## Mateusz Kwaśnicki

Faculty of Pure and Applied 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

*Characterisation of the class of bell-shaped functions*[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*[arXiv]*A new class of bell-shaped functions*, Trans. Amer. Math. Soc., to appear [arXiv]

## Publications

*Minimal Hermite-type eigenbasis of the discrete Fourier transform*with Alexey Kuznetsov, J. Fourier Anal. Appl. 25(3) (2019): 1053–1079 [online|MR|zb]*Fluctuation theory for Lévy processes with completely monotone jumps*, Electron. J. Probab. 24 (2019), no. 40 [online|MR|zb]

(supersedes:*Rogers functions and fluctuation theory*[arXiv])*On the $\ell^p$ norm of the discrete Hilbert transform*with Rodrigo Bañuelos, Duke Math. J. 168(3) (2019): 471–504 [online|MR|zb]*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]*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]*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!*Spectral analysis of stable processes on the positive half-line*with Alexey Kuznetsov, Electron. J. Probab. 23 (2018), no. 10 [online|MR|zb]*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]*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]*Martin kernels for Markov processes with jumps*with Tomasz Juszczyszyn, Potential Anal. 47(3) (2017): 313–335 [online|MR|zb]*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]*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.*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.*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]*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]*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]*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]*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]*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]*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.*Suprema of Lévy processes*with Jacek Małecki and Michał Ryznar, Ann. Probab. 41(3B) (2013): 2047–2065 [online|MR|zb]*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 $\cdots \exp(-\tfrac{1}{\pi} \int_0^\infty \cdots ) \cdots$.*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]*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 $\cos(\pi \alpha/2)$.*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!*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 $C_\alpha (c_\alpha^{-1} \lambda^{\alpha/2} y)^{1/2} K_{\alpha/2}((c_\alpha^{-1} \lambda^{\alpha/2} y)^{1/\alpha})$.*Boundary Harnack inequality for*with Kamil Kaleta, Probab. Math. Stat. 30(2) (2010): 353–368 [online|MR|zb]*α*-harmonic functions on the Sierpiński triangle*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]*Eigenvalues of the Cauchy process on an interval have at most double multiplicity*, Semigroup Forum 79(1) (2009): 183–192 [online|MR|zb]*Intrinsic ultracontractivity for stable semigroups on unbounded open sets*, Potential Anal. 31(1) (2009): 57–77 [online|MR|zb]*Spectral gap estimate for stable processes on arbitrary bounded open sets*, Probab. Math. Statist. 28(1) (2008): 163–167 [online|MR|zb]*Estimates and structure of*with Krzysztof Bogdan and Tadeusz Kulczycki, Prob. Theory Rel. Fields 140(3–4) (2008): 345–381 [online|MR|zb]*α*-harmonic functions

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)