A functional equation and its application to the characterization of gamma distributions

The functional equation $$ f\left(x\right)g\left(y\right)=p\left(x+y\right)q\left(\frac{x}{y} \right) $$ is investigated for almost all ${\left(x,\,y\right)\in\mathbb{R}^{2}_{+}}$ and for the measurable functions ${f,\,g,\,p,\,q:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}}$ . This equation is related to the Lukács characterization of gamma distribution.     

Local estimates for parabolic equations with nonlinear gradient terms

In this paper we deal with local estimates for parabolic problems in ${\mathbb{R}^N}$ with absorbing first order terms, whose model is $$\left\{\begin{array}{l@{\quad}l}u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{\rm in}\, (0,T) \times \mathbb{R}^N\,,\\u(0,x)= u_0 (x) &{\rm in}\, \mathbb{R}^N \,,\quad\end{array}\right.$$ where ${T >0 , \, N\geq 2,\, 1 < q \leq 2,\, f(t,x)\in L^1\left( 0,T; L^1_{\rm loc} \left(\mathbb{R}^N\right)\right)}$ and ${u_0\in L^1_{\rm loc}\left(\mathbb{R}^{N}\right)}$ .     

Existence and extremal solutions of parabolic variational–hemivariational inequalities

We consider quasilinear parabolic variational–hemivariational inequalities in a cylindrical domain $Q=\Omega \times (0,\tau )$ of the form $$\begin{aligned} u\in K:\ \langle u_t+Au, v-u\rangle +\int _Q j^o(x,t, u;v-u)\,dxdt\ge 0,\ \ \forall \ v\in K, \end{aligned}$$ where $K\subset X_0=L^p(0,\tau ;W_0^{1,p}(\Omega ))$ is some closed and convex subset, $A$ is a time-dependent quasilinear elliptic operator, and $s\mapsto j(\cdot ,\cdot ,s)$ is assumed to be locally Lipschitz with $(s,r)\mapsto j^o(x,t, s;r)$ denoting its generalized directional derivative at $s$ in the direction $r$ . The main goal of this paper is threefold: first, an existence and comparison principle is proved; second, the existence of extremal solutions within some sector of appropriately defined sub-supersolutions is shown; third, the equivalence of the above parabolic variational–hemivariational inequality with an associated multi-valued parabolic variational inequality of the form $$\begin{aligned} u\in K:\ \langle u_t+Au, v-u\rangle +\int _Q \eta \, (v-u)\,dxdt\ge 0,\ \ \forall \ v\in K \end{aligned}$$ with $\eta (x,t)\in \partial j(x,t, u(x,t))$ is established, where $s\mapsto \partial j(x,t, s)$ denotes Clarke’s generalized gradient of the locally Lipschitz function $s\mapsto j(\cdot ,\cdot ,s)$ .     

Inequalities for the harmonic numbers

We present various inequalities for the harmonic numbers defined by ${H_n=1+1/2 +\ldots +1/n\,(n\in{\bf N})}$ . One of our results states that we have for all integers n ???2: $$\alpha \, \frac{\log(\log{n}+\gamma)}{n^2} \leq H_n^{1/n} -H_{n+1}^{1/(n+1)} < \beta \, \frac{\log(\log{n}+\gamma)}{n^2}$$ with the best possible constant factors $$\alpha= \frac{6 \sqrt{6}-2 \sqrt[3]{396}}{3 \log(\log{2}+\gamma)}=0.0140\ldots \quad\mbox{and} \quad\beta=1.$$ Here, ?? denotes Euler??s constant.     

Recurrence in β-expansion over formal Laurent series

In this paper, we study the quantitative recurrence and hitting sets of β-transformation T _β on the unit disk I of formal Laurent series field $$E_\phi:= \{x\in I: \|T_\beta^nx - x\| < \|\beta\|^{-\phi(n)}\,\,\,{\rm infinitely\,often}\}$$ and $$F_\phi:=\{x\in I: \|T_\beta^nx-x_0\|<\|\beta\|^{-\phi(n)}\,\,\,{\rm infinitely\,often}\},$$ where x ₀ is any fixed point in I and ${\phi}$ is any positive function defined on ${\mathbb{N}}$ with ${\phi(n)\to\infty}$ as n → ∞. We completely determine the Hausdorff dimensions of these sets: $$\dim_{\rm H} E_{\phi}=\dim_{\rm H}F_\phi=\frac{1}{1+\liminf\limits_{n\to\infty}\frac{\phi (n)}{n}}.$$      

Integrability of vector-valued lacunary series with applications to function spaces

Let ${\mathcal L(r) = \sum_{n=0}^\infty a_nr^{\lambda_n}}$ be a lacunary series converging for 0 <  r < 1, with coefficients in a quasinormed space. It is proved that $$\int_0^1 F(1-r,\|\mathcal L(r)\|)(1-r)^{-1}\,{\rm d}r < \infty $$ if and only if $$ \sum_{n=0}^\infty F(1/\lambda_n,\|a_n\|) < \infty, $$ where F is a “normal function” of two variables. In the case when p ≥ 1 and F(x, y) =  x y ^p , this reduces to a theorem of Gurariy and Matsaev. As an application we prove that if ${f(r\zeta) = \sum_{n=0}^\infty r^{\lambda_n}f_{\lambda_n}(\zeta)}$ is a function harmonic in the unit ball of ${\mathbb R^N,}$ then $$\int_0^1M_p^q(r,f)(1-r)^{q\alpha-1} \,{\rm d}r <\infty\quad (p,\,q,\,\alpha >0 ) $$ if and only if $$\sum_{n=0}^\infty \|f_{\lambda_n} \|^q_{L^p(\partial B_N)}(1/\lambda_n)^{q\alpha} <\infty. $$      

Stability of the Derivative of a Canonical Product

With each sequence \(\alpha =(\alpha _n)_{n\in \mathbb{N }}\) of pairwise distinct and non-zero points which are such that the canonical product $$\begin{aligned} P_\alpha (z) := \lim _{r\rightarrow \infty }\prod _{|\alpha _n|\le r}\left( 1-\frac{z}{\alpha _n}\right) \end{aligned}$$ converges, the sequence $$\begin{aligned} \alpha ^{\prime } := \bigl (P_\alpha ^{\prime }(\alpha _n)\bigr )_{n\in \mathbb{N }} \end{aligned}$$ is associated. We give conditions on the difference \(\beta -\alpha \) of two sequences which ensure that \(\beta ^{\prime }\) and \(\alpha ^{\prime }\) are comparable in the sense that $$\begin{aligned} \exists \,c,C>0:\quad c|\alpha ^{\prime }_n| \le |\beta ^{\prime }_n| \le C|\alpha ^{\prime }_n|, \quad n\in \mathbb{N }. \end{aligned}$$ The values \(\alpha ^{\prime }_n\) play an important role in various contexts. As a selection of applications we present: an inverse spectral problem, a class of entire functions and a continuation problem.     

Exponential decay estimates for singular integral operators

The following subexponential estimate for commutators is proved $$\begin{aligned} |\{x\in Q: |[b,T]f(x)|>tM^2f(x)\}|\le c\,e^{-\sqrt{\alpha \, t\Vert b\Vert _{BMO}}}\, |Q|, \qquad t>0. \end{aligned}$$ where $c$ and $\alpha $ are absolute constants, $T$ is a Calderón–Zygmund operator, $M$ is the Hardy Littlewood maximal function and $f$ is any function supported on the cube $Q\subset \mathbb{R }^n$ . We also obtain that $$\begin{aligned} |\{x\in Q: |f(x)-m_f(Q)|>tM_{\lambda _n;Q}^\#(f)(x) \}|\le c\, e^{-\alpha \,t}|Q|,\qquad t>0, \end{aligned}$$ where $m_f(Q)$ is the median value of $f$ on the cube $Q$ and $M_{\lambda _n;Q}^\#$ is Strömberg’s local sharp maximal function with $\lambda _n=2^{-n-2}$ . As a consequence we derive Karagulyan’s estimate: $$\begin{aligned} |\{x\in Q: |Tf(x)|> tMf(x)\}|\le c\, e^{-c\, t}\,|Q|\qquad t>0, \end{aligned}$$ from [21] improving Buckley’s theorem [3]. A completely different approach is used based on a combination of “Lerner’s formula” with some special weighted estimates of Coifman–Fefferman type obtained via Rubio de Francia’s algorithm. The method is flexible enough to derive similar estimates for other operators such as multilinear Calderón–Zygmund operators, dyadic and continuous square functions and vector valued extensions of both maximal functions and Calderón–Zygmund operators. In each case, $M$ will be replaced by a suitable maximal operator.     

An Osgood type regularity criterion for the liquid crystal flows

In this paper, we prove an Osgood type regularity criterion for the model of liquid crystals, which says that the condition $$\sup_{2 \leq q< \infty} \int \nolimits_0^T \frac{\| \bar{S}_{q} \nabla {\bf u}(t)\|_{L^\infty}}{q \, {\rm \ln} \, q} {\rm d} t<\infty$$ implies the smoothness of the solution. Here, ${{\bar S_q=\sum\nolimits_{k=-q}^q \dot {\triangle}_k}}$ with ${\dot{\triangle}_k}$ being the frequency localization operator.     

A quadratic quasi-linear Klein–Gordon equation in two space dimensions

We consider the quasi-linear Klein–Gordon equations in two space dimensions $$\left(\partial_{t}^{2} - \Delta + 1\right) u=\mathcal{N} (u)$$ in ${(t, x) \in \mathbf{R} \times \mathbf{R}^{2}}$ with a quadratic nonlinearity ${\mathcal{N} (u)}$ , which is linear with respect to the second-order derivatives of unknown functions.     

Sharpening the Norm Bound in the Subspace Perturbation Theory

Let $A$ be a (possibly unbounded) self-adjoint operator on a separable Hilbert space $\mathfrak H .$ Assume that $\sigma $ is an isolated component of the spectrum of $A$ , that is, $\mathrm{dist}(\sigma ,\Sigma )=d>0$ where $\Sigma =\mathrm spec (A)\setminus \sigma .$ Suppose that $V$ is a bounded self-adjoint operator on $\mathfrak H $ such that $\Vert V\Vert <d/2$ and let $L=A+V$ , $\mathrm{Dom }(L)=\mathrm{Dom }(A).$ Denote by $P$ the spectral projection of $A$ associated with the spectral set $\sigma $ and let $Q$ be the spectral projection of $L$ corresponding to the closed $\Vert V\Vert $ -neighborhood of $\sigma .$ Introducing the sequence $$\begin{aligned} \varkappa _n=\frac{1}{2}\left(1-\frac{(\pi ^2-4)^n}{(\pi ^2+4)^n}\right), \quad n\in \{0\}\cup {\mathbb N }, \end{aligned}$$ we prove that the following bound holds: $$\begin{aligned} \arcsin (\Vert P-Q\Vert )\le M_\star \left(\frac{\Vert V\Vert }{d}\right), \end{aligned}$$ where the estimating function $M_\star (x)$ , $x\in \bigl [0,\frac{1}{2}\bigr )$ , is given by $$\begin{aligned} M_\star (x)=\frac{1}{2}\,\,n_{_\#}(x)\,\arcsin \left(\frac{4\pi }{\pi ^2+4}\right) +\frac{1}{2}\,\arcsin \left(\frac{\pi ( x-\varkappa _{n_{_\#}(x)})}{1-2\varkappa _{n_{_\#}(x)})}\right), \end{aligned}$$ with $n_{_\#}(x)=\max \left\{ n\,\bigr |\,\,n\in \{0\}\cup {\mathbb N }\,, \varkappa _n\le x\right\} $ . The bound obtained is essentially stronger than the previously known estimates for $\Vert P-Q\Vert .$ Furthermore, this bound ensures that $\Vert P-Q\Vert <1$ and, thus, that the spectral subspaces $\mathrm{Ran }(P)$ and $\mathrm{Ran }(Q)$ are in the acute-angle case whenever $\Vert V\Vert <c_\star \,d$ , where $$\begin{aligned} c_\star =16\,\,\frac{\pi ^6-2\pi ^4+32\pi ^2-32}{(\pi ^2+4)^4}=0.454169\ldots . \end{aligned}$$ Our proof of the above results is based on using the triangle inequality for the maximal angle between subspaces and on employing the a priori generic $\sin 2\theta $ estimate for the variation of a spectral subspace. As an example, the boundedly perturbed quantum harmonic oscillator is discussed.     

Some new refined Hardy type inequalities with general kernels and measures

We state and prove some new refined Hardy type inequalities using the notation of superquadratic and subquadratic functions with an integral operator A _k defined by $$ A_kf(x):=\frac{1}{K(x)} \int\limits_{\Omega_2} k(x,y)f(y)d\mu_2(y), $$ where ${k: \Omega_1 \times \Omega_2 \to \mathbb{R}}$ is a general nonnegative kernel, (Ω₁, μ ₁) and (Ω₂, μ ₂) are measure spaces and $$ K(x):=\int\limits_{\Omega_2} k(x,y)d\mu_2(y), \, x \in \Omega_1. $$ The relations to other results of this type are discussed and, in particular, some new integral identities of independent interest are obtained.     

Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then

1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B _n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B _n . Among others, we obtain
2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.

     

Quasiconvex variational functionals in Orlicz–Sobolev spaces

We prove a C ^1,α partial regularity result for minimizers of variational integrals of the type $$ J[u]:=\int\limits_\Omega f(\nabla u){\rm d}x, \, \, u:\Omega\subset \mathbb{R}^n \to \mathbb{R}^N, $$ where the integrand f is strictly quasiconvex and satisfies suitable growth conditions in terms of Young functions.     

Sharp weak type estimates for Riesz transforms

Let $d$ be a given positive integer and let $\{R_j\}_{j=1}^d$ denote the collection of Riesz transforms on $\mathbb {R}^d$ . For $1<p<\infty $ , we determine the best constant $C_p$ such that the following holds. For any locally integrable function $f$ on $\mathbb {R}^d$ and any $j\in \{1,\,2,\,\ldots ,\,d\}$ , $$\begin{aligned} ||(R_jf)_+||_{L^{p,\infty }(\mathbb {R}^d)}\le C_p||f||_{L^{p,\infty }(\mathbb {R}^d)}. \end{aligned}$$ A related statement for Riesz transforms on spheres is also established. The proofs exploit Gundy–Varopoulos representation of Riesz transforms and appropriate inequality for orthogonal martingales.     

Conical stochastic maximal L^p-regularity for 1\leqslant p<\infty

Let \(A = -\mathrm{div} \,a(\cdot ) \nabla \) be a second order divergence form elliptic operator on \({\mathbb R}^n\) with bounded measurable real-valued coefficients and let \(W\) be a cylindrical Brownian motion in a Hilbert space \(H\) . Our main result implies that the stochastic convolution process $$\begin{aligned} u(t) = \int _0^t e^{-(t-s)A}g(s)\,dW(s), \quad t\geqslant 0, \end{aligned}$$ satisfies, for all \(1\leqslant p<\infty \) , a conical maximal \(L^p\) -regularity estimate $$\begin{aligned} {\mathbb E}\Vert \nabla u \Vert _{ T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n)}^p \leqslant C_p^p {\mathbb E}\Vert g \Vert _{ T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n;H)}^p. \end{aligned}$$ Here, \(T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n)\) and \(T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n;H)\) are the parabolic tent spaces of real-valued and \(H\) -valued functions, respectively. This contrasts with Krylov’s maximal \(L^p\) -regularity estimate $$\begin{aligned} {\mathbb E}\Vert \nabla u \Vert _{L^p({\mathbb R}_+;L^2({\mathbb R}^n;{\mathbb R}^n))}^p \leqslant C^p {\mathbb E}\Vert g \Vert _{L^p({\mathbb R}_+;L^2({\mathbb R}^n;H))}^p \end{aligned}$$ which is known to hold only for \(2\leqslant p<\infty \) , even when \(A = -\Delta \) and \(H = {\mathbb R}\) . The proof is based on an \(L^2\) -estimate and extrapolation arguments which use the fact that \(A\) satisfies suitable off-diagonal bounds. Our results are applied to obtain conical stochastic maximal \(L^p\) -regularity for a class of nonlinear SPDEs with rough initial data.     

New upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane

In the projective planes PG(2, q), more than 1230 new small complete arcs are obtained for ${q \leq 13627}$ and ${q \in G}$ where G is a set of 38 values in the range 13687,..., 45893; also, ${2^{18} \in G}$ . This implies new upper bounds on the smallest size t ₂(2, q) of a complete arc in PG(2, q). From the new bounds it follows that $$t_{2}(2, q) < 4.5\sqrt{q} \, {\rm for} \, q \leq 2647$$ and q = 2659,2663,2683,2693,2753,2801. Also, $$t_{2}(2, q) < 4.8\sqrt{q} \, {\rm for} \, q \leq 5419$$ and q = 5441,5443,5449,5471,5477,5479,5483,5501,5521. Moreover, $$t_{2}(2, q) < 5\sqrt{q} \, {\rm for} \, q \leq 9497$$ and q = 9539,9587,9613,9623,9649,9689,9923,9973. Finally, $$t_{2}(2, q) <5 .15\sqrt{q} \, {\rm for} \, q \leq 13627$$ and q = 13687,13697,13711,14009. Using the new arcs it is shown that $$t_{2}(2, q) < \sqrt{q}\ln^{0.73}q {\rm for} 109 \leq q \leq 13627\, {\rm and}\, q \in G.$$ Also, as q grows, the positive difference ${\sqrt{q}\ln^{0.73} q-\overline{t}_{2}(2, q)}$ has a tendency to increase whereas the ratio ${\overline{t}_{2}(2, q)/(\sqrt{q}\ln^{0.73} q)}$ tends to decrease. Here ${\overline{t}_{2}(2, q)}$ is the smallest known size of a complete arc in PG(2,q). These properties allow us to conjecture that the estimate ${t_{2}(2,q) < \sqrt{q}\ln ^{0.73}q}$ holds for all ${q \geq 109.}$ The new upper bounds are obtained by finding new small complete arcs in PG(2,q) with the help of a computer search using randomized greedy algorithms. Finally, new forms of the upper bound on t ₂(2,q) are proposed.     

Laplace Equation on a Domain With a Cuspidal Point in Little Hölder Spaces

In this paper, we give new results about existence, uniqueness and regularity properties for solutions of Laplace equation $$\Delta u = h \quad {\rm in} \, \Omega$$ where Ω is a cusp domain. We impose nonhomogeneous Dirichlet conditions on some part of ?Ω. The second member h will be taken in the little Hölder space ${h^{2 \sigma}(\bar{\Omega})}$ with ${\sigma \, \in \, ]0, \, 1/2[}$ . Our approach is based essentially on the study of an abstract elliptic differential equation set in an unbounded domain. We will use the continuous interpolation spaces and the generalized analytic semigroup theory.     

An extension of one direction in Marty’s normality criterion

We prove the following extension of one direction in Marty’s theorem: If $k$ is a natural number, $\alpha >1$ and $\mathcal{F }$ is a family of functions meromorphic on a domain $D$ all of whose poles have multiplicity at least $\frac{k}{\alpha -1}$ , then the normality of $\mathcal{F }$ implies that the family $$\begin{aligned} \left\{ \frac{|f^{(k)}|}{1+|f|^\alpha }\,:\, f\in \mathcal{F }\right\} \end{aligned}$$ is locally uniformly bounded.     

On the asymptotic behavior of certain solutions of the Dirichlet problem for the equation -\Delta _p u=\lambda |u|^{q-2}u

Let $p>1$ . We study the behavior of certain positive and nodal solutions of the problem $$\begin{aligned} \left\{ \,\, \begin{array}{lll} -\Delta _p u=\lambda |u|^{q-2}u \ \ &{}\mathrm{in} \ \ &{}{\varOmega } \\ u=0 &{}\mathrm{in} \ \ &{}\partial {\varOmega } \end{array}\right. \end{aligned}$$ on varying of the parameters $\lambda >0$ and $q>1$ .