Interlacing properties and bounds for zeros of $${}_2\phi _1$$ hypergeometric and little q-Jacobi polynomials

Let \(\displaystyle \{p_n\}_{n=0}^{\infty }\), where \(p_n\) is a polynomial of degree n, be a sequence of polynomials orthogonal with respect to a positive probability measure. If \(x_{1,n} < \cdots < x_{n,n}\) denotes the zeros of \(p_n\) while \(x_{1,n-1} < \cdots < x_{n-1,n-1}\) are the zeros of \(p_{n-1}\), the inequality

$$\begin{aligned} x_{1,n} < x_{1,n-1} < x_{2,n} < \cdots < x_{n-1,n}< x_{n-1,n-1}< x_{n,n}, \end{aligned}$$

known as the interlacing property, is satisfied. We use a consequence of a generalised version of Markov’s monotonicity results to investigate interlacing properties of zeros of contiguous basic hypergeometric polynomials associated with little q-Jacobi polynomials and determine inequalities for extreme zeros of the above two polynomials. It is observed that the new bounds which are obtained in this paper give more precise upper bounds for the smallest zero of little q-Jacobi polynomials, improving previously known results by Driver and Jordaan (Math Model Nat Phenom 8(1):48–59, 2013), and in some cases, those by Gupta and Muldoon (J Inequal Pure Appl Math 8(1):7, 2007). Numerical examples are given in order to illustrate the accuracy of our bounds.

     

Large conformal metrics with prescribed sign-changing Gauss curvature

Let \((M,g)\) be a two dimensional compact Riemannian manifold of genus \(g(M)>1\). Let \(f\) be a smooth function on \(M\) such that

$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$

Let \(p_1,\ldots ,p_n\) be any set of points at which \(f(p_i)=0\) and \(D^2f(p_i)\) is non-singular. We prove that for all sufficiently small \(\lambda >0\) there exists a family of “bubbling” conformal metrics \(g_\lambda =e^{u_\lambda }g\) such that their Gauss curvature is given by the sign-changing function \(K_{g_\lambda }=-f+\lambda ^2\). Moreover, the family \(u_\lambda \) satisfies

$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$

and

$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$

where \(\delta _{p}\) designates Dirac mass at the point \(p\).

     

Uniform Bounds on the Relative Error in the Approximation of Upper Quantiles for Sums of Arbitrary Independent Random Variables

Fix any \(n\ge 1\). Let \(\tilde{X}_1,\ldots ,\tilde{X}_n\) be independent random variables. For each \(1\le j \le n\), \(\tilde{X}_j\) is transformed in a canonical manner into a random variable \(X_j\). The \(X_j\) inherit independence from the \(\tilde{X}_j\). Let \(s_y\) and \(s_y^*\) denote the upper \(\frac{1}{y}{\underline{\text{ th }}}\) quantile of \(S_n=\sum _{j=1}^nX_j\) and \(S^*_n=\sup _{1\le k\le n}S_k\), respectively. We construct a computable quantity \(\underline{Q}_y\) based on the marginal distributions of \(X_1,\ldots ,X_n\) to produce upper and lower bounds for \(s_y\) and \(s_y^*\). We prove that for \(y\ge 8\)

$$\begin{aligned} 6^{-1} \gamma _{3y/16}\underline{Q}_{3y/16}\le s^*_{y}\le \underline{Q}_y \end{aligned}$$

where

$$\begin{aligned} \gamma _y=\frac{1}{2w_y+1} \end{aligned}$$

and \(w_y\) is the unique solution of

$$\begin{aligned} \Big (\frac{w_y}{e\ln (\frac{y}{y-2})}\Big )^{w_y}=2y-4 \end{aligned}$$

for \(w_y>\ln (\frac{y}{y-2})\), and for \(y\ge 37\)

$$\begin{aligned} \frac{1}{9}\gamma _{u(y)}\underline{Q}_{u(y)}<s_y \le \underline{Q}_y \end{aligned}$$

where

$$\begin{aligned} u(y)=\frac{3y}{32} \left( 1+\sqrt{1-\frac{64}{3y}}\right) . \end{aligned}$$

The distribution of \(S_n\) is approximately centered around zero in that \(P(S_n\ge 0) \ge \frac{1}{18}\) and \(P(S_n\le 0)\ge \frac{1}{65}\). The results extend to \(n=\infty \) if and only if for some (hence all) \(a>0\)

$$\begin{aligned} \sum _{j=1}^{\infty }E\{(\tilde{X}_j-m_j)^2\wedge a^2\}<\infty . \end{aligned}$$

(1)

     

Congruences modulo $$10^3$$ for Euler numbers

For a non-negative integer \(n\), let \(E_n\) be the \(n\) th Euler number. In this note, for any positive integer \(n\), we prove the following congruences:

$$\begin{aligned} {\left\{ \begin{array}{ll} E_{4n} \equiv 380n-375 \pmod {10^3}, \\ E_{4n+2} \equiv -460n+399 \pmod {10^3}. \end{array}\right. } \end{aligned}$$

Our proof is based on induction on \(n\) and elementary direct calculations.

     

Congruences for 7 and 49-regular partitions modulo powers of 7

Let \(b_{k}(n)\) denote the number of k-regular partitions of n. In this paper, we prove Ramanujan-type congruences modulo powers of 7 for \(b_{7}(n)\) and \(b_{49}(n)\). For example, for all \(j\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{7}\Bigg (7^{2j-1}n+\frac{3\cdot 7^{2j-1}-1}{4}\Bigg )\equiv 0\pmod {7^{j}} \end{aligned}$$

and

$$\begin{aligned} b_{49}\Big (7^{j}n+7^{j}-2\Big )\equiv 0\pmod {7^{j}}. \end{aligned}$$

     

On a problem of Bourgain concerning the $$L_p$$ norms of exponential sums

For \(n \ge 1\) let

$$\begin{aligned} {\mathcal {A}}_n := \bigg \{ P: P(z) = \sum \limits _{j=1}^n{z^{k_j}}: 0 \le k_1 < k_2 < \cdots < k_n, k_j \in {\mathbb {Z}} \bigg \}, \end{aligned}$$

that is, \({\mathcal {A}}_n\) is the collection of all sums of \(n\) distinct monomials. These polynomials are also called Newman polynomials. Let

$$\begin{aligned} M_{p}(Q) := \left( \int _{0}^{1}{\left| Q(e^{i2\pi t}) \right| ^p\,dt} \right) ^{1/p}, \qquad p > 0. \end{aligned}$$

We define

$$\begin{aligned} S_{n,p} := \sup _{Q \in {\mathcal {A}}_n}{\frac{M_p(Q)}{\sqrt{n}}} \qquad \text{ and } \qquad S_p := \liminf _{n \rightarrow \infty }{S_{n,p}} \le \Sigma _p := \limsup _{n \rightarrow \infty }{S_{n,p}}. \end{aligned}$$

We show that

$$\begin{aligned} \Sigma _p \ge \Gamma (1+p/2)^{1/p}, \qquad p \in (0,2). \end{aligned}$$

The special case \(p=1\) recaptures a recent result of Aistleitner [1], the best known lower bound for \(\Sigma _1\).

     

Optimal bounds for convergence of expected spectral distributions to the semi-circular law

Let \(\mathbf {X}=(X_{jk})_{j,k=1}^n\) denote a Hermitian random matrix with entries \(X_{jk}\), which are independent for \(1\le j\le k\le n\). We consider the rate of convergence of the empirical spectral distribution function of the matrix \(\mathbf {X}\) to the semi-circular law assuming that \(\mathbf{E}X_{jk}=0\), \(\mathbf{E}X_{jk}^2=1\) and that

$$\begin{aligned} \sup _{n\ge 1}\sup _{1\le j,k\le n}\mathbf{E}|X_{jk}|^4=:\mu _4<\infty , \end{aligned}$$

and

$$\begin{aligned} \sup _{1\le j,k\le n}|X_{jk}|\le D_0n^{\frac{1}{4}}. \end{aligned}$$

By means of a recursion argument it is shown that the Kolmogorov distance between the expected spectral distribution of the Wigner matrix \(\mathbf {W}=\frac{1}{\sqrt{n}}\mathbf {X}\) and the semicircular law is of order \(O(n^{-1})\).

     

Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Let \(\Phi _{n}(x)=e^x-\sum _{j=0}^{n-2}\frac{x^j}{j!}\) and \(\alpha _{n} =n\omega _{n-1}^{\frac{1}{n-1}}\) be the sharp constant in Moser’s inequality (where \(\omega _{n-1}\) is the area of the surface of the unit \(n\)-ball in \(\mathbb {R}^n\)), and \(dV\) be the volume element on the \(n\)-dimensional hyperbolic space \((\mathbb {H}^n, g)\) (\(n\ge {2}\)). In this paper, we establish the following sharp Moser–Trudinger type inequalities with the exact growth condition on \(\mathbb {H}^n\):

For any \(u\in {W^{1,n}(\mathbb {H}^n)}\) satisfying \(\Vert \nabla _{g}u\Vert _{n}\le {1}\), there exists a constant \(C(n)>0\) such that

$$\begin{aligned} \int _{\mathbb {H}^n}\frac{\Phi _{n}(\alpha _{n}|u|^{\frac{n}{n-1}})}{(1+|u|)^{\frac{n}{n-1}}}dV \le {C(n)\Vert u\Vert _{L^n}^{n}}. \end{aligned}$$

The power \(\frac{n}{n-1}\) and the constant \(\alpha _{n}\) are optimal in the following senses:

1. (i)
   If the power \(\frac{n}{n-1}\) in the denominator is replaced by any \(p<\frac{n}{n-1}\), then there exists a sequence of functions \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha _{n}(|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV \rightarrow {\infty }. \end{aligned}$$
    
2. (ii)
   If \(\alpha >\alpha _{n}\), then there exists a sequence of function \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha (|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV\rightarrow {\infty }, \end{aligned}$$

   for any \(p\ge {0}\).
    

This result sharpens the earlier work of the authors Lu and Tang (Adv Nonlinear Stud 13(4):1035–1052, 2013) on best constants for the Moser–Trudinger inequalities on hyperbolic spaces.

     

Congruences for 5-regular partitions modulo powers of 5

Let \(b_{5}(n)\) denote the number of 5-regular partitions of n. We find the generating functions of \(b_{5}(An+B)\) for some special pairs of integers (A, B). Moreover, we obtain infinite families of congruences for \(b_{5}(n)\) modulo powers of 5. For example, for any integers \(k\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{5}\left( 5^{2k-1}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}) \end{aligned}$$

and

$$\begin{aligned} b_{5}\left( 5^{2k}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}). \end{aligned}$$

     

Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions,

$$\begin{aligned} Q^\Omega _\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text { on } \partial \Omega , \end{aligned}$$

in a class of bounded smooth domains \(\Omega \in \mathbb {R}^\nu \); here \(n\) is the outward unit normal and \(\alpha >0\) is a constant. We show that for each \(j\in \mathbb {N}\) and \(\alpha \rightarrow +\infty \), the \(j\)th eigenvalue \(E_j(Q^\Omega _\alpha )\) has the asymptotics

$$\begin{aligned} E_j(Q^\Omega _\alpha )=-\alpha ^2 -(\nu -1)H_\mathrm {max}(\Omega )\,\alpha +{\mathcal O}(\alpha ^{2/3}), \end{aligned}$$

where \(H_\mathrm {max}(\Omega )\) is the maximum mean curvature at \(\partial \Omega \). The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of \(H_\mathrm {max}\). In particular, we show that the ball is the strict minimizer of \(H_\mathrm {max}\) among the smooth star-shaped domains of a given volume, which leads to the following result: if \(B\) is a ball and \(\Omega \) is any other star-shaped smooth domain of the same volume, then for any fixed \(j\in \mathbb {N}\) we have \(E_j(Q^B_\alpha )>E_j(Q^\Omega _\alpha )\) for large \(\alpha \). An open question concerning a larger class of domains is formulated.

     

Contributions to a conjecture of Mueller and Schmidt on Thue inequalities

Let \(F(X,Y)=\sum \nolimits _{i=0}^sa_iX^{r_i}Y^{r-r_i}\in {\mathbb {Z}}[X,Y]\) be a form of degree \(r=r_s\ge 3\), irreducible over \({\mathbb {Q}}\) and having at most \(s+1\) non-zero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality

$$\begin{aligned} |F(X,Y)|\le h \end{aligned}$$

is \(\ll s^2h^{2/r}(1+\log h^{1/r})\). They conjectured that \(s^2\) may be replaced by s. Let

$$\begin{aligned} \Psi = \max _{0\le i\le s} \max \left( \sum _{w=0}^{i-1} \frac{1}{r_i-r_w},\sum _{w= i+1}^{s}\frac{1}{r_w-r_i}\right) . \end{aligned}$$

Then we show that \(s^2\) may be replaced by \(\max (s\log ^3s, se^{\Psi })\). We also show that if \(|a_0|=|a_s|\) and \(|a_i|\le |a_0|\) for \(1\le i\le s-1\), then \(s^2\) may be replaced by \(s\log ^{3/2}s\). In particular, this is true if \(a_i\in \{-1,1\}\).

     

On the Dilation of Truncated Toeplitz Operators

An operator \(S_{\varphi ,\psi }^{u}\in \mathcal {L}(L^2)\) is called the dilation of a truncated Toeplitz operator if for two symbols \(\varphi ,\psi \in L^{\infty }\) and an inner function u,

$$\begin{aligned} S_{\varphi ,\psi }^{u}f=\varphi P_uf+\psi Q_uf \end{aligned}$$

holds for \(f\in {L}^{2}\) where \(P_{u}\) denotes the orthogonal projection of \(L^2\) onto the model space \(\mathcal { K}_{u}^2=H^2{\ominus }{{u}H^2}\) and \(Q_u=I-P_u.\) In this paper, we study properties of the dilation of truncated Toeplitz operators on \(L^{2}\). In particular, we provide conditions for the dilation of truncated Toeplitz operators to be normal. As some applications, we give several examples of such operators.

     

Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials

We consider Gaussian elliptic random matrices X of a size \(N \times N\) with parameter \(\rho \), i.e., matrices whose pairs of entries \((X_{ij}, X_{ji})\) are mutually independent Gaussian vectors with \(\mathbb {E}\,X_{ij} = 0\), \(\mathbb {E}\,X^2_{ij} = 1\) and \(\mathbb {E}\,X_{ij} X_{ji} = \rho \). We are interested in the asymptotic distribution of eigenvalues of the matrix \(W =\frac{1}{N^2} X^2 X^{*2}\). We show that this distribution is determined by its moments, and we provide a recurrence relation for these moments. We prove that the (symmetrized) asymptotic distribution is determined by its free cumulants, which are Narayana polynomials of type B:

$$\begin{aligned} c_{2n} = \sum _{k=0}^n {\left( {\begin{array}{c}n\\ k\end{array}}\right) }^2 \rho ^{2k}. \end{aligned}$$

     

Scaling Limits for Mixed Kernels

Let \(\mu \) and \(\nu \) be measures supported on \(\left( -1,1\right) \) with corresponding orthonormal polynomials \(\left\{ p_{n}^{\mu }\right\} \) and \( \left\{ p_{n}^{\nu }\right\} \), respectively. Define the mixed kernel

$$\begin{aligned} K_{n}^{{\mu },\nu }\left( x,y\right) =\sum _{j=0}^{n-1}p_{j}^{\mu }\left( x\right) p_{j}^{\nu }\left( y\right) . \end{aligned}$$

We establish scaling limits such as

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{\pi \sqrt{1-\xi ^{2}}\sqrt{\mu ^{\prime }\left( \xi \right) \nu ^{\prime }\left( \xi \right) }}{n}K_{n}^{\mu ,\nu }\left( \xi +\frac{a\pi \sqrt{1-\xi ^{2}}}{n},\xi +\frac{b\pi \sqrt{1-\xi ^{2}}}{n}\right) \\&\quad =S\left( \frac{\pi \left( a-b\right) }{2}\right) \cos \left( \frac{\pi \left( a-b\right) }{2}+B\left( \xi \right) \right) , \end{aligned}$$

where \(S\left( t\right) =\frac{\sin t}{t}\) is the sinc kernel, and \(B\left( \xi \right) \) depends on \({\mu },\nu \) and \(\xi \). This reduces to the classical universality limit in the bulk when \(\mu =\nu \). We deduce applications to the zero distribution of \(K_{n}^{{\mu },\nu }\), and asymptotics for its derivatives.

     

Congruences for (3,11)-regular bipartitions modulo 11

In this note we investigate the function \(B_{k,\ell }(n)\), which counts the number of \((k,\ell )\)-regular bipartitions of n. We shall prove an infinite family of congruences modulo 11: for \(\alpha \ge 2\) and \(n\ge 0\),

$$\begin{aligned} B_{3,11}\left( 3^{\alpha }n+\frac{5\cdot 3^{\alpha -1}-1}{2}\right) \equiv 0\ (\mathrm{mod\ }11). \end{aligned}$$

     

On the sum of the reciprocals of the differences between consecutive primes

Let \(p_n\) denote the n-th prime number, and let \(d_n=p_{n+1}-p_{n}\). Under the Hardy–Littlewood prime-pair conjecture, we prove

$$\begin{aligned} \sum _{n\le X}\frac{\log ^{\alpha }d_n}{d_n}\sim {\left\{ \begin{array}{ll} \quad \frac{X\log \log \log X}{\log X}~\qquad \quad ~ &{}\alpha =-1,\\ \frac{X}{\log X}\frac{(\log \log X)^{1+\alpha }}{1+\alpha }\qquad &{}\alpha >-1, \end{array}\right. } \end{aligned}$$

and establish asymptotic properties for some series of \(d_n\) without the Hardy–Littlewood prime-pair conjecture.

     

The identification problem for complex-valued Ornstein–Uhlenbeck operators in $$L^p(\mathbb {R}^d,\mathbb {C}^N)$$

In this paper we study perturbed Ornstein–Uhlenbeck operators

$$\begin{aligned} \left[ \mathcal {L}_{\infty } v\right] (x)=A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle -B v(x),\,x\in \mathbb {R}^d,\,d\geqslant 2, \end{aligned}$$

for simultaneously diagonalizable matrices \(A,B\in \mathbb {C}^{N,N}\). The unbounded drift term is defined by a skew-symmetric matrix \(S\in \mathbb {R}^{d,d}\). Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain \(\mathcal {D}(A_p)\) of the generator \(A_p\) belonging to the Ornstein–Uhlenbeck semigroup coincides with the domain of \(\mathcal {L}_{\infty }\) in \(L^p(\mathbb {R}^d,\mathbb {C}^N)\) given by

$$\begin{aligned} \mathcal {D}^p_{\mathrm {loc}}(\mathcal {L}_0)=\left\{ v\in W^{2,p}_{\mathrm {loc}}\cap L^p\mid A\triangle v + \left\langle S\cdot ,\nabla v\right\rangle \in L^p\right\} ,\,1<p<\infty . \end{aligned}$$

One key assumption is a new \(L^p\)-dissipativity condition

$$\begin{aligned} |z|^2\mathrm {Re}\,\left\langle w,Aw\right\rangle + (p-2)\mathrm {Re}\,\left\langle w,z\right\rangle \mathrm {Re}\,\left\langle z,Aw\right\rangle \geqslant \gamma _A |z|^2|w|^2\;\forall \,z,w\in \mathbb {C}^N \end{aligned}$$

for some \(\gamma _A>0\). The proof utilizes the following ingredients. First we show the closedness of \(\mathcal {L}_{\infty }\) in \(L^p\) and derive \(L^p\)-resolvent estimates for \(\mathcal {L}_{\infty }\). Then we prove that the Schwartz space is a core of \(A_p\) and apply an \(L^p\)-solvability result of the resolvent equation for \(A_p\). In addition, we derive \(W^{1,p}\)-resolvent estimates. Our results may be considered as extensions of earlier works by Metafune, Pallara and Vespri to the vector-valued complex case.

     

Congruences involving $$g_n(x)=\sum \limits _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k$$

Define \(g_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k\) for \(n=0,1,2,\ldots \). Those numbers \(g_n=g_n(1)\) are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving \(g_n(x)\). For example, for any prime \(p>5\) we have

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p. \end{aligned}$$

This is similar to Wolstenholme’s classical congruences

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{1}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{1}{k^2}\equiv 0\pmod p \end{aligned}$$

for any prime \(p>3\).

     

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

     

Solvability of Minimal Graph Equation Under Pointwise Pinching Condition for Sectional Curvatures

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan–Hadamard manifold M whose radial sectional curvatures outside a compact set satisfy an upper bound

$$\begin{aligned} K(P)\le - \frac{\phi (\phi -1)}{r(x)^2} \end{aligned}$$

and a pointwise pinching condition

$$\begin{aligned} |K(P) |\le C_K|K(P') | \end{aligned}$$

for some constants \(\phi >1\) and \(C_K\ge 1\), where P and \(P'\) are any 2-dimensional subspaces of \(T_xM\) containing the (radial) vector \(\nabla r(x)\) and \(r(x)=d(o,x)\) is the distance to a fixed point \(o\in M\). We solve the asymptotic Dirichlet problem with any continuous boundary data for dimensions \(n=\dim M>4/\phi +1\).