Coefficients of multiplication formulas for classical orthogonal polynomials

In this paper using both analytic and algorithmic approaches, we derive the coefficients \(D_m(n,a)\) of the multiplication formula

$$\begin{aligned} p_n(ax)=\sum _{m=0}^nD_m(n,a)p_m(x) \end{aligned}$$

or the translation formula

$$\begin{aligned} p_n(x+a)=\sum _{m=0}^nD_m(n,a)p_m(x), \end{aligned}$$

where \(\{p_n\}_{n\ge 0}\) is an orthogonal polynomial set, including the classical continuous orthogonal polynomials, the classical discrete orthogonal polynomials, the \(q\)-classical orthogonal polynomials, as well as the classical orthogonal polynomials on a quadratic lattice and a \(q\)-quadratic lattice. We give a representation of the coefficients \(D_m(n,a)\) as a single, double or triple sum whereas in many cases we get simple representations.

     

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 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.

     

A Sharp Weak-Type $$(\infty ,\infty )$$ Inequality for the Hilbert Transform

The paper is devoted to sharp weak type \((\infty ,\infty )\) estimates for \({\mathcal {H}}^{\mathbb {T}}\) and \({\mathcal {H}}^{\mathbb {R}}\), the Hilbert transforms on the circle and real line, respectively. Specifically, it is proved that

$$\begin{aligned} \left\| {\mathcal {H}}^{\mathbb {T}}f\right\| _{W({\mathbb {T}})}\le \Vert f\Vert _{L^\infty ({\mathbb {T}})} \end{aligned}$$

and

$$\begin{aligned} \left\| {\mathcal {H}}^{\mathbb {R}}f\right\| _{W({\mathbb {R}})}\le \Vert f\Vert _{L^\infty ({\mathbb {R}})}, \end{aligned}$$

where \(W({\mathbb {T}})\) and \(W({\mathbb {R}})\) stand for the weak-\(L^\infty \) spaces introduced by Bennett, DeVore and Sharpley. In both estimates, the constant \(1\) on the right is shown to be the best possible.

     

Algebraic independence of certain Mahler functions

A monotonicity-type result for functions \(f\ : \ \mathbb {N}_a\rightarrow \mathbb {R}\) satisfying the sequential fractional difference inequality

$$\begin{aligned} \Delta _{1+a-\mu }^{\nu }\Delta _{a}^{\mu }f(t)\ge 0, \end{aligned}$$

for \(t\in \mathbb {N}_{2+a-\mu -\nu }\), where \(0<\mu <1\), \(0<\nu <1\), and \(1<\mu +\nu <2\), is proved, subject to the restriction that

$$\begin{aligned} \mu <2(1-\nu ). \end{aligned}$$

We demonstrate that this result is sharp in the sense that the restriction \(\mu <2(1-\nu )\) cannot be improved.

     

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:

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

     

On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

In this paper, we investigate solutions of the hyperbolic Poisson equation \(\Delta _{h}u(x)=\psi (x)\), where \(\psi \in L^{\infty }(\mathbb {B}^{n}, {\mathbb R}^n)\) and

$$\begin{aligned} \Delta _{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)\left( 1-|x|^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned}$$

is the hyperbolic Laplace operator in the n-dimensional space \(\mathbb {R}^n\) for \(n\ge 2\). We show that if \(n\ge 3\) and \(u\in C^{2}(\mathbb {B}^{n},{\mathbb R}^n) \cap C(\overline{\mathbb {B}^{n}},{\mathbb R}^n )\) is a solution to the hyperbolic Poisson equation, then it has the representation \(u=P_{h}[\phi ]-G_{ h}[\psi ]\) provided that \(u\mid _{\mathbb {S}^{n-1}}=\phi \) and \(\int _{\mathbb {B}^{n}}(1-|x|^{2})^{n-1} |\psi (x)|\,d\tau (x)<\infty \). Here \(P_{h}\) and \(G_{h}\) denote Poisson and Green integrals with respect to \(\Delta _{h}\), respectively. Furthermore, we prove that functions of the form \(u=P_{h}[\phi ]-G_{h}[\psi ]\) are Lipschitz continuous.

     

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.

     

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 \(\Sigma _1\).

     

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.

     

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)

     

Smoothness and Asymptotic Properties of Functions with General Monotone Fourier Coefficients

In this paper we study trigonometric series with general monotone coefficients, i.e., satisfying

$$\begin{aligned} \sum \limits _{k=n}^{2n} |a_k - a_{k+1}| \le C \sum \limits _{k=[{n}/{\gamma }]}^{[\gamma n]} \frac{|a_k|}{k}, \quad n\in \mathbb {N}, \end{aligned}$$

for some \(C \ge 1\) and \(\gamma >1\). We first prove the Lebesgue-type inequalities for such series

$$\begin{aligned} n|a_n|\le C \omega (f,1/n). \end{aligned}$$

Moreover, we obtain necessary and sufficient conditions for the sum of such series to belong to the generalized Lipschitz, Nikolskii, and Zygmund spaces. We also prove similar results for trigonometric series with weak monotone coefficients, i.e., satisfying

$$\begin{aligned} |a_n | \le C \sum \limits _{k=[{n}/{\gamma }]}^{\infty } \frac{|a_k|}{k}, \quad n\in \mathbb {N}, \end{aligned}$$

for some \(C \ge 1\) and \(\gamma >1\). Sharpness of the obtained results is given. Finally, we study the asymptotic results of Salem–Hardy type.

     

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 \(p>3\).

     

Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations

$$\begin{aligned} L^{\hbar }_{A,V} u = f(|u|^2)u \quad \hbox {in}\quad \mathbb {R}^N \end{aligned}$$

(0.1)

where \(N \ge 3\), \(L^{\hbar }_{A,V}\) is the Schrödinger operator with a magnetic field having source in a \(C^1\) vector potential A and a scalar continuous (electric) potential V defined by

$$\begin{aligned} L^{\hbar }_{A,V}= -\hbar ^2 \Delta -\frac{2\hbar }{i} A \cdot \nabla + |A|^2- \frac{\hbar }{i}\mathrm{div}A + V(x). \end{aligned}$$

(0.2)

Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain \(\Omega \subset \mathbb {R}^N\) such that

$$\begin{aligned} m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \end{aligned}$$

and we set \(K = \{ x \in \Omega \ | \ V(x) = m_0\}\). For \(\hbar >0\) small we prove the existence of at least \({\mathrm{cupl}}(K) + 1\) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as \(\hbar \rightarrow 0\).

     

Least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in $$\mathbb {R}^{N}$$

In this article we study the problem

$$\begin{aligned} \Delta ^{2}u-\left( a+b\int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}dx\right) \Delta u+V(x)u=\left| u\right| ^{p-2}u\ \text { in }\mathbb {R}^{N}, \end{aligned}$$

where \(\Delta ^{2}:=\Delta (\Delta )\) is the biharmonic operator, \(a,b>0\) are constants, \(N\le 7,\) \(p\in (4,2_{*})\) for \(2_{*}\) defined below, and \(V(x)\in C(\mathbb {R}^{N},\mathbb {R})\). Under appropriate assumptions on V(x), the existence of least energy sign-changing solution is obtained by combining the variational methods and the Nehari method.

     

Spectrality of infinite convolutions with three-element digit sets

Let \(0< \rho <1\) and let \(\{a_n, b_n\}_{n=1}^\infty \) be a sequence of integers with bounded from upper and lower. Associated with them there exists a unique Borel probability measure \(\mu _{\rho , \{0, a_n, b_n\}}\) generated by the following infinite convolution product

$$\begin{aligned} \mu _{\rho , \{0, a_n, b_n\}}=\delta _{\rho \{0, a_1, b_1\}} *\delta _{\rho ^2 \{0, a_2, b_2\}} *\delta _{\rho ^3 \{0, a_3, b_3\}} *\cdots \end{aligned}$$

in the weak convergence, where \(\delta _E=\frac{1}{\# E}\sum _{e \in E} \delta _e\) and \(\hbox {gcd}(a_n, b_n)=1\) for all \(n \in {{\mathbb {N}}}\). In this paper, we show that \(L^2(\mu _{\rho , \{0, a_n, b_n\}})\) admits an exponential orthonormal basis if and only if \(\rho ^{-1} \in 3{{\mathbb {N}}}\) and  \(\{a_n, b_n\} \equiv \{1, 2\} \ (\mathrm {mod} \ 3)\) for all \(n \in {{\mathbb {N}}}\).

     

Positive solutions of nonlinear multi-point boundary value problems

This paper deals with the existence of positive solutions of nonlinear differential equation

$$\begin{aligned} u^{\prime \prime }(t)+ a(t) f(u(t) )=0,\quad 0<t <1, \end{aligned}$$

subject to the boundary conditions

$$\begin{aligned} u(0)=\sum _{i=1}^{m-2} a_i u (\xi _i) ,\quad u^{\prime } (1) = \sum _{i=1}^{m-2} b_i u^{\prime } (\xi _i), \end{aligned}$$

where \( \xi _i \in (0,1) \) with \( 0< \xi _1<\xi _2< \cdots<\xi _{m-2} < 1,\) and \(a_i,b_i \) satisfy   \(a_i,b_i\in [0,\infty ),~~ 0< \sum _{i=1}^{m-2} a_i <1,\) and \( \sum _{i=1}^{m-2} b_i <1. \) By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality

$$\begin{aligned} \displaystyle \inf _{0 \le t \le 1} u(t) \ge \gamma \Vert u\Vert _\infty . \end{aligned}$$

     

The nonlocal Liouville-type equation in $$\mathbb {}$$ and conformal immersions of the disk with boundary singularities

In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension 1. More precisely, given a sequence \(u_k :\mathbb {R}\rightarrow \mathbb {R}\) of solutions to

$$\begin{aligned} (-\Delta )^\frac{1}{2} u_k =K_ke^{u_k}\quad \text {in} \quad \mathbb {R}, \end{aligned}$$

(1)

with \(K_k\) bounded in \(L^\infty \) and \(e^{u_k}\) bounded in \(L^1\) uniformly with respect to k, we show that up to extracting a subsequence \(u_k\) can blow-up at (at most) finitely many points \(B=\{a_1,\ldots , a_N\}\) and that either (i) \(u_k\rightarrow u_\infty \) in \(W^{1,p}_{{{\mathrm{loc}}}}(\mathbb {R}{\setminus } B)\) and \(K_ke^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}K_\infty e^{u_\infty }+ \sum _{j=1}^N \pi \delta _{a_j}\), or (ii) \(u_k\rightarrow -\infty \) uniformly locally in \(\mathbb {R}{\setminus } B\) and \(K_k e^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}\sum _{j=1}^N \alpha _j \delta _{a_j}\) with \(\alpha _j\ge \pi \) for every j. This result, resting on the geometric interpretation and analysis of (1) provided in a recent collaboration of the authors with T. Rivière and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis–Merle and Li–Shafrir on the 2-dimensional Liouville equation, but providing sharp quantization estimates (\(\alpha _j=\pi \) and \(\alpha _j\ge \pi \)) which are not known in dimension 2 under the weak assumption that \((K_k)\) be bounded in \(L^\infty \) and is allowed to change sign.

     

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 \(p\).

     

Some applications of the regularity principle in sequence spaces

The Hardy–Littlewood inequalities for m-linear forms have their origin with the seminal paper of Hardy and Littlewood (Q J Math 5:241–254, 1934). Nowadays it has been extensively investigated and many authors are looking for the optimal estimates of the constants involved. For \(m<p\le 2m\) it asserts that there is a constant \(D_{m,p}^{\mathbb {K}}\ge 1\) such that

$$\begin{aligned} \left( \sum _{j_{1},\ldots ,j_{m}=1}^{n}\left| T(e_{j_{1}},\ldots ,e_{j_{m} })\right| ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\le D_{m,p} ^{\mathbb {K}}\left\| T\right\| , \end{aligned}$$

for all m-linear forms \(T:\ell _{p}^{n}\times \cdots \times \ell _{p} ^{n}\rightarrow \mathbb {K}=\mathbb {R}\) or \(\mathbb {C}\) and all positive integers n. Using a regularity principle recently proved by Pellegrino, Santos, Serrano and Teixeira, we present a straightforward proof of the Hardy–Littlewood inequality and show that:

1. (1)
   If \(m<p_{1}\le p_{2}\le 2m\) then \(D_{m,p_{1}}^{\mathbb {K}}\le D_{m,p_{2}}^{\mathbb {K}}\);
    
2. (2)
   \(D_{m,p}^{\mathbb {K}}\le D_{m-1,p}^{\mathbb {K}}\) whenever \(m<p\le 2\left( m-1\right) \) for all \(m\ge 3\).