Mateusz Kwaśnicki

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

e-mail:
ORCID: 0000-0003-3896-8124

Research articles

For an author’s copy of any of the following papers, please send me an email at
[MathSciNet | zbMATH | arXiv | Scholar | Scopus | Web of Science | ORCID | MathOverflow]

Preprints

• Mateusz Kwaśnicki, Phanuel Mariano, Hugo Panzo, Jing Wang,
  Large time behavior of Lévy processes and their nonlocal Schrödinger semigroups.
  [arXiv]
• Mateusz Kwaśnicki,
  Bell-shaped functions are Poisson kernels.
  [arXiv]
• Mateusz Kwaśnicki, Jacek Wszoła,
  Dirichlet–Neumann bracketing for nonlocal operators.
  [arXiv]
• Maciej Kucharski, Mateusz Kwaśnicki,
  Shorter proof of dimension-free $L^p$ estimates for maximal Riesz transforms.
  [arXiv]
• Mateusz Kwaśnicki,
  Maximal inequalities for derivatives of spherical means.
  [arXiv]
• Mateusz Kwaśnicki, Jacek Wszoła,
  Generalised eigenvector expansion of infinite Toeplitz matrices with absolutely/completely monotone entries.
  [arXiv]
• Mateusz Kwaśnicki, Jack Thompson,
  Volumetric density estimates for nonlocal minimal surfaces.
  [arXiv]
• Maciej Kucharski, Mateusz Kwaśnicki, Błażej Wróbel,
  On the dimension-free control of higher order truncated Riesz transforms by higher order Riesz transforms.
  Ann. Scuola Norm. Super. Pisa Cl. Sci., in press
  [online]

Publications

49. Jan Christoph Schlegel, Mateusz Kwaśnicki, Akaki Mamageishvili,
    Axioms for Constant Function Market Makers.
    Math. Finance 2026 (2026), in press
    [online | Article Share]
    Extended abstract:
    Proc. 24th ACM Conf. Econom. Comput. (EC ’23), July 9–12, 2023, London, UK. ACM, New York, NY, USA
    [online]
48. Mateusz Kwaśnicki,
    Poisson kernels on the half-plane are bell-shaped.
    Bull. London Math. Soc. 58 (2026), no. e70303: 1–21
    [online | MR | zb | Article Share] Show/hide notes
    Notes: The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • In the abstract, the parenthesised expression should read “it is convex, then concave and then convex again.”
    • In the Caffarelli–Silvestre result on page 2, there should be no factor $\alpha$ in the denominator, and the argument of $f$ should be $x$ rather than $x - x'$ here and above. The correct expressions are \[ u(x, y) = \frac{\Gamma(\tfrac{d + 1}{2})}{\pi^{(d + 1) / 2}} \int_{\mathbb R^d} \frac{y^\alpha}{(\lvert x - x' \rvert^2 + y^2)^{(d + 1) / 2}} \, f(x') dx' \]
    and \[ u(x, y) = \frac{\Gamma(\tfrac{d + \alpha}{2})}{\pi^{d / 2} \Gamma(\tfrac{\alpha}{2})} \int_{\mathbb R^d} \frac{y^\alpha}{(\lvert x - x' \rvert^2 + y^2)^{(d + \alpha) / 2}} \, f(x') dx' . \]
    • In Equation (1.5), $f(x)$ should read $f(x')$.
    • In the last two displayed equations in Section&1.5, on page 7, in the expressions for $P(x)$ and $C$, $\mu p$ needs to be changed to $\mu q$, and the factor $q$ is missing in the argument of the characteristic function. The correct expressions are \[ P(x) = C \exp\biggl(-\frac{(1 - \mu) \mu q}{x - \mu q}\biggr) \frac{1}{\lvert x - \mu q \rvert^{\mu + 1}} \mathbb 1_{(0, \infty)}\biggl( \frac{x - \mu q}{(1 - \mu) \mu q} \biggr) \] and \[ C = \frac{(\lvert 1 - \mu \rvert \mu q)^\mu}{\Gamma(\mu)} . \]
    • In Section 2.1, in the definition of absolute monotonicity, $x \in (0, \infty)$ should be changed to $x \in (-\infty, 0)$.
    • In Equation (2.2), $\xi \to \infty$ should be changed to $\xi \to 0^+$.
    • In the last displayed equation in the proof of Proposition 2.4, on page 11, $x$ should be changed to $s$ in the integral on the right-hand side.
    • In the first paragraph of Section 3.1, on page 12, the “unambiguous expression” should have a minus sign, the correct one is $\varphi_\xi'(0^+) - a(\{0\}) \xi^2$.
    • In the second displayed equation on page 13, the boundary condition for the Neumann solution should have no $\tfrac12$, the correct expression is \[ (\varphi_\xi^{\mathrm N})'(0^+) = a(\{0\}) \xi^2 . \]
    • In Proposition 3.4, $y$ needs to be non-zero, the correct condition is $y \in (0, R)$.
    • The last paragraph of the proof of Proposition 3.4 has an error: the inequality $\lvert \varphi_\xi^{\mathrm D}(y) \rvert^2 \ge 2 y$ need not hold. However, [18, Lemma A.3] in fact shows that $\lvert \varphi_\xi^{\mathrm D} \rvert$ is convex, and hence \[ \lvert \eta(\xi) \rvert = \lvert \varphi_\xi^{\mathrm D}(y) \rvert \ge y . \] This leads to the same conclusion that $\eta$ does not converge to zero.
    • In the last displayed equation in Section 3.3, on page 16, $R$ should be changed to $\check R$.
    • Proposition 3.5 should say “weakly bell-shaped” rather than “bell-shaped.”
47. Rodrigo Bañuelos, Daesung Kim, Mateusz Kwaśnicki,
    Sharp $\ell^p$ inequalities for discrete singular integrals.
    J. Funct. Anal. 290(9) (2026), no. 111359: 1–80
    [online | MR | zb | Share Link]
46. Mateusz Kwaśnicki, Jacek Wszoła,
    Two-sided bell-shaped sequences.
    Math. Z. 311 (2025), no. 62: 1–42
    [online | MR | zb]
45. Serena Dipierro, Mateusz Kwaśnicki, Jack Thompson, Enrico Valdinoci,
    The nonlocal Harnack inequality for antisymmetric functions: an approach via Bochner’s relation and harmonic analysis.
    Comm. Partial Differ. Equ. 50(8) (2025): 1074–1098
    [online | MR | zb]
44. Michał Gutowski, Mateusz Kwaśnicki,
    Beurling–Deny formula for Sobolev–Bregman forms.
    Nonlinear Anal. 257 (2025): no. 113808: 1–14
    [online | MR | zb | Share Link]
43. Mateusz Kwaśnicki,
    Suprema of Lévy processes with completely monotone jumps: spectral-theoretic approach.
    J. Spectral Theory 15(1) (2025): 13–122
    [online | MR | zb] Show/hide notes
    Notes: The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • When $\inf Z_f = 0$, Theorem 1.1(b) and Corollary 5.8 should require that \[ \liminf_{r \to 0^+} \operatorname{Arg} \zeta_f(r) > -\frac{\pi}{2} . \] This is one of the assumption of Lemma 3.24 used in the proof. Similarly, when $\inf Z_f = 0$, Theorem 1.1(a) needs \[ \limsup_{r \to 0^+} \operatorname{Arg} \zeta_f(r) < \frac{\pi}{2} . \] This error affects examples in Sections 6.1 and 6.3. More precisely, in Section 6.1 the new condition is essentially independent of the Brownian component and needs to be imposed separately. In Section 6.3, Theorem 1.1 no longer applies when $\alpha \ge 1$, and for $\alpha < 1$ the correct condition is $\lvert \vartheta \rvert \ne \tfrac{\pi}{2}$.
    • In Theorems 1.1 and 1.2, and in Corollary 5.5, $t \ge 0$ should be changed to $t > 0$.
    • In the definition of $h_2(\xi)$ in Equation (3.16c), $\sqrt{c_r}$ should be changed to $1 / \sqrt{c_r}$. The expressions for the Wiener–Hopf factors $h_2^\pm(\xi)$ should therefore have $c_r^{-1 / 4}$, not $c_r^{1 / 4}$.
    • In Equation (3.22) the middle term should have $1 + \lvert \xi \rvert$ in the denominator. The proof of this estimate is flawed. The correct argument uses \[ \sin \frac{\vartheta}{2} \le \frac{\lvert (\zeta - \xi) - (\zeta + \bar \xi) \rvert}{\lvert \zeta - \xi \rvert + \lvert \zeta + \bar \xi \rvert} \le C(\zeta) \frac{\operatorname{Re} \xi}{1 + \lvert \xi \rvert} . \]
    • In the last displayed equation on page 26, the roles of $\Pi_+$ and $\Pi_-$ are interchanged. This error propagates to the remaining part of Section 4.4. Additionally, the factor $\frac{2}{\pi}$ is missing in $\frac{2}{\pi} \Pi_+^* \Pi_- v = v$ and $\frac{2}{\pi} \Pi_-^* \Pi_+ u = u$.
    • As stated, the second and the third assertion of Proposition 4.4 are wrong and should be deleted. This does not affect other parts of the paper, which only use the first assertion.
    • In Lemma 4.7, the distance to the spine $\Gamma_f$ should be replaced by the distance to the symmetrised spine $\Gamma_f^\star$. The proof is valid for real $\xi$, and for complex $\xi$, $\lvert \xi + i \zeta \rvert^2$ should be changed to $\lvert \xi + i \zeta \rvert \lvert \xi - i \bar \zeta \rvert$.
    • In Steps 3 of the proofs of Propositions 5.3 and 5.6, after an application of Proposition 4.6, the factor $\tfrac{2}{\pi}$ should no longer be there. This affects Equations (5.8) and (5.9) in the proof of Proposition 5.3, as well as the final two expressions for $I$ in the proof of Proposition 5.6.
    • In the final part of the proof of Corollary 5.5, the inequality $\lvert e^{-t f(r)} \rvert \le e^{-t \lvert f(r) \rvert}$ is false, it has the wrong direction. A corrected argument uses instead $\lvert e^{-t f(r)} \rvert = e^{-t \operatorname{Re} f(r)}$ together with the Harnack inequality for the positive harmonic function $\operatorname{Re}(f(\xi) / \xi)$: \[ \operatorname{Re} f(r) = r \operatorname{Re} \frac{f(r)}{r} \ge c_1 r \operatorname{Re} \frac{f(\zeta_f(r))}{\zeta(r)} = c_1 \lambda_f(r) \operatorname{Re} \frac{r}{\zeta(r)} \ge c_2 \lambda_f(r) \] for some $c_1, c_2 > 0$ (depending on $\varepsilon$).
    • In Section 6.4, the condition for $\nu(z)$ should have $A_\pm \ell(\lvert z \rvert^{-1}) / \lvert z \rvert^{1 + \alpha}$ with $\ell$ in the numerator, not the denominator. Later, the derivation of&nsbp;(6.5) is flawed: it should begin with the integral of $(r e^{i \vartheta} + s)^{-4}$, not $(r e^{i \vartheta} + s)^{-3}$ (and appropriate modifications are needed when $A_+ = 0$ or $A_- = 0$).
42. Tomasz Grzywny, Mateusz Kwaśnicki,
    Liouville’s theorems for Lévy operators.
    Math. Ann. 391 (2025): 5857–5910
    [online | MR | zb | Sharedit] Show/hide notes
    Notes: The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • Example 5.21 is flawed and should be removed. Currently, $L$ can be concentrated on a proper subgroup, and $Y$ need not satisfy the second part of condition (a) of Definition 5.13.
    • In the argument preceding Corollary 5.17, the inequality $Y(x - y) \le c (1 + |x|)^q Y(-y)$ need not hold. The correct proof of $h \circledast (f * g) = (h \circledast f) * g$ uses the following instead of (5.13): for every Schwartz class $\varphi$, we have \[ \begin{aligned} |f| * |g| * |h| * |\varphi|(x) & \le \|f / Y\|_\infty \|g / Y\|_\infty \|\varphi(\cdot + x) / (1 + \lvert\cdot\rvert)^q \|_\infty |h| * Y * Y * Y(0) \\ & \le \|f / Y\|_\infty \|g / Y\|_\infty \|\varphi / (1 + \lvert\cdot\rvert)^q\|_\infty (1 + |x|)^q |h| * Y(0) \\ & = C (1 + |x|)^q < \infty . \end{aligned} \]
    • Remark 5.4 should require $\mathscr F \check L$ to be not only smooth in $\mathbb R^d \setminus \{0\}$, but also non-zero, and sufficiently regular near the origin and at infinity, so that $L$ preserves the Lizorkin space.
    • In Step 4 of the proof of Theorem 3.2, the set $A$ should be assumed to be equal to the closure of its interior.
0. Mateusz Kwaśnicki,
   Harmonic extension technique: probabilistic and analytic perspectives.
   unpublished lecture notes
   [arXiv]
41. Rodrigo Bañuelos, Mateusz Kwaśnicki,
    The $\ell^p$ norm of the Riesz–Titchmarsh transform for even integer $p$.
    J. London Math. Soc. 109(4) (2024), no. e12888: 1–21
    [online | MR | zb]
40. Mateusz Kwaśnicki,
    Fleming–Viot couples live forever.
    Probab. Theory Relat. Fields 188 (2024): 1385–1408
    [online | MR | zb] Show/hide notes
    The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • For the second part of Theorem 1.5, the irreducibility condition needs to be strengthened to the following one: for almost every $x$, $G(x, \cdot)$ and $m$ are mutually absolutely continuous. Then $\mu$ and $m \times m$ are mutually absolutely continuous, as implicitly required at the end of the proof of this theorem.
    • In the proof of Lemma 3.2, in the displayed equation across pages 1395–1396, the increments $\mathbb 1(Y(t + s)) - \mathbb 1(Y(t))$ etc. should be reversed. In particular, $g$ approximates $-L \mathbb 1$ rather than $L \mathbb 1$.
    • Also in the proof of Lemma 3.2, the derivation of Equation (3.6) requires additionally continuity of $f$ (and uses almost sure left-continuity of $X(t)$ at a fixed time $t$). Extension of the result to arbitrary bounded Borel $f$ is then immediate.
0. Mateusz Kwaśnicki,
   Simplicity of eigenvalues of the fractional Laplace operator in an interval.
   unpublished
   [arXiv]
39. Mateusz Kwaśnicki, Jacek Wszoła,
    Bell-shaped sequences.
    Studia Math. 271(2) (2023): 151–185
    [online | MR | zb] Show/hide notes
    Note: Theorem 8.5.3 (instead of 11.5.3) in Karlin’s monograph should be referenced in Section 4.
38. 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] Show/hide notes
    Notes:

    • In Definition 1.1 the process $Y(t)$ should be assumed transitive on $[0, R)$, as explained in the second paragraph of Section 3.

    The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • The integrability condition in Theorem 4.1(a) is wrong. The correct version asserts integrability of $\lvert \varphi_\xi^\prime(y) + i \xi b(y) \varphi_\xi(y) \rvert^2$.
    • The above modification affects the proof of Theorem 1.2. The last two integrals on the right-hand side of Equation (4.18) need to be combined. Since \[ \mathbb E_Y^0 \int_0^{T_R} \bigl\lvert \hat X(s) \varphi^\prime(Y(s)) + i \xi \hat X(s) \varphi(Y(s)) b(Y(s)) \bigr\rvert^2 ds \le \int_0^R \bigl\lvert \varphi^\prime(y) + i \xi \varphi(y) b(y) \bigr\rvert^2 dy < \infty , \] the Itô integral \[ \int_0^{L_0^{-1}(u) \wedge T_R} \bigl( \hat X(s) \varphi^\prime(Y(s)) + i \xi \hat X(s) \varphi(Y(s)) b(Y(s)) \bigr) d\dot Y(s) \] exists and has zero mean.
    • In the first displayed equation in Remark 1.5, the latter integral is wrong. The correct expression is \[ Z(t) = X\biggl(\int_{[0, R)} L_y(L_0^{-1}(t)) a(dy)\biggr) - \frac{1}{2} \int_{(0, R)} L_y(L_0^{-1}(t)) db(y) - \frac{1}{2} \, b(0^+) u \]
    • The normalisation of the local time is inconsistent throughout the paper. The factor $2$ or $\tfrac{1}{2}$ should be added to:
      • the right-hand side $2 (R - y) \wedge (R - z)$ of the displayed equation in Proposition 2.11;
      • formula $\varphi_\xi^{\prime\prime}(dy) = 2 \xi \varphi_\xi(y) a(dy)$ in Equation 3.1, and $\psi(\xi) = -\tfrac{1}{2} \varphi_\xi^\prime(0^+) + \tilde a(\{0\}) \xi$ in the following line;
      • the mean $2 R$ of $L_0(T_R)$, and the kiling rate $\gamma = 1 / (2 R)$ of $Z(t)$, on page 427;
      • the trace $Z(u) = -\tfrac{1}{2} q u$ in Section 5.1;
      • the Lévy density $\nu(dx) = \frac{p}{2 \pi} \frac{1}{\lvert x \rvert^2} dx$ and the drift coefficient $\tfrac{1}{2} q$ when $\alpha = 1$ in Section 5.2.
    • In Equation (3.1), the condition $\varphi_\xi(R^-) = 0$ for $R = \infty$ needs to be replaced by boundedness of $\varphi_\xi$.
    • Equation (4.7) should contain an additional factor $\tfrac{1}{2}$ and the term $\tfrac{1}{2} a(\{0\}) f^{\prime\prime}(x)$ is missing. The correct expression is \[ K f(x) = \tfrac{1}{2} \partial_y u(x, 0) + \tfrac{1}{2} a(\{0\}) f^{\prime\prime}(x) . \]
    • In Equation (4.13), $dY(s)$ should be $d\dot Y(s)$.
    • In Equation (4.14), $\tfrac{i \xi}{2}$ should be changed to $-i \xi$.
    • At the beginning of Section 5, there are several typos: $\tilde x = x - B(y)$ rather than $\tilde x = x + B(y)$; the profile lines are integral curves of $\vec v_2$, not $\vec v_1$; the apparent drift is $(-\tfrac{1}{2} b'(y), 0)$, not $(b'(y), 0)$; the boundary contribution is $-\tfrac{1}{2} b(0^+) L_0(t)$, not $-b(0^+) L_0(t)$. Apparent drifts given later in this section also need a factor $\tfrac{1}{2}$.
    • Square is missing in the definition of $a$ in Section 5.2. The correct expression is \[ a(dy) = \frac{1}{(1 + 2 m y)^2} dy . \] The corresponding generator again is missing the factor $\tfrac{1}{2}$.
    • In the proof of Corollary 1.4, every occurrence of $(1 - e^{-\gamma v})$ should be $e^{-\gamma v}$.
    • In the same proof, on page 428, intervals $[\alpha_n, \beta_n]$ should be $[A(\alpha_n), A(\beta_n)]$.
    • In Section 5.3, the constants $C_+$ and $C_-$ require an additinal factor $1 / (2 \lvert \Gamma(-\alpha) \rvert)$. When $\alpha > 1$, the jump direction is negative rather than positive. Also, the description of vertical and horizontal speeds has them interchanged.
    • In Section 5.4, “meromorphic” should be replaced by “rational.”
37. 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] Show/hide notes
    Notes: The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • In Section 2.1, the derivation of the expression for $\partial_y u_x(x, 0)$ uses a wrong substitution. The correct expression is \[ \begin{aligned} \partial_y u_3(x, 0) & = \lim_{y \to 0^+} \frac{u_3(x, y) - u_3(x, 0)}{y} \\ & = \lim_{y \to 0^+} \frac{u_0(x - \tau(y), \sigma^{-1}(y)) - u_0(x, 0)}{y} \\ & = \lim_{z \to 0^+} \frac{u_0(x - \tau(\sigma(z)), z) - u_0(x, 0)}{\sigma(z)} \, . \end{aligned} \] This expression should appear in Equation (1.2): \[ K f(x) = \lim_{y \to 0^+} \frac{u(x - \tau(\sigma(y)), y) - u(x, 0)}{\sigma(y)} . \]
    • In Equations (2.3) and (4.1), the measure $\tilde a(dy) = a(dy) - (b(y))^2 dy$ should be replaced by $a(dy) - 2 (b(y))^2 dy$. The definition of $\tilde a$ remains unchanged, this error only affects these two equations.
    • Definitions 2.9 and 2.11 should allow infinite $R$ and $\dot R$: $R \in (0, \infty]$ and $\dot R \in (0, \infty]$.
    • The last equality in Equation (2.23) is missing $\dot a(y)$. The correct expression is \[ \tilde d(\dot \sigma(y)) = \dot a(y) \dot b^\prime(y) . \]
    • In Section 5.3, the operator $K$ with symbol $\psi(i \xi)$ acts in the negative direction, and the last displayed equation in this section is wrong. The correct expression is \[ K f(x) = -\check \beta f^\prime(x) - \gamma f(x) + \int_0^\infty (f(x - z) - f(x)) \nu(z) dz . \] Similarly, the last displayed equation in Section 5.5 should read \[ K f(x) = \alpha^\sharp f^{\prime\prime}(x) + \check \beta^\sharp f^\prime(x) + \int_0^\infty (f(x - z) - f(x) + z f^\prime(x)) \nu^\sharp(z) dz . \]
    • In Section 5.4, $k$ should be assumed not to be constant 0; otherwise, $k^\sharp(\xi) = \xi^2 / k(\xi)$ is not well-defined. The discussion of boundary conditions when $\dot R < \infty$ is incomplete: if one of the two PDEs for $\dot u$ and $\dot u^\sharp$ has a Dirichlet boundary condition at $y = \dot R$, the other one has a Neumann boundary condition at $y = \dot R$. In this case the expressions for $R$ and $R^\sharp$ need to be modified appropriately: the one corresponding to the Neumann boundary condition should be infinite rather than given by the integral of $1 / \dot a(y)$ or $\dot a(y)$ over $[0, \dot R)$.
    • In Section 5.6, on page 31, both occurrencies of $\dot q \partial_y$ should be $\dot q \partial_x$ in the third PDE for $\dot u$. The correct form is \[ \partial_{xx} \dot u + y^{\mu - 1} (\dot p \partial_y - \dot q \partial_x) \bigl(y^{1 - \mu} (\dot p \partial_y + \dot q \partial_x) \dot u\bigr) = 0 . \] Three lines below, the factor $\frac{\pi}{2}$ is missing in the condition for $A$ and $B$. The correct statement is $\lvert \operatorname{Arg}(A + i B) \rvert \le \frac{\pi}{2} \operatorname{min}\{\mu, 2 - \mu\}$.
    • In the proof of Lemma A.1, the estimates of $M$ have typos. The correct inequalities are \[ M(0) + \int_0^t M(s) ds \le M(t) , \] \[ \int_{[0, t)} M(s) a(ds) \le \frac{M(t)}{4 C^2} \] and \[ \int_0^t M(s) \lvert b(s) \rvert ds \le \frac{M(t)}{8 C} . \]
    • Also in the proof of Lemma A.1, the space $X$ need not be complete. Instead one should drop the condition that $\varphi^{\prime\prime}$ be a measure, and simply require that $\lVert \varphi \rVert_X$ be finite. The fact that the second order derivative of the fixed point $\varphi$ is a measure then follows from the equation $\varphi = I \varphi$.
    • The left-hand side of Equation (A.5) is missing a derivative: it should be $\tfrac{1}{2} (\lvert \varphi \rvert^2)^{\prime\prime}$. The displayed equation in the proof of Lemma A.3 needs a correction when the measure $a$ has atoms: $(\lvert \varphi^\prime \rvert^2)^\prime$ has a jump of size $2 \xi^2 a(\{t\}) \operatorname{Re}(\varphi(t) \overline{\varphi^\prime(t)}) + \lvert \xi^2 a(\{t\}) \varphi(t) \rvert^2$ at $t$. Monotonicity claim remains valid.
    • In the proof of Lemma A.4, on page 36, both ocurrences of $\varphi_D(t) + \beta \varphi_N(t)$ should be $\varphi_N(t) + \beta \varphi_D(t)$.
36. Mateusz Kwaśnicki, Thomas Simon,
    Characterisation of the class of bell-shaped functions.
    Math. Zeitschrift 301(3) (2022): 2659–2683
    [online | MR | zb] Show/hide notes
    The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • In Remark 1.2(b), $\varphi(s) ds$ is the vague limit of $-\pi^{-1} \operatorname{Im} \log \Phi(t + i s) ds$; a minus sign is missing.
    • In Proposition 1.8(c) the condition should be $c = -\int_{-1}^1 s^{-1} \varphi(s) ds$, not $c = 0$.
    • In Corollary 1.12, $p$ should be the limit of $\alpha_n$, not $1 / \alpha_n$.
    • The final example in Section 1.3 should be $f(x) = x^{-3 / 2} e^{-1 / x} \mathbb 1_{(0, \infty)}(x)$, the factor $x^{-3 / 2}$ is missing. The corresponding expression for $f_p(x)$ should read \[ f_p(x) = (1 - \tfrac{3}{2} p x^{-1} + p x^{-2}) x^{-3 / 2} e^{-1 / x} \mathbb 1_{(0, \infty)}(x) . \]
    • In Lemma 2.3, the limit $\Phi(\xi)$ may have a removable discontinuity at $\xi = 0$. The stated result is correct after removing this discontinuity. Also, “non-zero limit $\Phi(\xi)$” should be understood as “not identically equal to zero for $\xi \ne 0$.”
    • In Lemmas 2.4 and 3.2, $\xi^n$ should read $(i \xi)^n$.
    • In Lemma 3.1, the domain is $\mathbb R \setminus \{0\}$, not $i \mathbb R \setminus \{0\}$.
    • In the last displayed equation on page 2675, the denominator should read $s (s + i \xi)$, not $s^2 (s + i \xi)$.
    • The first displayed equation on page 2675 should read \[ \begin{aligned} \varphi_n(s) & \le \int_1^2 (1 + \varphi(s + t)) dt \\ & \le 1 + (s + 2)^3 \int_1^2 \varphi_n(s + t) \min\{s + t, (s + t)^{-3}\} dt \le 1 + (s + 2)^3 C_1 . \end{aligned} \]
    • On page 2678, in the paragraph after Equation (3.14), a minus sign is missing: $\operatorname{Im} (1 + x z)^{-1} = -x \lvert 1 + x z \rvert^{-2} \operatorname{Im} z$.
    • On page 2679, in the first displayed equation, $(\cos(\xi x) - 1)$ should read $(1 - \cos(\xi x))$.
    • In the proof of Proposition 1.10, a condition (c) rather than (b) of Theorem 1.1 should be referenced. Also, vague vs. weak compactness should be clearly distinguished: the present argument only shows the former, and a short calculation is needed to prove the latter.
35. Andreas E. Kyprianou, Mateusz Kwaśnicki, Sandra Palau, 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]
34. Mateusz Kwaśnicki,
    Random walks are determined by their trace on the positive half-line.
    Ann. Henri Lebesgue 3 (2020): 1389–1397
    [online | MR | zb] Show/hide notes
    Notes: The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • On page 1392, in the paragraph preceding Theorem 2.2, the definition of bounded type should require “nonnegative harmonic majorant,” not merely a “harmonic majorant.”
    • Also on page 1392, in Theorem 2.2, the term $f_s$ should be called “a ratio of singular inner functions” rather than “a singular inner function.” The same remark applies to the proof of Lemma 2.4.
    • Again on page 1392, in Theorem 2.2, $z_1, z_2, \ldots$ should be the sequence of zeroes excluding $-i$ (which is already treated separately), and the measure $\sigma$ may be infinite ($(1 + x^2)^{-1} \sigma(dx)$ is, however, finite).
    • On page 1393, also in Theorem 2.2, the claim that the limit $f(x) = \lim_{y \to 0^-} f(x + i y)$ is zero almost everywhere with respect to $\sigma$ is false in general and should be deleted (it is true when $f$ is bounded rather than of bounded type). This error does not affect other parts of the text.
    • On page 1394, in the proof of Lemma 2.4, the definition of $\varphi_{n, \mathrm s}$ should read $\varphi_{n, \mathrm s} = h_{n, \mathrm s} / g_{\mathrm s}$.
    • In the last displayed equation on page 1394, in the proof of Lemma 2.4, there should be no minus sign in front of $a_{\varphi, n} \operatorname{Im} z$.
    • On page 1395, in the proof of Theorem 1.1, the equality $\eta_+ = \mu_+ = \mu_-$ should read $\eta_+ = \mu_+ = \nu_+$.
    • On page 1395, in Section 3, the definitions of $\chi_n$ from [Ula92] are given with a wrong sign. The correct ones are $\chi_1 = \nu - \mu$ and $\chi_2 = \nu^{*2} - \mu^{*2}$. This error does not affect other parts of the text. Additionally, these measures are supported in $[0, \infty)$ rather than $(0, \infty)$.
    • An alternative, simpler proof of Lemma 2.4 can be given, which entirely avoids the inner-outer factorisation. Observe that if $\operatorname{Im} z < 0$, then $\lvert h_n(z) \rvert \le \lVert \mu_+^{*n} * \mu_- \rVert \le \lVert \mu_+ \rVert^n \lVert \mu_- \rVert$, and so whenever $g(z) \ne 0$, we have $\lvert \varphi(z) \rvert \le \lVert \mu_+ \rVert \lVert \mu_- \rVert^{1 / n} \lvert g(z) \rvert^{-1 / n}$. Passing to the limit as $n \to \infty$, we find that $\varphi$ is bounded in the set of $z$ such that $\operatorname{Im} z < 0$ and $g(z) \ne 0$. This already implies that all isolated singularities of $\varphi$ in the lower complex half-plane are removable, and the argument involving Painlevé’s theorem given in the paper applies to the singularities at the real axis.
33. Mateusz Kwaśnicki,
    A new class of bell-shaped functions.
    Trans. Amer. Math. Soc. 373(4) (2020): 2255–2280
    [online | MR | zb] Show/hide notes
    The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • In Corollary 1.2, condition (a) should additionally require that the completely monotone functions $x \nu(x)$ and $x \nu(-x)$ converge to zero as $x \to \infty$.
    • In the proof of Lemma 3.4, the definition of $s_k$ should use strict inequalities: $\varphi(s) < k$ and $\varphi(s) > k$.
    • In the proof of Theorem 3.7, the definition of $f_k(x)$ for $x < 0$ should read $\int_{[0, k]} e^{-s x} \mu_-(ds)$. In the definition of $f_k(x)$ for $x > 0$, $dx$ should be $ds$.
    • Comments about bell-shaped functions in the broad sense mention “exactly $n$” sign changes where they meant to say “at most $n$.” This affects Remark 3.9, Example 3.10 and Remark 5.6, which therefore should be taken with extra care.
    • On page 2265, $e^y f(y)$ should read $e^y g(y)$.
    • In Equation (5.5), the last integral should have a minus sign: $\int_{(0, \infty)} (\frac{-1}{z - s} - \frac{1}{s}) s \mu_-(ds)$.
    • Equation (5.8) and the last displayed equation in the proof of Lemma 5.4 should have an extra minus sign on the left-hand side: $\lim_{t \to 0^+} (-t \operatorname{Im} F(i t))$.
    • In the first line on page 2269, $(1 + s)$ should be $(1 + \lvert s \rvert)$.
    • Condition (5.15) in Lemma 5.4 should require the first integral to be finite, not allowing $-\infty$. In the proof, $\operatorname{Re} F(i t) \ge C / (2 t)$ should be changed to $\lvert \operatorname{Re} F(i t) \rvert \ge C / (2 t)$. This still leads to a contradiction, because, by continuity, $\operatorname{Re} F(i t)$ has a constant sign.
    • In the proof of Lemma 5.4, continuity of $\mathcal L h$ is insufficient. One needs to use $\operatorname{Re} \mathcal L h(i t) = 1 + O(t^2)$ and $\operatorname{Im} \mathcal L h(i t) = O(t)$ instead.
    • The trivial case $g = 0$ should be excluded in the statement of Lemma 5.4.
    • In Remark 5.6, the boundary limit is equal to $-\pi \varphi(-s)$.
    • On page 2276, the number of conitnuous derivatives is $n - 2$, not $n - 1$.
    • The left-hand side of the second displayed equation on page 2277 should read $\tfrac{1}{2} e^{-2 x} (x \nu(x))'$.
    • On page 2278, the substitution should read $x = x_j$, not $x = t^{-1 / 2} x_j$.
    • In item (b) in Section 6.4, the condition should read $\lvert \varphi(s) \rvert < 1$, not $\lvert \varphi(s) \rvert < \pi$. Also, the harmonic function $\operatorname{arg} \mathcal L f$ is not bounded.
    • In item (c) in Section 6.4, $x^{-p - 2 n}$ should be changed to $x^{p + 2 n}$. The next sentence is correct for $p > 1$; that Theorem 1.1 also covers the case $p \in (0, 1]$ needs to be verified separately.
    • In Section 6.5, a factor $\tfrac{1}{\pi}$ is missing in the definition of $\varphi(s)$.
32. 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]
31. Mateusz Kwaśnicki,
    Fluctuation theory for Lévy processes with completely monotone jumps.
    Electron. J. Probab. 24 (2019), no. 40: 1–40
    [online | MR | zb] Show/hide notes
    Notes:

    • In the last lines of equations (5.3), (5.8) and (5.10), $f^-(-i \xi_2)$ should read $f^-(i \xi_2)$. Similar corrections apply to the last displayed equation on page 29.
    • In the expression for $I$ in line 7 on page 26, $ds$ is missing.
    • In the displayed equation in line 14 on page 31, $\lambda(r)$ should read $\lambda_f(r)$.
    • 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.
    • In the proof of Theorem 3.3, in the first displayed equation on page 10, a factor $i$ is missing on the right-hand side: the integrand should be $\operatorname{Im} \tfrac{i}{\xi + i s} \tilde{\varphi}(s)$.
    • A factor $\pi$ is missing on the left-hand side of equation in line 14 on page 16: the correct expression is $\tfrac{\pi |f^\prime(\xi)|}{f(\xi)}$.
    • On page 29, line 22, formula (5.18) (rather than (5.17)) should be referenced.
    • On page 21, line 1, $\operatorname{Im} f(\xi) > 0$ should read $\frac{\operatorname{Im} f(\xi)}{\operatorname{Re} \xi} > 0$ (with the left-hand side extended continuously to $\operatorname{Re} \xi = 0$). Similar change should be made in line 2 on the same page.

    The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • The definition of $\overline{T}_t$ in Section 2.2 should use “$X_s = \overline{X}_t$ or $X_{s-} = \overline{X}_t$” instead of just “$X_s = \overline{X}_t$”. Similar remark applies to the definition of $\underline{T}_t$.
    • In the displayed formula after Equation (2.11), $e^{-\xi_2 x}$ should read $e^{\xi_2 x}$, and the left-hand side in the next displayed formula should additionally have $\kappa^\circ(\tau_1)$ in the numerator and $\kappa^\circ(\tau_2)$ in the denominator.
    • In the first paragraph of the proof of Theorem 3.3 and in Remark 3.4(c), the factor $\tfrac{1}{\pi}$ is missing in the expressions for the Bernstein measures of $\nu(x)$ and $\nu(-x)$.
    • In Remark 3.4(f), Equation (3.5), the expression for $b$ is incorrect. Instead, the value of $b$ can be reconstructed using Equation (3.1) at $\xi = 1$, with $a$, $c$ and $\mu$ given by the other reconstruction formulae. In the same equation, the correct expression for $\tilde{b}$ is $\operatorname{Im} f(r) / r$, not $\operatorname{Im} f(r)$.
    • Proposition 3.12(d) should require $g$ to be non-degenerate rather than non-zero. When $\alpha = -1$, Proposition 3.12(f) should additionally require that $f(\xi) + g(\xi)$ is non-zero.
    • In Proposition 3.14, Equations (3.14) and (3.15), the integrals need to be over $(-\infty, \infty)$ rather than $(0, \infty)$.
    • In the paragraph after Equation (3.16), in the expression for $f(\infty^-)$, the factor $\tfrac{1}{\pi}$ in front of the integral is missing.
    • In the proof of Proposition 3.15, $\tilde \mu(dt)$ needs to be the push-forward of $\lvert s \rvert^{-1} \mu(ds)$ rather than $\mu(ds)$.
    • Remark 3.16 requires three amendments. Convergence of $b_n$ to $b$ need not hold, and it should be replaced by convergence of $\operatorname{Im} f_n(1)$ to $\operatorname{Im} f(1)$. The factors $\tfrac{1}{\pi}$ should be changed to $\pi$ in the claims about vague convergence of measures. Finally, the last paragraph requires that the limit $f$ is non-zero.
    • In the proof of Proposition 3.17, $(y + s)^2$ should be changed to $r^2 (y + s)^2$ in the second displayed formula, and $\tilde b r$ should read $\lvert \tilde b \rvert r$ in the first line on page 15.
    • As stated, Proposition 3.18 may fail for imaginary $\xi$. The proof is correct under an additional assumption that $\operatorname{Re} \xi \ne 0$. When $\operatorname{Re} \xi = 0$, an additional assumption is needed: $\xi = \zeta_f(r)$ for some $r > 0$. The proof in this case is very similar, but the reasoning following Equation (3.23) needs appropriate modifications. When $\xi = \zeta_f(r) = i r$, then the limit of $\operatorname{Arg} f(z) / \operatorname{Re} z$ as $z \to \xi$ is nonpositive, and so instead of equality, we only have \[ \int_0^\infty \frac{1}{r^2 + 2 r s \sin \alpha + s^2} \, \varphi(s) ds \ge \int_{-\infty}^0 \frac{1}{r^2 + 2 r s \sin \alpha + s^2} \, \varphi(s) ds . \] However, this is sufficient for the remaining part of the proof. When $\xi = \zeta_f(r) = -i r$, the argument is completely analogous.
    • In the last displayed equation in the proof of Theorem 5.5, $\log c$ is missing on the right-hand side.
    • At the end of the proof of Theorem 5.7, $g(0^+) = \pi$ rather than $-\pi$.
    • In the proof of Lemma 7.1, there is a sign error in the second displayed equation on page 34: $8 t \sin \alpha$ should go with a plus sign before the last equality. The middle parts of the two inequalities used in the estimate of $\lvert B \rvert$ are interchanged. Later, $4 \Theta'(R)$ should read $4 \lvert \Theta'(R) \rvert$.
    • In the proof of Lemma 7.1, in the second displayed equation, $\lvert R_1 - R_0 \rvert$ should read $\lvert \Theta(R_1) - \Theta(R_0) \rvert$. Later, the integral from $R_0$ to $R_1$ should be changed to its absolute value to correctly cover the case $R_0 > R_1$.
30. 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] Show/hide notes
    Notes: The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • On page 479, there is a factor of $\pi$ missing: $(y_0 h(x_0, y_0))^{1 / p}$ should be $(\pi y_0 h(x_0, y_0))^{1 / p}$.
    • Also on page 479, in the application of Fatou’s lemma, the limit should be changed to the lower limit.
    • On pages 487–478, in Lemmas 3.3 and 3.4, $y_0 \ge 2 \pi n$ should be changed to $y_0 \ge 2 \pi |n|$.
    • On page 489, it should be specified that inequality (3.4) holds when $y \ge 1$.
    • On page 492, there should be no minus sign before $\partial_x p_0$ in the definition of $I_5$.
    • On page 493, there should be no third power in the denominator of the second term on the right-hand side of the displayed equation in Lemma 3.10: $(y^2 + \pi^2 n^2)$, not $(y^2 + \pi^2 n^2)^3$.
    • The same remark applies to the proof of Lemma 3.12 on page 496.
    • On page 499, in the integral equal to $\mathscr F_n$, the numerator should be $2 y^3$, not $y^3$.
29. 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]
28. 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: 159–193
    [online | MR | google]
27. 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] Show/hide notes
    Notes:

    • 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 $\Delta$).
    • 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 $\varphi_\lambda(s) = f_N(s) - \psi(\lambda) f_D(s)$.
    • 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!
    • 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 $\eta(|\xi|^2)$. $\square$
    • The relation between strings and canonical systems is discussed in [M. Kaltenbäck, J. Winkler, H. Woracek, Strings, dual strings, and related canonical systems, Math. Nachr. 280(13–14) (2007): 1518–1536]. A citation of this article is missing in the last paragraph on page 1348.
    • On page 1364, the two operators $\partial_t^2 + \frac{a^\prime(t)}{a(t)} \partial_t$ and $\partial_t^2 - \frac{a^\prime(t)}{a(t)} \partial_t$ are not formally adjoint. Instead, connections with Darboux transformation and Siegmund duality can be mentioned here.
26. 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]
25. 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]
24. 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] Show/hide notes
    Notes:

    • In Remark 1.6(c), the constants in the upper and lower estimates are comparable. In order to obtain truly comparable bounds, one should assume that, for example, $f(x) = 0$ for $x \in \mathbb{R}^d \setminus B(x_0, r)$.
    • In Remark 1.8(b), the word bounded is missing in the first sentence of the second paragraph: If $X_t$ is a subordinate Brownian motion, then, by Theorem 1.7, bounded harmonic functions are in fact smooth.
    • In the proof of Theorem 1.7 on pages 22–23, ball $B_{(k + 1)r}$ should be changed to $B_{2 k r}$ (twice). Thanks to Krzysztof Bogdan for pointing out these inaccuracies!
23. Tomasz Juszczyszyn, Mateusz Kwaśnicki,
    Martin kernels for Markov processes with jumps.
    Potential Anal. 47(3) (2017): 313–335
    [online | MR | zb] Show/hide notes
    Notes:

    • Non-alphabetical order of the authors in the published version is a mistake which occurred during the production process. Preprint on arXiv lists the authors in the correct, alphabetical, order.
    • In the definition of accessible and inaccessible points in Remark 1 on page 318 and in Theorem 3(b) on page 320, the term $\nu(x_0, y)$ or $\nu(z, y)$ in the integrand is missing. The definitions given later in Sections 4.2 and 4.3 are correct.
22. 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] Show/hide notes
    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 $$C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{(f(x, y))^2}{|(x, y) - (x^\prime, y^\prime)|^{d + 1 + \alpha}} dy dy^\prime = C_{d, \alpha} \frac{(g(x))^2}{|x - y|^{d + \alpha}} .$$ Furthermore, by Cauchy–Schwarz, $$\begin{aligned}&C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{f(x, y) f(x^\prime, y^\prime)}{|(x, y) - (x^\prime, y^\prime)|^{d + 1 + \alpha}} dy dy^\prime \\ & \qquad \leqslant \biggl(C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{(f(x, y))^2}{|(x, y) - (x^\prime, y^\prime)|^{d + 1 + \alpha}} dy dy^\prime\biggr)^{1/2} \times \\ & \qquad \qquad \biggl(C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{(f(x^\prime, y^\prime))^2}{|(x, y) - (x^\prime, y^\prime)|^{d + 1 + \alpha}} dy dy^\prime\biggr)^{1/2} \\ & \qquad = C_{d, \alpha} \frac{g(x) g(x^\prime)}{|x - x^\prime|^{d + \alpha}}. \end{aligned}$$ It follows that $$C_{d + 1, \alpha} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{(f(x, y) - f(x^\prime, y^\prime))^2}{|(x, y) - (x^\prime, y^\prime)|^{d + 1 + \alpha}} dy dy^\prime \geqslant C_{d, \alpha} \frac{(g(x) - g(x^\prime))^2}{|x - x^\prime|^{d + \alpha}}.$$ Integration with respect to $x, x^\prime$ leads to the desired inequality $\mathscr{E}(f, f) \geqslant \mathscr{E}(g, g)$. Thanks to Jiya Rose Johnson for pointing out this error!
21. Bartłomiej Dyda, Alexey Kuznetsov, Mateusz Kwaśnicki,
    Fractional Laplace operator and Meijer G-function.
    Constructive Approx. 45(3) (2017): 427–448
    [online | MR | zb] Show/hide notes
    Notes:

    • 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.
    • 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!
    • Factor $(2 \pi)^{-\ell}$ is missing on 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.
    • 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!
20. Mateusz Kwaśnicki,
    Ten equivalent definitions of the fractional Laplace operator.
    Frac. Calc. Appl. Anal. 20(1) (2017): 7–51
    [online | MR | zb] Show/hide notes
    Notes:

    • $p \in (1, \infty)$ should read $p \in (1, \tfrac{d}{\alpha})$ on page 22, line 20.

    The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • In the Balakrishnan’s and Bochner’s definitions, the Bochner integrals need not converge absolutely on $(0, \infty)$. Instead, improper Bochner integrals (convergent in norm) need to be considered, with a singularity at $t = 0$ for Bochner’s definition and at $s = \infty$ for Balakrishnan’s definition. This affects Theorems 1.1, 5.5 and 5.7, and Definitions 2.3 and 2.4. Similar remark applies to the pointwise Bochner’s definition and Theorem 6.1.
    • With the modified Bochner’s definition, the proof of Lemma 3.6 requires a tweak: the dominated convergence theorem can only be applied after integration by parts and the substitution $t^{-2 / \alpha} r = s$.
    • In Theorem 1.1(e) and (f), $B(x, r)$ should be changed do $B(0, r)$ in the domains of integrals. There is a missing minus sign in Theorem 1.1(g): $\langle L f, \varphi \rangle = -\mathcal E(f, \varphi)$.
    • In Remark 1.2, in the definition of $\tilde \nu_r$, $(r^2 - |z|^2)$ should be changed to $(|z|^2 - r^2)$.
    • In Section 2.3, in the expression for the resolvent kernel, a factor $s^{d / 2 - 1}$ is missing, and $x$ should be changed to $|x|$.
    • In Section 2.6, in the application of weak Young’s inequality, condition $p < \frac{d}{\alpha}$ should be replaced by $1 < p < \frac{d}{\alpha}$.
    • In Section 2.7, the norm convergence of $P_t f$ to $0$ requires $p \in (1, \infty)$, not $p \in [1, \infty)$.
    • In Section 2.8, minus signs are missing in the expression for $\alpha = 1$: $\varphi_\lambda(y) = \exp(-\sqrt{\lambda} \, y)$, and in the scaling property: $q_t(z) = t^{-d / \alpha} q_1(t^{-1 / \alpha) z)$.
    • In the proof of Lemma 3.4, the limit of $c_{d, \alpha} r^{1 + \alpha} m'(r)$ is equal to $-\alpha 2^{2 \alpha - 1} \pi^{-d / 2} \Gamma(\tfrac{d}{2} + \alpha) / \Gamma(-\alpha)$, not the given expression $2^{2 \alpha - d / 2 - 1} \Gamma(\tfrac{d}{2} + \alpha) / \Gamma(-\alpha)$.
    • In the proof of Lemma 3.7, $\frac{p_t(z)}{z}$ should be $\frac{p_t(z)}{t}$.
    • In Equation (4.2), $|z|^d$ needs to be changed to $|z - y|^d$ in the denominator.
    • In the second displayed equation on page 34, $r^d$ is missing in the denominator of the second term, and the third parameter of the hypergeometric function should by $1 + \frac{d - \alpha}{2}$, not $\frac{d - \alpha}{2}$.
    • In Example 6.2, $f = \partial_{x_d} g$ should read $f = (\alpha - 1)^{-1} \partial_{x_1} g$.
    • In the proof of Lemma 6.1, $(\lambda f - g)$ should be changed to $(g - \lambda f)$.
    • In the proof of Proposition 6.1, in the definition of $g_2$, $r$ should be changed to $R$.
    • In Section 7, property (d), $p_t(x + z)$ should read $p_t(z - x)$.
    • In the proof of Lemma 7.1, a minus sign is missing in $1 - e^{-\lambda \tau_{B(x, r)}}$.
19. 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] Show/hide notes
    The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • In the abstract, the Klein–Gordon operator corresponds to $\psi(\xi^2) = (1 + \xi^2)^{1 / 2} - 1$, not $\psi(\xi) = (1 + \xi^2)^{1 / 2} - 1$.
    • There is an extra factor $\tfrac{1}{\pi}$ in various places in the text. This originates in the Stieltjes inversion formula in the last displayed equation in the proof of Lemma 2.1. The error propagates to Equation (6–7) in Lemma 2.1, both estimates in Lemma 2.13.
    • Other corrections to the proof of Lemma 2.1 are as follows. In the second displayed equation on page 923, the divided difference of $y g(y)$ is bounded by $\tilde C_4 G(y)$, not $C_4 y^{-1} G(y)$, with a possibly larger constant $\tilde C_4$ the should be used in the remainin part of the proof. The middle expression in the next displayed equation should have $y$ rather than $y^{-1}$ (the right-hand side is correct). Since $g$ may be signed on the real axis, $\lvert g(y) \rvert$ should be used in estimates. In the last displayed equation has a stray integral $\int_0^\infty$.
    • In Section&nsbp;2.1, on page 900, the definition of $\psi^\dagger$ should have $\xi^2 + r^2$, not $\xi^2 + r$, in the denominator. On the next page, in the expression for $\psi_\mu(\xi)$ when $\psi(0) > 0$, $1 - \xi^2 / \mu^2$ should read $1 - \xi / \mu^2$.
    • In Section 2.2, on page 901, $(1 + \eta)^\alpha$ should read $(1 + \frac{\eta}{\xi})^\alpha$, and the definition of $\mathcal E(f, g)$ needs a complex conjugate over $\mathcal F g(\xi)$.
    • In the proof of Proposition 2.3, the linear map $x \mapsto (x - b_n) / a_n$ should map $I$ to its neigbouhood, not vice versa. Consequently, $a_n \le 1$ instead of $a_n \ge 1$.
    • The existence of eigenfunctions discussed in the first paragraph of Section 2.5 can extends to arbitrary $\psi$. This is not needed for the main results, but eigenfunctions for general $\psi$ appear in auxiliary statements.
    • On page 905, in the third expression for $\gamma_\mu(d\xi)$, the arguments should be $-e^{-i \varepsilon} \xi$, not $e^{-i \varepsilon} \xi$.
    • In Section 2.7, $\psi$ should be regularly varying of order $\frac{\alpha}{2}$, not $\alpha$.
    • In Lemma 2.7, the words “increasing” and “decreasing” should be interchanged. Also, when $\psi$ is linear, then $P = 1$ and $Q = 0$. In this case we understand that $Q / (1 - P) = 1$.)
    • In the proof of Lemma 2.10, the limits of integration $\xi_z$ and $\xi_{1 / z}$ are interchanged.
    • In Example 2.11, the denominator $\mu^{\alpha - \beta} w + 1$ should read $\mu^{\alpha - \beta} w + z^{\alpha - \beta}$.
    • After Equation (20), $0 \le q''(x) \le 1 / b^2$ should read $\lvert q''(x) \rvert \le 1 / b^2$.
    • In Lemma 3.2, in the last displayed equation, the term $M_2 c$ should be incorporated into the second term: \[ \lvert \mathcal A g(x) \rvert \le \biggl( \frac{M_0}{b^2} + \frac{2 M_1}{b} + M_2 \biggr) \biggl( c + \int_0^b z^2 \nu(z) dz \biggr) + 2 M_0 \int_0^b \nu(z) dz + \nu(b) M_{-1} . \] An appropriate amendment is needed in the paragraph preceding the statement of Lemma 3.4.
    • In the proof of Lemma 3.3, in the estimate of $\lvert I(x, z)$, absolute value is missing around $\frac{\partial^2}{\partial s^2} I(x, s)$. Later, $q$ is not integrable, but nevertheless $\tilde q$ is. In the next sentence, $\mathcal F q$ and $\mathcal F q^{(3)}$ should read $\mathcal F \tilde q$ and $\mathcal F \tilde q^{(3)}$. Finally, the proof should also show that $\mathcal A u(x)$ is in $L^2(D)$, not merely well-defined in $D$. This follows directly from the proof of Lemma 3.4.
    • The concavity argument in the proof of Lemma 3.9 is incorrect. Instead, one needs to repeat the preceding argument for $\psi(\tilde \mu_n^2) - \psi((\tilde \mu_n - \varepsilon)^2)$.
    • In the proof of Lemma 3.6, in the estimate of $-G_\mu'(x)$, the minus sign is missing.
    • In the proof of Lemma 3.11, lower indices of sums in the displayed equation after Equation (27) should be $N + 1$ rather than $N$, and “concave” should read “convex.”
    • The proof of Proposition 3.12 needs three corrections. In the estimate of $\lvert \lambda_j - \tilde \lambda_n \rvert$, the maximum should be the minimum, and $C_1$ should be $C_1 a$. The final estimate should read \[ \begin{aligned} \lVert \tilde \varphi_n - \alpha_{n, n} \varphi_n \lVert_{L^2(D)}^2 & = \sum_{j \ne n} \lvert \alpha_{n, j} \rvert^2 \le \biggl( \frac{C_1 a}{\frac{n \pi}{2 a} \psi'((\frac{n \pi}{2 a})^2)} \biggr)^2 \sum_{j \ne n} (\lambda_j - \tilde \lambda_n)^2 \lvert \alpha_{n, j} \rvert^2 \\ & \le \biggl( \frac{C_1 a}{\frac{n \pi}{2 a} \psi'((\frac{n \pi}{2 a})^2)} \biggr)^2 \lVert A_D \tilde \varphi_n - \tilde \lambda_n \tilde \varphi_n \rVert_{L^2(D)}^2 \le \biggl( \frac{C_1 a}{\frac{n \pi}{2 a} \psi'((\frac{n \pi}{2 a})^2)} \biggr)^2 \frac{C_2(a, b, \psi)}{n^2} . \end{aligned} \]
    • In the proof of Proposition 3.13, $\cos \vartheta_\lambda$ should read $\sin \vartheta_\mu$. Also, $G_\mu(0)$ should read $G_\mu(0^+)$ throughout the proof.
    • In the first displayed equation of the proof of Proposition 3.14, $T_D(t) \varphi_n$ should read $T_D(t) \tilde \varphi_n$. Also in the following line, $\varphi_n$ should be $\tilde \varphi_n$. In the final part of the argument, $\beta_n = O(1)$ needs to be replaced by $1 / \beta_n = O(1)$, and a reference to Proposition 3.12 is missing.
18. 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]
17. 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]
16. 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] Show/hide notes
    Notes:

    • In Example 5.5, an additional condition that Hunt’s hypothesis (H) holds, is missing. It is satisfied in each of the bullet points discussed there. See paper no. 23 for further discussion.

    The following errors were found using the large language model GPT-6 Astra (OpenAI):

    • In the proof of Lemma 3.2, the closed ball $\overline{B}(x_0, q)$ needs to be changed into the open one $B(x_0, q)$ in a few places.
    • In Equation (3.8), $c_{(3.9)} \varrho$ should be changed to $2 c_{(3.9)} \varrho$.
    • $C^{1,\beta}$ boundary might be insufficient in Example 5.6, fourth bullet point, and Example 5.11. The argument is, however, correct if the boundary is $C^2$.
    • Mixtures of stable processes do not satisfy condition (g), and so should not be mentioned in Example 5.8.
    • The last paragraph in Example 5.11 is wrong and should be removed: $f$ and $g$ need not be harmonic with respect to the nontruncated process.
    • Step 3 in the proof of Theorem A.1 should treat the case $j = 1$ separately.
15. 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]
14. Nikolay Kuznetsov, Tadeusz Kulczycki, Mateusz Kwaśnicki, Alexander Nazarov, Sergey Poborchi, Iosif Polterovich, Bartłomiej Siudeja,
    The Legacy of Vladimir Andreevich Steklov.
    Notices Amer. Math. Soc. 61(1) (2014): 9–23
    [online | MR | zb]
0. Mateusz Kwaśnicki,
   Rogers functions and fluctuation theory.
   unpublished
   [arXiv])
13. Kamil Kaleta, Mateusz Kwaśnicki, Jacek Małecki,
    One-dimensional quasi-relativistic particle in the box.
    Rev. Math. Phys. 25(8) (2013) 1350014
    [online | MR | zb] Show/hide notes
    Note: In the first displayed formula in Lemma 4.2 the norm on the left-hand side should not be squared.
12. Mateusz Kwaśnicki, Jacek Małecki, Michał Ryznar,
    Suprema of Lévy processes.
    Ann. Probab. 41(3B) (2013): 2047–2065
    [online | MR | zb]
11. Mateusz Kwaśnicki, Jacek Małecki, Michał Ryznar,
    First passage times for subordinate Brownian motions.
    Stoch. Proc. Appl. 123 (2013): 1820–1850
    [online | MR | zb] Show/hide notes
    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 $\cdots \exp(-\tfrac{1}{\pi} \int_0^\infty \cdots ) \cdots$.
10. Tadeusz Kulczycki, Mateusz Kwaśnicki,
    On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains.
    Proc. London Math. Soc. 105(5) (2012): 921–952
    [online | MR | zb]
9. 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] Show/hide notes
   Notes:

   • In Sect. 1 every occurrence of $2 \xi \Psi^{\prime\prime}(\xi) \le \Psi^\prime(\xi)$ should be $\xi \Psi^{\prime\prime}(\xi) \le \Psi^\prime(\xi)$ (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 $\cos(\pi \alpha/2)$.
8. Mateusz Kwaśnicki,
   Eigenvalues of the fractional Laplace operator in the interval.
   J. Funct. Anal. 262(5) (2012): 2379–2402
   [online | MR | zb] Show/hide notes
   Notes:

   • 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!
   • Square is missing in Lemma 2 and its proof: $\lVert\tilde\varphi_n\rVert_2$ should be $\lVert\tilde\varphi_n\rVert_2^2$ (twice). Thanks to Jacek Wszoła for pointing out this typo!
   • On page 2385, the correct reference is [14, Lemma 4.21], not [14, Lemma 16].
7. Mateusz Kwaśnicki,
   Spectral analysis of subordinate Brownian motions on the half-line.
   Studia Math. 206(3) (2011): 211–271
   [online | MR | zb] Show/hide notes
   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 $C_\alpha (c_\alpha^{-1} \lambda^{\alpha/2} y)^{1/2} K_{\alpha/2}((c_\alpha^{-1} \lambda^{\alpha/2} y)^{1/\alpha})$.
6. Kamil Kaleta, Mateusz Kwaśnicki,
   Boundary Harnack inequality for $\alpha$-harmonic functions on the Sierpiński triangle.
   Probab. Math. Stat. 30(2) (2010): 353–368
   [online | MR | zb]
5. Tadeusz Kulczycki, Mateusz Kwaśnicki, Jacek Małecki, 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] Show/hide notes
   The following errors were found using the large language model GPT-6 Astra (OpenAI):

   • In Equation (11.5), $c_{n, i} c_{m, j} G_{i, j}$ should be changed to $c_{m, i} c_{n, j} G_{n - 2 i, m - 2 j}$. In order to have normalised functions, $c_{n, i}$ should have an additional factor $\sqrt{n + \tfrac{1}{2}}$.
   • On page 610, in the displayed formula above Equation (8.7), the estimate of $|(-\Delta)^{1 / 2} h(z)|$ given in the text would lead to constant 2 in the numerator. The estimate holds with constant 1, but requires a more careful proof: write $(x - z)^{-2} = \int_0^\infty s e^{s z - s x} ds$, change the order of integration, evaluate \[ \int_1^\infty e^{-s x} \sin(\mu_n(1 + x) + \pi / 8) dx = e^{-s} \, \frac{\mu_n \cos(2 \mu_n + \frac{\pi}{8}) + s \sin(2 \mu_n + \frac{\pi}{8})}{s^2 + \mu_n^2} , \] and use $|\mu_n \cos \gamma + s \sin \gamma| / (s^2 + \mu_n^2) \le 1 / \mu_n$.
   • At the bottom of page 616, $T g_j = 0$ should read $P_N T g_j = 0$.
   • At the end of Section 11.2, the signs are wrong: $T f_n(x)$ is equal to $-g_{n - 1}(x) - g_{n + 1}(x)$, and the factor $(-1)^{1 + (m + n) / 2}$ in the expression for $b_{m, n}$ should be deleted. The claim that all eigenvalues are less than $N + 2$ is wrong.
   • There are sign errors in Remark 4: $H g'(s) = s g'(s)$ and $H h(s) = (1 / s) H g'(1 / s)$ without minus signs, and \[ \frac{\mathbb P_x(\tau_D \in dt)}{dt} = \frac{x^2}{t} g'\biggl(\frac{x}{t}\biggr) = -\frac{c}{t} e^{-B(-t / x)} \] in the last displayed equation.
4. 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]
3. Mateusz Kwaśnicki,
   Intrinsic ultracontractivity for stable semigroups on unbounded open sets.
   Potential Anal. 31(1) (2009): 57–77
   [online | MR | zb] Show/hide notes
   The following errors were found using the large language model GPT-6 Astra (OpenAI):

   • Whenever horn-shaped regions are discussed, it should be assumed that $d \ge 2$.
   • Square is missing in $(r_n^2 - \lvert x - x_n\rvert^2)$ twice in Example 3.
   • In the proof of Lemma 1, in the third displayed equation on page 67, $P_t$ should be changed to $P_{t / 2}$.
   • In the proof of Lemma 2, the factor $\lambda_1$ is missing in the estimate of $\varphi_1(y)$: the right-hand side should be $\lambda_1 \lVert \varphi_1 \rVert_\infty \mathbf E_y(\tau_D)$.
2. Mateusz Kwaśnicki,
   Spectral gap estimate for stable processes on arbitrary bounded open sets.
   Probab. Math. Statist. 28(1) (2008): 163–167
   [online | MR | zb] Show/hide notes
   The following errors were found using the large language model GPT-6 Astra (OpenAI):

   • In the proof of Proposition 2.1, the constant in Plancherel’s formula should be $(2 \pi)^d$, not $(2 \pi)$, which leads to the same bound with a better constant.
   • On page 194, the assertion that no analogous bound holds even for connected sets requires $d \ge 2$, it is false in dimension one.
   • On the same page, the lower bound written as $c (\operatorname{diam D})^{-2}$ should be $c (w(D))^{2 - \alpha} (\operatorname{diam D})^{-2}$, where $w(D)$ denotes the minimal width of an infinite strip which contains $D$.
1. Krzysztof Bogdan, Tadeusz Kulczycki, Mateusz Kwaśnicki,
   Estimates and structure of $\alpha$-harmonic functions.
   Probab. Theory Relat. Fields 140(3–4) (2008): 345–381
   [online | MR | zb] Show/hide notes
   Note:

   • The first displayed formula in p. 352 should read $w = r^{-2} (r^2 - |x|^2) (r^2 - |v|^2) / |x - v|^2$.

   The following errors were found using the large language model GPT-6 Astra (OpenAI):

   • In Equation (20), $\varphi(x)$ should be outside of the integral (this matters when $\omega^x_D$ has mass less than one).
   • As stated, Remark 3 is incorrect: the lower bound does not hold with a constant nonincreasing in $p$, let alone independent of $p$.
   • In the proof of Lemma 10, every occurrence of $\lVert g_2 \rVert_\infty$ should be replaced by $\sup \{ \lvert g_2(x) \rvert : x \in D_1 \setminus D_2 \}$.
   • In the proof of Proposition 1, the sets $U$ and $V$ should be defined using conditions $\operatorname{dist}(x, K) < 2 \operatorname{dist}(x, L)$ and $\operatorname{dist}(x, L) < 2 \operatorname{dist}(x, K)$, respectively. Also, Lemma 9 should be applied to $f(x)$, not $\omega^x_D(K)$.
   • In the first paragraph on page 375, the normalisation factor is missing in $M(\cdot, \infty) = (s_D(x_0))^{-1} s_D$.
   • In the proof of Proposition 3, the middle integral in the first displayed equation should be over $D_r$, not $D_{3 r}$, and $\nu(v, y)$ should be $\nu(v, z)$.

Papers of my PhD students

• Jacek Wszoła,
  Eigenvalues of Hermitian Toeplitz matrices with Fisher–Hartwig symbols.
  Preprint, 2026
  [arXiv]
• Jacek Wszoła, Krzysztof Burnecki, Marek Teuerle, Martyna Zdeb,
  Design and valuation of multi-region CoCoCat bonds.
  Preprint, 2025
  [arXiv]
• Michał Gutowski,
  Littlewood–Paley estimates for pure-jump Dirichlet forms.
  Preprint, 2025
  [arXiv]
• Jacek Wszoła,
  First passage locations for two-dimensional lattice random walks and the bell-shape.
  Electron. J. Probab. 31 (2026), no. 44: 1–28
  [online]
• Krzysztof Bogdan, Michał Gutowski, Katarzyna Pietruska-Pałuba,
  Polarized Hardy–Stein identity.
  J. Funct. Anal. 288(7) (2025): no. 110827: 1–39
  [online]
• Tadeusz Kulczycki, Jacek Wszoła,
  On the interior Bernoulli free boundary problem for the fractional Laplacian on an interval.
  Collectanea Math. 76 (2025): 11–34
  [online]
• Michał Gutowski,
  Hardy–Stein identity for pure-jump Dirichlet forms.
  Bull. Pol. Acad. Sci. Math. 71 (2023): 65–84
  [online | MR | zb]
• Tomasz Juszczyszyn,
  Decay rate of harmonic functions for non-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

• Boundary traces of planar diffusions [pdf]
  Seminar on Stochastic Processes (Schenectady, USA, 2026)
• Boundary traces of diffusions [pdf]
  SPA conference (Wrocław, Poland, 2025)
• Liouville’s theorems for Lévy operators [pdf]
  IM PAN Colloquium (Warsaw, Poland, 2024)
• Discrete Hilbert transform [pdf]
  Ryll-Nardzewski Day (Wrocław University of Science and Technology, Poland, 2023)
• The interplay between spectral theory and Lévy processes [pdf]
  Recent Developments in Stochastic Processes (Sofia, Bulgaria, 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)