula should read \begin{aligned}\vartheta_\lambda & \ge \tfrac{1}{\pi} (\mathrm{arcsin}(...))^2 \\ & \qquad + \tfrac{1}{\pi} (\mathrm{arcsin}(...))^2 \\ & \qquad \qquad - \tfrac{1}{\pi} (\mathrm{arcsin}(...))^2\end{aligned} (factor $\tfrac{1}{\pi}$ is missing); in formula (7.2) a minus sign in the exponent is missing, the correct form is $\cdots \exp(-\tfrac{1}{\pi} \int_0^\infty \cdots ) \cdots$.
24. On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains with Tadeusz Kulczycki, Proc. London Math. Soc. 105(5) (2012): 921–952 [online|MR|zb]
25. Spectral theory for one-dimensional symmetric Lévy processes killed upon hitting the origin, Electron. J. Probab. 17 (2012), no. 83 [online|MR|zb]
Note: In Sect. 1 every occurrence of $2 \xi \Psi''(\xi) \le \Psi'(\xi)$ should be $\xi \Psi''(\xi) \le \Psi'(\xi)$ (5 times). In Example 5.1, in the last line on page 24, $\sin(\pi \alpha)$ and $\cos(\pi \alpha)$ should read $\sin(\pi \alpha/2)$ and $\cos(\pi \alpha/2)$.
26. Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal. 262(5) (2012): 2379–2402 [online|MR|zb]
Note: Columns 3, 4, 6 and 7 in Table 3 are wrong! Corrected values: 0.541, 0.586, 0.745, 1.110, 1.655, 2.103, 2.278 (column 3); 1.084, 1.174, 1.491, 2.222, 3.311, 4.207, 4.557 (column 4); 0.552, 0.610, 0.824, 1.360, 2.243, 3.029, 3.348 (column 6); 1.106, 1.222, 1.650, 2.721, 4.488, 6.060, 6.698 (column 7). Thanks to Juan Pablo Borthagaray for pointing this out!
27. Spectral analysis of subordinate Brownian motions on the half-line, Studia Math. 206(3) (2011): 211–271 [online|MR|zb]
Note: In displayed formula (4.14) $\frac{\lambda^2}{\lambda^2 + \xi^2}$ should be $\frac{\lambda}{\lambda^2 + \xi^2}$; in Example 6.1 in displayed formulae for $\gamma_\lambda(\xi)$ and $\gamma(s)$ a minus sign in the exponent is missing, the correct form is $\cdots \exp(-\tfrac{1}{\pi} \int_0^\infty \cdots ) \cdots$; in the displayed formula for $a(y)$ in page 267 the factor $c_\alpha^2$ should be replaced with $c_\alpha^{2/\alpha}$; the definition of $g_\lambda(y)$ in the following line should be $C_\alpha (c_\alpha^{-1} \lambda^{\alpha/2} y)^{1/2} K_{\alpha/2}((c_\alpha^{-1} \lambda^{\alpha/2} y)^{1/\alpha})$.
28. Boundary Harnack inequality for α -harmonic functions on the Sierpiński triangle with Kamil Kaleta, Probab. Math. Stat. 30(2) (2010): 353–368 [online|MR|zb]
29. Spectral properties of the Cauchy process on half-line and interval with Tadeusz Kulczycki, Jacek Małecki and Andrzej Stós, Proc. London Math. Soc. 101(2) (2010): 589–622 [online|MR|zb]
30. Eigenvalues of the Cauchy process on an interval have at most double multiplicity, Semigroup Forum 79(1) (2009): 183–192 [online|MR|zb]
31. Intrinsic ultracontractivity for stable semigroups on unbounded open sets, Potential Anal. 31(1) (2009): 57–77 [online|MR|zb]
32. Spectral gap estimate for stable processes on arbitrary bounded open sets, Probab. Math. Statist. 28(1) (2008): 163–167 [online|MR|zb]
33. Estimates and structure of $\alpha$-harmonic functions with Krzysztof Bogdan and Tadeusz Kulczycki, Prob. Theory Rel. Fields 140(3–4) (2008): 345–381 [online|MR|zb]
Note: The first displayed formula in p. 352 should read $w = r^{-2} (r^2 - |x|^2) (r^2 - |v|^2) / |x - v|^2$.

## Selected conference presentations

• Bell-shaped functions [pdf]
7th Lévy conference (Karlovasi, Greece, 2019)
• Discrete Hilbert transform [pdf]
Gordin Prize lecture (Vilnius, Lithuania, 2018)
• Fractional Laplace operator and operators with unimodal and isotropic kernels [pdf]
Young PDEers at work (Warsaw, Poland, 2018)
• Random walks are completely determined by their trace on the positive half-line [pdf]
Mexico-Poland 1st Meeting in Probability (Guanajuato, Mexico, 2017)
• Fractional Laplace operator in the unit ball [pdf|video]
Stable processes conference (Oaxaca, Mexico, 2016)
• Fractional Laplacian: explicit calculations and applications [pdf]
Nonlocal operators and PDEs (Będlewo, Poland, 2016)
• Rogers functions and fluctuation theory [pdf]
7th Conference on Lévy Processes (Wrocław, Poland, 2013)
• Recent progress in the study of suprema of Lévy processes [pdf]
German–Polish Conference on Probability and Mathematical Statistics (Toruń, Poland, 2013)
• Two-term asymptotics for Lévy operators in intervals [pdf]
6nd International Conference on Stochastic Analysis and Its Applications (Będlewo, Poland, 2012)
• Spectral decomposition of integro-differential operators related to one-dimensional Lévy processes in domains [pdf]
Harmonic Analysis and Probability (Angers, France, 2012)
• Boundary Harnack Inequality For Jump-Type Processes [pdf]
Foundations of Stochastic Analysis (Banff, Canada, 2011)
• Spectral Theory for Subordinate Brownian Motions in Half-line [pdf]
Yangyang Summer School (Yangyang, Korea, 2011)
• Cauchy process on half-line [pdf]
3rd International Conference on Stochastic Analysis and Its Applications (Beijing, China, 2009)
• Boundary Harnack inequality for stable processes on the Sierpiński gasket [pdf]
Fractal Geometry and Stochastics 4 (Greifswald, Germany, 2008)
• Intrinsic ultracontractivity for isotropic stable processes in unbounded domains [pdf]
2nd International Conference on Stochastic Analysis and Its Applications (Seoul, Korea, 2008)