Mean Square Estimates for Coefficients of Symmetric Power <Emphasis Type="Italic">L</Emphasis>-Functions

Let L(sym ^j f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that

$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$

and

$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$

     

Regularity results for non-autonomous functionals with $$\varvec{{L}\log {L}}$$-growth and Orlicz Sobolev coefficients

We study the regularity properties of local minimizers of non-autonomous convex integral functionals of the type

$$\begin{aligned} \mathcal {F}( u, \Omega )= \int _{\Omega } \! f(x,Du) \, \,dx, \end{aligned}$$

when the integrand f has almost linear growth with respect to the gradient variable and the dependence on the x-variable is controlled by a function which belongs to a suitable Orlicz Sobolev space.

     

An Example of an Unbounded Maximal Singular Operator

Suppose that an even integrable function Ω on the unit sphere S ¹ in R ² with mean value zero satisfies

$\mathop{\mathrm{essup}}\limits_{\xi\in \mathbf{S}^{1}}\biggl|\int_{\mathbf{S}^{1}}\Omega(\theta)\log\frac{1}{|\theta\cdot\xi|}\,d\theta\biggr|<+\infty,$

then the singular integral operator T _Ω given by convolution with the distribution p.v.?Ω(x/|x|)|x|^?2, initially defined on Schwartz functions, extends to an L ²-bounded operator. We construct examples of a function Ω satisfying the above conditions and of a continuous bounded integrable function f such that

$\limsup_{\epsilon\to 0^+}\biggl|\int_{\epsilon<|y|}\Omega(y/|y|)|y|^{-2}f(x-y)dy\biggr|=\infty\quad \hbox{a. e.}$

     

On some invariance of the quotient mean with respect to Makó–Páles means

Given a continuous strictly monotone function \(\varphi \) defined on an open real interval I and a probability measure \(\mu \) on the Borel subsets of [0, 1], the Makó–Páles mean is defined by

$$\begin{aligned} {\mathcal {M}}_{\varphi ,\mu }(x,y):=\varphi ^{-1}\left( \int ^1_0\varphi (tx+(1-t)y)\, d\mu (t)\right) ,\quad x,y\in I. \end{aligned}$$

Under some conditions on the functions \(\varphi \) and \(\psi \) defined on I, the quotient mean is given by

$$\begin{aligned} Q_{\varphi ,\psi }(x,y):=\left( \frac{\varphi }{\psi }\right) ^{-1}\left( \frac{\varphi (x)}{\psi (y)}\right) , \quad x,y\in I. \end{aligned}$$

In this paper, we study some invariance of the quotient mean with respect to Makó–Páles means.

     

$${\sigma_2}$$-Energy as a polyconvex functional on a space of self-maps of annuli in the multi-dimensional calculus of variations

Let \({\mathbb{X} \subset \mathbb {R}^n}\) be a bounded Lipschitz domain and consider the energy functional

$${{\mathbb F}_{\sigma_2}}[u; \mathbb{X}] := \int_\mathbb{X} {\mathbf F}(\nabla u) \, dx,$$

over the space of admissible maps

$${{\mathcal {A}_\varphi}(\mathbb{X}) :=\{u \in W^{1,4}(\mathbb{X}, {\mathbb{R}^n}) : {\rm det}\, \nabla u > 0\, {\rm for}\, {\mathcal {L}^n}{\rm -a.e. in}\, \mathbb{X}, u|_{\partial \mathbb{X}} =\varphi \}},$$

where the integrand \({{\mathbf F}\colon \mathbb M_{n\times n}\to \mathbb{R}}\) is quasiconvex and sufficiently regular. Here our attention is paid to the prototypical case when \({{\mathbf F}(\xi):=\frac{1}{2}\sigma_2(\xi)+\Phi(\det\xi)}\). The aim of this paper is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of \({\mathbb F_{\sigma_2}}\) and the relation it bares to the domain topology. In contrast, for constructing explicitly and directly solutions to the system of Euler–Lagrange equations associated to \({{\mathbb F}_{\sigma_2}}\), we use a topological class of maps referred to as generalised twists and relate the problem to extremising an associated energy on the compact Lie group \({\mathbf {SO}(n)}\). The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions amongst such maps whereas in odd dimensions this number reduces to one.

     

Some <Emphasis Type="Italic">q</Emphasis>-inequalities for Hausdorff operators

We calculate the sharp bounds for some q-analysis variants of Hausdorff type inequalities of the form

$$\int_0^{ + \infty } {{{\left( {\int_0^{ + \infty } {\frac{{\phi \left( t \right)}}{t}f\left( {\frac{x}{t}} \right){d_q}t} } \right)}^p}{d_q}x} \leqslant {C_\phi }\int_0^b {{f^p}\left( t \right)} {d_q}t$$

. As applications, we obtain several sharp q-analysis inequalities of the classical positive integral operators, including the Hardy operator and its adjoint operator, the Hilbert operator, and the Hardy-Littlewood-Pólya operator.

     

Numerical Radius Inequalities for Certain 2 × 2 Operator Matrices

We prove several numerical radius inequalities for certain 2 × 2 operator matrices. Among other inequalities, it is shown that if X, Y, Z, and W are bounded linear operators on a Hilbert space, then

$$w\left( \left[\begin{array}{cc} X &; Y \\ Z &; W \end{array} \right] \right) \geq \max \left(w(X),w(W),\frac{w(Y+Z)}{2},\frac{w(Y-Z)}{2}\right) $$

and

$$w\left( \left[\begin{array}{cc}X &; Y \\ Z &; W\end{array} \right] \right) \leq \max \left( w(X), w(W)\right)+\frac{w(Y+Z)+w(Y-Z)}{2}. $$

As an application of a special case of the second inequality, it is shown that

$$\frac{\left\Vert X\right\Vert }{2}+\frac{\left\vert \left\Vert\operatorname{Re}{X}\right\Vert -\frac{\left\Vert X\right\Vert}{2}\right\vert }{4}+\frac{ \left\vert \left\Vert \operatorname{Im}{X}\right\Vert -\frac{\left\Vert X\right\Vert}{2}\right\vert }{4} \leq w(X), $$

which is a considerable improvement of the classical inequality \({\frac{ \left\Vert X\right\Vert }{2}\leq w(X)}\) . Here w(·) and || · || are the numerical radius and the usual operator norm, respectively.

     

On the existence of periodic motions of the excited inverted pendulum by elementary methods

Using purely elementary methods, necessary and sufficient conditions are given for the existence of 2T-periodic and 4T-periodic solutions around the upper equilibrium of the mathematical pendulum when the suspension point is vibrating with period 2T. The equation of the motion is of the form

$$\ddot{\theta}-\frac{1}{l}(g+a(t)) \theta=0,$$

where l, g are constants and

$$a(t) := \begin{cases} A &\text{if } 2kT\leq t < (2k+1)T,\\ -A &\text{if } (2k+1)T\leq t < (2k+2)T,\end{cases}\quad (k=0,1,\dots);$$

A, T are positive constants. The exact stability zones for the upper equilibrium are presented.

     

Optimization of the control by elastic boundary forces at two ends of a string in an arbitrarily large time interval

We further develop the method, devised earlier by the authors, which permits finding closed-form expressions for the optimal controls by elastic boundary forces applied at two ends, x = 0 and x = l, of a string. In a sufficiently large time T, the controls should take the string vibration process, described by a generalized solution u(x, t) of the wave equation

$$u_{tt} (x,t) - u_{tt} (x,t) = 0,$$

from an arbitrary initial state

$$\{ u(x,0) = \varphi (x), u_t (x,0) = \psi (x)$$

to an arbitrary terminal state

$$\{ u(x,T) = \hat \varphi (x), u_t (x,T) = \hat \psi (x).$$

     

Blow Up Criteria for the Incompressible Nematic Liquid Crystal Flows

In this paper, we investigate blow up criteria for the local smooth solutions to the 3D incompressible nematic liquid crystal flows via the components of the gradient velocity field \(\nabla u\) and the gradient orientation field \(\nabla d\). More precisely, we show that \(0< T_{ \ast}<+\infty\) is the maximal time interval if and only if

$$\begin{aligned} & \int_{0}^{T_{\ast}} \bigl\Vert \Vert \partial_{i}u\Vert _{L_{x_{i}} ^{\gamma}} \bigr\Vert _{L_{x_{j}x_{k}}^{\alpha}}^{\beta}+ \|\nabla d\| _{L^{\infty}}^{\frac{8}{3}}\mathrm{d}t=\infty, \\ &\quad\text{ with } \frac{2}{\alpha}+\frac{2}{\beta}\leq\frac{3\alpha +2}{4\alpha}, \text{ and } 1\leq\gamma\leq\alpha,2< \alpha\leq+\infty, \end{aligned}$$

or

$$\begin{aligned} \int_{0}^{T_{\ast}}\|\partial_{3}u_{3} \|^{\beta}_{L^{\alpha}}+\| \nabla d\|^{\frac{8}{3}}_{L^{\infty}} \mathrm{d}t=\infty,\quad\text{with } \frac{3}{\alpha}+\frac{2}{\beta}\leq \frac{3(\alpha+2)}{4 \alpha}, \text{ and } 2< \alpha\leq\infty, \end{aligned}$$

where \(i,j,k\in\{1,2,3\}\), \(i\neq j\), \(i\neq k\), and \(j\neq k\).

     

Approximation orders of the unit in the <Emphasis Type="Italic">β</Emphasis>-dynamical systems

For any real number β > 1, let S _n (β) be the partial sum of the first n items of the β-expansion of 1. It was known that the approximation order of 1 by S _n (β) is β ^?n for Lebesgue almost all β > 1. We consider the size of the set of β > 1 for which 1 can be approximated with the other orders \({\beta^{-\varphi(n)}}\) , where \({\varphi}\) is a positive function defined on \({\mathbb N}\) . More precisely, the size of the sets

$$\left\{\beta\in \mathfrak{B}:\limsup_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$

and

$$\left\{\beta\in \mathfrak{B}:\liminf_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$

are determined, where \({\mathfrak{B}=\{ \beta>1:\beta \text{ is not a simple Parry number}\}}\) .

     

Geometric constant term functor(s)

We study the Eisenstein series and constant term functors in the framework of geometric theory of automorphic functions. Our main result says that for a parabolic \(P\subset G\) with Levi quotient M, the !-constant term functor

$$\begin{aligned}{\text {CT}}_!:{\text {D-mod}}({\text {Bun}}_G)\rightarrow {\text {D-mod}}({\text {Bun}}_M)\end{aligned}$$

is canonically isomorphic to the *-constant term functor

$$\begin{aligned} {\text {CT}}^-_*:{\text {D-mod}}({\text {Bun}}_G)\rightarrow {\text {D-mod}}({\text {Bun}}_M), \end{aligned}$$

taken with respect to the opposite parabolic \(P^-\).

     

A note on skew characters of symmetric groups

In this paper we establish the following estimate:

$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant \frac{{{c_T}}}{{{\varepsilon ^2}}}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right){M_{L{{\left( {\log L} \right)}^{1 + \varepsilon }}}}} \omega \left( x \right)dx$$

where ω ≥ 0, 0 < ε < 1 and Φ(t) = t(1 + log⁺(t)). This inequality relies upon the following sharp L ^p estimate:

$${\left\| {\left[ {b,T} \right]f} \right\|_{{L^p}\left( \omega \right)}} \leqslant {c_T}{\left( {p'} \right)^2}{p^2}{\left( {\frac{{p - 1}}{\delta }} \right)^{\frac{1}{{p'}}}}{\left\| b \right\|_{BMO}}{\left\| f \right\|_{{L^p}\left( {{M_{L{{\left( {{{\log }_L}} \right)}^{2p - 1 + {\delta ^\omega }}}}}} \right)}}$$

where 1 < p < ∞, ω ≥ 0 and 0 < δ < 1. As a consequence we recover the following estimate essentially contained in [18]:

$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant {c_T}{\left[ \omega \right]_{{A_\infty }}}{\left( {1 + {{\log }^ + }{{\left[ \omega \right]}_{{A_\infty }}}} \right)^2}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right)M} \omega \left( x \right)dx.$$

We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols.

     

The hole probability for Gaussian entire functions

Consider the random entire function

$f(z) = \sum\limits_{n = 0}^\infty {{\phi _n}{a_n}{z^n}} $

, where the ? _n are independent standard complex Gaussian coefficients, and the a _n are positive constants, which satisfy

$\mathop {\lim }\limits_{x \to \infty } {{\log {a_n}} \over n} = - \infty $

.

We study the probability P _H (r) that f has no zeroes in the disk{|z| < r} (hole probability). Assuming that the sequence a _n is logarithmically concave, we prove that

$\log {P_H}(r) = - S(r) + o(S(r))$

, where

$S(r) = 2 \cdot \sum\limits_{n:{a_n}{r^n} \ge 1} {\log ({a_n}{r^n})} $

, and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.

     

Dirichlet-to-Neumann semigroup with respect to a general second order eigenvalue problem

In this paper we study a Dirichlet-to-Neumann operator with respect to a second order elliptic operator with measurable coefficients, including first order terms, namely, the operator on \(L^2(\partial \Omega )\) given by \(\varphi \mapsto \partial _{\nu }u\) where u is a weak solution of

$$\begin{aligned} \left\{ \begin{aligned}&-\mathrm{div}\, (a\nabla u) +b\cdot \nabla u -\mathrm{div}\, (cu)+du =\lambda u \ \ \text {on}\ \Omega ,\\&u|_{\partial \Omega } =\varphi . \end{aligned} \right. \end{aligned}$$

Under suitable assumptions on the matrix-valued function a, on the vector fields b and c, and on the function d, we investigate positivity, sub-Markovianity, irreducibility and domination properties of the associated Dirichlet-to-Neumann semigroups.

     

Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

We study nonlinear elliptic equations in divergence form

$$\text {div }{\mathcal A}(x,Du)=\text {div } G.$$

When \({\mathcal A}\) has linear growth in D u, and assuming that \(x\mapsto {\mathcal A}(x,\xi )\) enjoys \(B^{\alpha }_{\frac {n}\alpha , q}\) smoothness, local well-posedness is found in \(B^{\alpha }_{p,q}\) for certain values of \(p\in [2,\frac {n}{\alpha })\) and \(q\in [1,\infty ]\). In the particular case \({\mathcal A}(x,\xi )=A(x)\xi \), G = 0 and \(A\in B^{\alpha }_{\frac {n}\alpha ,q}\), \(1\leq q\leq \infty \), we obtain \(Du\in B^{\alpha }_{p,q}\) for each \(p<\frac {n}\alpha \). Our main tool in the proof is a more general result, that holds also if \({\mathcal A}\) has growth s?1 in D u, 2 ≤ s ≤ n, and asserts local well-posedness in L ^q for each q > s, provided that \(x\mapsto {\mathcal A}(x,\xi )\) satisfies a locally uniform VMO condition.

     

On the uniform Kreiss resolvent condition

Let B be a Banach space with norm ‖ · ‖ and identity operator I. We prove that, for a bounded linear operator T in B, the strong Kreiss resolvent condition

$\parallel (T - \lambda I)^{ - k} \parallel \leqslant \frac{M}{{(|\lambda | - 1)^k }}, |\lambda | > 1,k = 1,2, \ldots ,$

implies the uniform Kreiss resolvent condition

$\left\| {\sum\limits_{k = 0}^n {\frac{{T^k }}{{\lambda ^{k + 1} }}} } \right\| \leqslant \frac{L}{{|\lambda | - 1}}, |\lambda | > 1, n = 0,1,2, \ldots .$

We establish that an operator T satisfies the uniform Kreiss resolvent condition if and only if so does the operator T ^m for each integer m ? 2.

     

Weak Type (1,1) Behavior for the Maximal Operator with <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Dini Kernel

Let M _Ω be the maximal operator with homogeneous kernel Ω. In the present paper, we show that if Ω satisfies the L ¹-Dini condition on ?? ^n?1, then the following weak type (1,1) behaviors

$$\lim\limits _{\lambda \rightarrow 0_{+}}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})=\frac {1}{n} \|\Omega \|_{1} \|f\|_{1},$$

$$\sup\limits_{\lambda >0}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})\lesssim {\bigg ((\log n)\|\Omega \|_{1}+{\int }_{0}^{1/n}\frac {\tilde {\omega }_{1}(\delta )}{\delta }d\delta \bigg )}\|f\|_{1}$$

hold for the maximal operator M _Ω and \(f\in L^{1}(\mathbb {R}^{n})\), here \(\tilde {\omega }_{1}\) denotes the L ¹ integral modulus of continuity of Ω defined by translation in \(\mathbb {R}^{n}\).     

Strong isochronism of a center and a focus for systems with homogeneous nonlinearities

We suggest a new approach to studying the isochronism of the system

${{dx} \mathord{\left/ {\vphantom {{dx} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - y + p_n (x,y),{{dy} \mathord{\left/ {\vphantom {{dy} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = x + q_n (x,y),$

where p _n and q _n are homogeneous polynomials of degree n. This approach is based on the normal form

${{dX} \mathord{\left/ {\vphantom {{dX} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - Y + XS(X,Y),{{dY} \mathord{\left/ {\vphantom {{dY} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = X + YS(X,Y)$

and its analog in polar coordinates. We prove a theorem on sufficient conditions for the strong isochronism of a center and a focus for the reduced system and obtain examples of centers with strong isochronism of degrees n = 4, 5. The present paper is the first to give examples of foci with strong isochronism for the system in question.

     

Combinatorial Properties of Rogers-Ramanujan Type Identities Arising from Hall-Littlewood Polynomials

Here we consider the q-series coming from the Hall-Littlewood polynomials,

$$\begin{array}{l}{R_{v} (a, b; q) = {\sum_{\mathop {\lambda}\limits_{\lambda_1 \leq a}}} q^{c | \lambda |} P_{2\lambda}(1, q, q^{2}, \ldots ; q^{2b+d}).}\end{array}$$

These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruence

$$p(5n+4) \equiv 0\quad ({\rm mod}\, 5).$$