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

Preprint articles

Tomasz Grzywny, Mateusz Kwaśnicki
Liouville’s theorems for Lévy operators
[arXiv]
Mateusz Kwaśnicki
Suprema of Lévy processes with completely monotone jumps: spectral-theoretic approach
[arXiv]
Jan Christoph Schlegel, Mateusz Kwaśnicki, Akaki Mamageishvili
Axioms for Constant Function Market Makers
[arXiv]
Rodrigo Bañuelos, Mateusz Kwaśnicki,
The $\ell^p$ norm of the Riesz–Titchmarsh transform for even integer $p$
[arXiv]
Rodrigo Bañuelos, Daesung Kim, Mateusz Kwaśnicki,
Sharp $\ell^p$ inequalities for discrete singular integrals
[arXiv]
Mateusz Kwaśnicki, Jacek Wszoła,
Bell-shaped sequences,
Studia Math., in press
[arXiv|online]

Publications

Mateusz Kwaśnicki,
Boundary traces of shift-invariant diffusions in half-plane,
Ann. Inst. Henri Poincaré Probab. Statist. 59(1) (2023): 411–436
[online | MR | zb]
Note: In Definition 1.1 the process $Y(t)$ should be assumed transitive on $[0, R)$, as explained in the second paragraph of Section 3.
Mateusz Kwaśnicki,
Harmonic extension technique for non-symmetric operators with completely monotone kernels,
Calc. Var. Partial. Differ. Equ. 61 (2022), no. 202: 1–40
[online | MR | zb]
Mateusz Kwaśnicki, Thomas Simon,
Characterisation of the class of bell-shaped functions,
Math. Zeitschrift 301(3) (2022): 2659–2683
[online | MR | zb]
Andreas E. Kyprianou, Mateusz Kwaśnicki, Sandra Palau and Tsogzolmaa Saizmaa,
Oscillatory attraction and repulsion from a subset of the unit sphere or hyperplane for isotropic stable Lévy processes,
In: L. Chaumont, A. E. Kyprianou (eds.), A Lifetime of Excursions Through Random Walks and Lévy Processes, Progress in Probability 78, Springer Nature Switzerland AG, 2021
[online | MR | zb]
Mateusz Kwaśnicki,
Random walks are determined by their trace on the positive half-line,
Ann. Henri Lebesgue 3 (2020): 1389–1397 [online | MR | zb]
Mateusz Kwaśnicki,
A new class of bell-shaped functions,
Trans. Amer. Math. Soc. 373(4) (2020): 2255–2280
[online | MR | zb]
Alexey Kuznetsov, Mateusz Kwaśnicki,
Minimal Hermite-type eigenbasis of the discrete Fourier transform,
J. Fourier Anal. Appl. 25(3) (2019): 1053–1079
[online | MR | zb]
Mateusz Kwaśnicki,
Fluctuation theory for Lévy processes with completely monotone jumps,
Electron. J. Probab. 24 (2019), no. 40: 1–40
[online | MR | zb]
Notes:
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. Similarly, 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.
4. In the proof of Theorem 3.3, in the first displayed equation on page 10, a factor $i$ is missing in the right-hand side: the integrand should be $\operatorname{Im} \tfrac{i}{\xi + i s} \tilde{\varphi}(s)$.
5. A factor $\pi$ is missing in the left-hand side of equation in line 14 on page 16: should be $\tfrac{\pi |f'(\xi)|}{f(\xi)}$.
6. On page 29, line 22, formula (5.18) (rather than (5.17)) should be referenced.
(supersedes: Rogers functions and fluctuation theory, unpublished [arXiv])
Rodrigo Bañuelos, Mateusz Kwaśnicki,
On the $\ell^p$ norm of the discrete Hilbert transform,
Duke Math. J. 168(3) (2019): 471–504
[online | MR | zb]
Mateusz Kwaśnicki, Richard S. Laugesen, Bartłomiej Siudeja
Pólya’s conjecture fails for the fractional Laplacian,
J. Spectral Theory 9(1) (2019): 127–135
[online | MR | zb]
Mateusz Kwaśnicki,
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]
Mateusz Kwaśnicki, Jacek Mucha,
Extension technique for complete Bernstein functions of the Laplace operator,
J. Evol. Equ. 18(3) (2018): 1341–1379
[online | MR | zb]
Notes:
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 the above errors!
4. Integrability of the Fourier transform of $f_n$ in the third paragraph of Section 7 does not seem to be known. A correct proof of continuity of $u_n$ proceeds as follows:
  - prove that for every $s$ the function $\varphi(|\xi|^2, s)$ is the Fourier transform of a sub-probability measure (this is clear with the probabilistic picture; analytically, a possible argument is given in a lemma below);
  - deduce that $u_n(s, \cdot)$ is continuous, and furthermore the family of functions $u_n(s, \cdot)$ is uniformly equicontinous;
  - use $\mathscr{L}^2(\mathbf{R}^d)$-continuity of $u_n(s, \cdot)$ with respect to $s$ to conclude joint continuity of $u_n(s, x)$.
  Lemma: For every $s$, $\varphi(|\xi|^2, s)$ is the Fourier transform of a mixture of exponentials.
  Sketch of the proof: By considering the string $\tilde{A}(E) = A(E + s)$, we find that $\partial_s (-\log \varphi(\lambda, s))$ is a complete Bernstein function of $\lambda$ for every $s$. Integration with respect to $s$ over $[0, s_0]$ proves that $\eta(\lambda) = -\log \varphi(\lambda, s_0)$ is a complete Bernstein function of $\lambda$ for every $s_0$. Thus, $\varphi(\lambda, s_0) = \exp(-\eta(\lambda))$ is the Laplace transform of a positive sub-probability measure on $[0, \infty)$, the distribution at time $1$ of the subordinator with Laplace exponent $\eta(\lambda)$. It follows that $\varphi(|\xi|^2, s_0)$ is the Fourier transform of a positive sub-probability measure on $\mathbb{R}^d$, namely the distribution at time $1$ of the subordinate Brownian motion with characteristic exponent $\eta(|\xi|^2)$. $\square$
Alexey Kuznetsov, Mateusz Kwaśnicki,
Spectral analysis of stable processes on the positive half-line,
Electron. J. Probab. 23 (2018), no. 10: 1–29
[online | MR | zb]
Kamil Kaleta, Mateusz Kwaśnicki, József Lőrinczi,
Contractivity and ground state domination properties for non-local Schrödinger operators,
J. Spectr. Theory 8 (2018): 165–189
[online | MR | zb]
Tomasz Grzywny, Mateusz Kwaśnicki,
Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes,
Stoch. Proc. Appl. 128(1) (2018): 1–38
[online | MR | zb]
Tomasz Juszczyszyn, Mateusz Kwaśnicki,
Martin kernels for Markov processes with jumps,
Potential Anal. 47(3) (2017): 313–335
[online | MR | zb]
Bartłomiej Dyda, Alexey Kuznetsov, Mateusz Kwaśnicki,
Eigenvalues of the fractional Laplace operator in the unit ball,
J. London Math. Soc. 95 (2017): 500–518
[online | MR | zb]
Note: In the first paragraph on page 509, the definition of the function $g$ should read $g(x) = \bigl(\int_{-1}^1 (f(x, y))^2 dy\bigr)^{1/2}$. Then indeed $Q(f) \geqslant Q(g)$, because $\|g\|_2 = \|f\|_2$, and $\mathscr{E}(f, f) \geqslant \mathscr{E}(g, g)$. The proof of the latter inequality goes as follows.
With the usual constant $C_{d, \alpha}$, we have $$C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{(f(x, y))^2}{|(x, y) - (x', y')|^{d + 1 + \alpha}} dy dy' = C_{d, \alpha} \frac{(g(x))^2}{|x - y|^{d + \alpha}} .$$ Furthermore, by Cauchy–Schwarz, $$\begin{aligned}&C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{f(x, y) f(x', y')}{|(x, y) - (x', y')|^{d + 1 + \alpha}} dy dy' \\ & \qquad \leqslant \biggl(C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{(f(x, y))^2}{|(x, y) - (x', y')|^{d + 1 + \alpha}} dy dy'\biggr)^{1/2} \times \\ & \qquad \qquad \biggl(C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{(f(x', y'))^2}{|(x, y) - (x', y')|^{d + 1 + \alpha}} dy dy'\biggr)^{1/2} \\ & \qquad = C_{d, \alpha} \frac{g(x) g(x')}{|x - x'|^{d + \alpha}}. \end{aligned}$$ It follows that $$C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{(f(x, y) - f(x', y'))^2}{|(x, y) - (x', y')|^{d + 1 + \alpha}} dy dy' \geqslant C_{d, \alpha} \frac{(g(x) - g(x'))^2}{|x - x'|^{d + \alpha}}.$$ Integration with respect to $x, x'$ leads to the desired inequality $\mathscr{E}(f, f) \geqslant \mathscr{E}(g, g)$. Thanks to Jiya Rose Johnson for pointing out this error!
Bartłomiej Dyda, Alexey Kuznetsov, Mateusz Kwaśnicki,
Fractional Laplace operator and Meijer G-function,
Constructive Approx. 45(3) (2017): 427–448
[online | MR | zb]
Notes:
1. 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.
2. There are two typos in the last paragraph on page 437: the expression for the Bessel function should read ${_0\mathbf{F}_1}(a \, | \, {-|x|^2}) = |x|^{1-a} J_{a-1}(2|x|)$, and the displayed formula should read \[ (-\Delta)^{\alpha/2} f(x) = \sqrt{\pi} \, \Gamma(\tfrac{d+\alpha}{2}) \, {_1\mathbf{F}_2}\Bigl(\begin{array}{c} \tfrac{d+\alpha}{2} \\ \tfrac{1+\alpha}{2}, \, \tfrac{d}{2} \end{array} \;\Big\vert\; {-\tfrac{1}{4} |x|^2}\Bigr) . \] Thanks to Yanzhi Zhang for pointing out the above errors!
3. Factor $(2 \pi)^{-\ell}$ is missing in the right-hand side of Bochner's relation (45) (due to different definitions of the Fourier transform here and in Stein's book) and, consequently, in several places in the proof of Proposition 3 (after 1st and 2nd sign in the second display on page 442, in the RHS of the third display, before and after 1st equality sign in the fifth display, and $(2 \pi)^{-2 \ell}$ is missing after the 2nd equality sign in the fifth display). The statement of Proposition 3 remains correct.
4. In formula (38) in Corollary 4, the factor $|x|^{\alpha - 2\rho}$ should be replaced by $|x|^{2\rho - \alpha}$. Same comment applies to the display after the statement of Corollary 4. Thanks to Vince Ervin for pointing out this error!
Mateusz Kwaśnicki,
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.
Kamil Kaleta, Mateusz Kwaśnicki, Jacek Małecki,
Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval,
J. Math. Anal. Appl. 439(2) (2016): 896–924
[online | MR | zb]
Tadeusz Kulczycki, Mateusz Kwaśnicki, Bartłomiej Siudeja,
The shape of the fundamental sloshing mode in axisymmetric containers,
J. Eng. Math. 99(1) (2016): 157–183
[online | MR | zb]
Tomasz Juszczyszyn, Mateusz Kwaśnicki,
Hitting times of points for symmetric Lévy processes with completely monotone jumps,
Electron. J. Probab. 20 (2015), no. 48: 1–24
[online | MR | zb]
Krzysztof Bogdan, Takashi Kumagai, Mateusz Kwaśnicki,
Boundary Harnack inequality for Markov processes with jumps,
Trans. Amer. Math. Soc. 367(1) (2015): 477–517
[online | MR | zb]
James Burridge, Alexey Kuznetsov, Andreas Kyprianou, Mateusz Kwaśnicki,
New families of subordinators with explicit transition probability semigroup,
Stoch. Proc. Appl. 124(10) (2014): 3480–3495
[online | MR | zb]
Nikolay Kuznetsov, Tadeusz Kulczycki, Mateusz Kwaśnicki, Alexander Nazarov, Sergey Poborchi, Iosif Polterovich and Bartłomiej Siudeja,
The Legacy of Vladimir Andreevich Steklov,
Notices Amer. Math. Soc. 61(1) (2014): 9–23
[online | MR | zb]
Kamil Kaleta, Mateusz Kwaśnicki, Jacek Małecki,
One-dimensional quasi-relativistic particle in the box,
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.
Mateusz Kwaśnicki, Jacek Małecki, Michał Ryznar,
Suprema of Lévy processes,
Ann. Probab. 41(3B) (2013): 2047–2065
[online | MR | zb]
Mateusz Kwaśnicki, Jacek Małecki, Michał Ryznar,
First passage times for subordinate Brownian motions,
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 $\cdots \exp(-\tfrac{1}{\pi} \int_0^\infty \cdots ) \cdots$.
Mateusz Kwaśnicki, Tadeusz Kulczycki,
On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains,
Proc. London Math. Soc. 105(5) (2012): 921–952
[online | MR | zb]
Mateusz Kwaśnicki,
Spectral theory for one-dimensional symmetric Lévy processes killed upon hitting the origin,
Electron. J. Probab. 17 (2012), no. 83: 1–29
[online | MR | zb]
Notes:
1. In Sect. 1 every occurrence of $2 \xi \Psi''(\xi) \le \Psi'(\xi)$ should be $\xi \Psi''(\xi) \le \Psi'(\xi)$ (5 times).
2. 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 $\cos(\pi \alpha/2)$.
Mateusz Kwaśnicki,
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 out this error!
Mateusz Kwaśnicki,
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 $C_\alpha (c_\alpha^{-1} \lambda^{\alpha/2} y)^{1/2} K_{\alpha/2}((c_\alpha^{-1} \lambda^{\alpha/2} y)^{1/\alpha})$.
Kamil Kaleta, Mateusz Kwaśnicki,
Boundary Harnack inequality for α-harmonic functions on the Sierpiński triangle,
Probab. Math. Stat. 30(2) (2010): 353–368
[online | MR | zb]
Tadeusz Kulczycki, Mateusz Kwaśnicki, Jacek Małecki and Andrzej Stós,
Spectral properties of the Cauchy process on half-line and interval,
Proc. London Math. Soc. 101(2) (2010): 589–622
[online | MR | zb]
Mateusz Kwaśnicki,
Eigenvalues of the Cauchy process on an interval have at most double multiplicity,
Semigroup Forum 79(1) (2009): 183–192
[online | MR | zb]
Mateusz Kwaśnicki,
Intrinsic ultracontractivity for stable semigroups on unbounded open sets,
Potential Anal. 31(1) (2009): 57–77
[online | MR | zb]
Mateusz Kwaśnicki,
Spectral gap estimate for stable processes on arbitrary bounded open sets,
Probab. Math. Statist. 28(1) (2008): 163–167
[online | MR | zb]
Krzysztof Bogdan, Tadeusz Kulczycki, Mateusz Kwaśnicki,
Estimates and structure of $\alpha$-harmonic functions,
Prob. Theory Rel. Fields 140(3–4) (2008): 345–381
[online | MR | zb]
Note: The first displayed formula in p. 352 should read $w = r^{-2} (r^2 - |x|^2) (r^2 - |v|^2) / |x - v|^2$.

Papers of my PhD students

Michał Gutowski,
Hardy-Stein identity for pure-jump Dirichlet forms,
Bull. Pol. Acad. Sci. Math., in press
[arXiv]
Tomasz Juszczyszyn,
Decay rate of harmonic functions fornon-symmetric strictly α-stable Lévy processes,
Studia Mathematica 260 (2021): 141–165
[online|MR|zb]
Jacek Mucha,
Spectral theory for one-dimensional (non-symmetric) stable processes killed upon hitting the origin,
Electron. J. Probab. 26 (2021), #14: 1–33
[online|MR|zb]

Selected conference presentations

The interplay between spectral theory and Lévy processes [pdf]
Recent Developments in Stochastic Processes (Sofia, Bulgaria, 2023)
Liouville’s theorems for Lévy operators [pdf]
NOMP (Będlewo, Poland, 2023)
Discrete Hilbert transforms on $\ell^p$ [pdf]
ETA Seminar (online, Rutgers University, USA, 2022)
Harmonic extensions, operators with completely monotone kernels, and traces of 2-D diffusions [pdf]
Non-local operators, probability and singularities webinar (2020)
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)