Small tight sets of hyperbolic quadrics

An x-tight set of a hyperbolic quadric Q ⁺(2n + 1, q) can be described as a set M of points with the property that the number of points of M in the tangent hyperplanes of points of M is as big as possible. We show that such a set is necessarily the union of x mutually disjoint generators provided that x ≤ q and n ≤ 3, or that x < q, n ≥ 4 and q ≥ 71. This unifies and generalizes many results on x-tight sets that are presently known, see (J Comb Theory Ser A 114(7):1293–1314 [1], J Comb Des 16(4):342–349 [5], Des Codes Cryptogr 50:187–201 [4], Adv Geom 4(3):279–286 [8], Bull Lond Math Soc 42(6):991–996 [11]).     

Blocking Sets in PG(r, q n )

Let $\mathcal S$ be a Desarguesian (n – 1)-spread of a hyperplane Σ of PG(rn, q). Let Ω and ${\bar B}$ be, respectively, an (n – 2)-dimensional subspace of an element of $\mathcal S $ and a minimal blocking set of an ((r – 1)n + 1)-dimensional subspace of PG(rn, q) skew to Ω. Denote by K the cone with vertex Ω and base ${\bar B}$ , and consider the point set B defined by $$B=\left(K\setminus\Sigma\right)\cup \{X\in \mathcal S\, : \, X\cap K\neq \emptyset\}$$ in the Barlotti–Cofman representation of PG(r, q ⁿ ) in PG(rn, q) associated to the (n – 1)-spread $\mathcal S$ . Generalizing the constructions of Mazzocca and Polverino (J Algebraic Combin, 24(1):61–81, 2006), under suitable assumptions on ${\bar B}$ , we prove that B is a minimal blocking set in PG(r, q ⁿ ). In this way, we achieve new classes of minimal blocking sets and we find new sizes of minimal blocking sets in finite projective spaces of non-prime order. In particular, for q a power of 3, we exhibit examples of r-dimensional minimal blocking sets of size q ⁿ⁺² + 1 in PG(r, q ⁿ ), 3 ≤ r ≤ 6 and n ≥ 3, and of size q ⁴ + 1 in PG(r, q ²), 4 ≤ r ≤ 6; actually, in the second case, these blocking sets turn out to be the union of q ³ Baer sublines through a point. Moreover, for q an even power of 3, we construct examples of minimal blocking sets of PG(4, q) of size at least q ² + 2. From these constructions, we also get maximal partial ovoids of the hermitian variety H(4, q ²) of size q ⁴ + 1, for any q a power of 3.     

Variable Exponent Herz Spaces

We introduce a new type of variable exponent function spaces  ? ^{p(·),q(·),α(·)}( ${\mathbb{R}^n}$ ) and H ^{p(·),q(·),α(·)}( ${\mathbb{R}^n}$ ) of Herz type, homogeneous and non-homogeneous versions, where all the three parameters are variable, and give comparison of continual and discrete approaches to their definition. Under the only assumption that the exponents p, q and α are subject to the log-decay condition at infinity, we prove that sublinear operators, satisfying the size condition known for singular integrals and bounded in L ^p(·)( ${\mathbb{R}^n}$ ), are also bounded in the nonhomogeneous version of the introduced spaces, which includes the case maximal and Calderón-Zygmund singular operators.     

Pointwise multipliers for Besov spaces of dominating mixed smoothness-Ⅱ

We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes S_(p,q)~rB(R~d) with respect to pointwise multiplication. In addition, if p≤q, we are able to describe the space of all pointwise multipliers for S_(p,q)~rB(R~d).     

The supertail of a subspace partition

Let V = V(n, q) be a vector space of dimension n over the finite field with q elements, and let d ₁ < d ₂ < ... < d _m be the dimensions that occur in a subspace partition ${\mathcal{P}}$ of V. Let σ _q (n, t) denote the minimum size of a subspace partition ${\mathcal P}$ of V, in which t is the largest dimension of a subspace. For any integer s, with 1 < s ≤ m, the set of subspaces in ${\mathcal{P}}$ of dimension less than d _s is called the s-supertail of ${\mathcal{P}}$ . The main result is that the number of spaces in an s-supertail is at least σ _q (d _s , d _s?1).     

Solutions of second order degenerate integro-differential equations in vector-valued function spaces

We study the well-posedness of the second order degenerate integro-differential equations(P2):(Mu)(t)+α(Mu)(t) = Au(t)+ft-∞ a(ts)Au(s)ds + f(t),0t2π,with periodic boundary conditions M u(0)=Mu(2π),(Mu)(0) =(M u)(2π),in periodic Lebesgue-Bochner spaces Lp(T,X),periodic Besov spaces B s p,q(T,X) and periodic Triebel-Lizorkin spaces F s p,q(T,X),where A and M are closed linear operators on a Banach space X satisfying D(A) D(M),a∈L1(R+) and α is a scalar number.Using known operatorvalued Fourier multiplier theorems,we completely characterize the well-posedness of(P2) in the above three function spaces.     

Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote ${\Delta_g=-{\rm div}_g\nabla}$ the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation $$\Delta_gu(x)+h(x)u(x)=A(x)u^p(x)+\frac{B(x)}{u^q(x)}$$ on (M, g). Here, p, q ≥ 0, A(x), B(x) and h(x) are smooth functions on (M, g). We also derive the Harnack differential inequality for the positive solutions of $$u_t(x,t)+\Delta_gu(x,t)+h(x)u(x,t)=A(x)u^p(x,t)+\frac{B(x)}{u^q(x,t)}$$ on (M, g) with initial data u(x, 0) > 0.     

Boundedness for pseudodifferential operators on multivariate α-modulation spaces

The α-modulation spaces M ^s,α _p,q (R ^d ), α∈[0,1], form a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that a pseudodifferential operator σ(x,D) with symbol in the Hörmander class S ^b _ρ,0 extends to a bounded operator σ(x,D):M ^s,α _p,q (R ^d )→M ^s-b,α _p,q (R ^d ) provided 0≤α≤ρ≤1, and 1<p,q<∞. The result extends the well-known result that pseudodifferential operators with symbol in the class S ^b _1,0 maps the Besov space B ^s _p,q (R ^d ) into B ^s-b _p,q (R ^d ).     

Fast Growing Solutions of Linear Differential Equations in the Unit Disc

For $n \in \mathbb{N}$ , the n-order of an analytic function f in the unit disc D is defined by $$\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}},$$ where log⁺ x  =  max{log x, 0}, log ⁺ ₁ x  =  log ⁺ x, log ⁺ _n+1 x  =  log ⁺ log ⁺ _n x, and M(r, f) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equation $$f^{{(k)}} + a_{{k - 1}} (z)f^{{(k - 1)}} + \cdots + a_{1} (z)f^{\prime} + a_{0} (z)f = 0,\quad \quad \quad (\dag)$$ where the coefficients are analytic in D, satisfy σ _M,n+1(f)  ≤  α if and only if σ _M,n (a _j )  ≤  α for all j  =  0, ..., k ? 1. Moreover, if q ∈{0, ..., k ? 1} is the largest index for which $\sigma _{M,n} ( a_{q}) = {\mathop {\max }\limits_{0 \leq j \leq k - 1} }{\left\{ {\sigma _{{M,n}} {\left( {a_{j} } \right)}} \right\}}$ , then there are at least k ? q linearly independent solutions f of ( $\dag$ ) such that σ _M,n+1(f) = σ _M,n (a _q ). Some refinements of these results in terms of the n-type of an analytic function in D are also given.     

Discretization and best approximation in Besov spaces

In this paper we prove a theorem about existence of best approximation in a class of spaces involving Besov spaces, via a discretization technique. It is a consequence of this theorem that rational functions and exponencial sums are proximinal subsets of B _∞,q ^a (π). It is also proved the proximinality of R _m ⁿ [a, b] in B _p,q ^a (π) for arbitrary p,q and a.     

The annihilator of Bergman space

LetD be a bounded plane domain (with some smoothness requirements on its boundary). LetB _p(D), 1≤p<∞, be the Bergmanp-space ofD. In a previous paper we showed that the “natural projection”P, involving the Bergman kernel forD, is a bounded projection fromL _p(D) ontoB _p(D), 1<p<∞. With this we have the decompositionL _p(D)=B _p(D)⊕B _q ^⊥ (D,p ^–1+q ^–=1, 1<p< ∞. Here, we show that the annihilatorB _q ^⊥ (D) is the space of allL _p-complex derivatives of functions belonging to Sobolev space and which vanish on the boundary ofD. This extends a result of Schiffer for the casep=2. We also study certain operators onL _p(D). Especially, we show that , whereI is the identity operator and ? is an operator involving the adjoint of the Bergman kernel. Other relationships relevant toB _q ^⊥ (D) are studied.     

Extensions of spaces of analytic functions via pointwise limits of bounded sequences and two integral operators on generalized Bloch spaces

In [2], operators $$P_\mu f(z):=-\frac{1}{(1-z)^{\mu+1}} \int \limits_1^z f(\zeta)(1-\zeta)^{\mu} \,d\zeta$$ P μ f ( z ) : = - 1 ( 1 - z ) μ + 1 ∫ 1 z f ( ζ ) ( 1 - ζ ) μ d ζ and $$Q_\mu f(z):=(1-z)^{\mu-1} \int\limits_0^z f(\zeta)(1-\zeta)^{-\mu} \,d \zeta\quad (z \in \mathbb{D})$$ Q μ f ( z ) : = ( 1 - z ) μ - 1 ∫ 0 z f ( ζ ) ( 1 - ζ ) - μ d ζ ( z ∈ D ) were investigated in the setting of the analytic Besov spaces B _p , 1 ≤ p ≤ ∞, and the little Bloch space B _∞,0. In particular, for X = B _p , 1 ≤ p < ∞, or X = B _∞,0, the spectra, essential spectra of P _μ , and Q _μ in ${\mathcal {L}(X),}$ L ( X ) , together with one sided analytic resolvents in the Fredholm regions of P _μ , and Q _μ were obtained along with an explicit strongly decomposable operator extending Q _μ and simultaneously lifting P _μ . In the current paper, we extend the spectral analysis to generalized Bloch spaces using a modification of a construction due to Aleman and Persson, [3].     

Wavelet estimations for density derivatives

Donoho et al. in 1996 have made almost perfect achievements in wavelet estimation for a density function f in Besov spaces Bsr,q(R). Motivated by their work, we define new linear and nonlinear wavelet estimators flin,nm, fnonn,m for density derivatives f(m). It turns out that the linear estimation E(‖flinn,m-f(m)‖p) for f(m) ∈ Bsr,q(R) attains the optimal when r≥ p, and the nonlinear one E(‖fnonn,m-f(m)‖p) does the same if r≤p/2(s+m)+1 . In addition, our method is applied to Sobolev spaces with non-negative integer exponents as well.     

Optimal Sobolev Type Inequalities in Lorentz Spaces

It is well known that the classical Sobolev embeddings may be improved within the framework of Lorentz spaces L ^p,q : the space $\mathcal{D}^{1,p}(\mathbb R^n)$ , 1?<?p?<?n, embeds into $L^{p^*,q}(\mathbb R^n)$ , p?≤?q?≤?∞. However, the value of the best possible embedding constants in the corresponding inequalities is known just in the case $L^{p^*,p}(\mathbb R^n)$ . Here, we determine optimal constants for the embedding of the space $\mathcal{D}^{1,p}(\mathbb R^n)$ , 1?<?p?<?n, into the whole Lorentz space scale $L^{p^{\ast}, q}(\mathbb R^n)$ , p?≤?q?≤?∞, including the limiting case q?=?p of which we give a new proof. We also exhibit extremal functions for these embedding inequalities by solving related elliptic problems.     

Transitive A 6-invariant k-arcs in PG(2, q)

For q = p ^r with a prime p ≥ 7 such that ${q \equiv 1}$ or 19 (mod 30), the desarguesian projective plane PG(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to the alternating group A ₆ of degree 6. For a projectivity group ${\Gamma \cong A_6}$ of PG(2, q), we investigate the geometric properties of the (unique) Γ-orbit ${\mathcal{O}}$ of size 90 such that the 1-point stabilizer of Γ in its action on ${\mathcal O}$ is a cyclic group of order 4. Here ${\mathcal O}$ lies either in PG(2, q) or in PG(2, q ²) according as 3 is a square or a non-square element in GF(q). We show that if q ≥ 349 and q ≠ 421, then ${\mathcal O}$ is a 90-arc, which turns out to be complete for q = 349, 409, 529, 601,661. Interestingly, ${\mathcal O}$ is the smallest known complete arc in PG(2,601) and in PG(2,661). Computations are carried out by MAGMA.     

A Liouville comparison principle for sub- and super-solutions of the equation w t −Δ p (w) = |w| q-1 w

We establish a Liouville comparison principle for entire weak sub- and super-solutions of the equation (*) w _t ?Δ _p (w) =  |w| ^q-1 w in the half-space ${\mathbb {S}= \mathbb {R}^1_+\times \mathbb {R}^n}$ , where n ≥ 1, q > 0, and ${ \Delta_p(w) := {\rm div}_x \left(|\nabla_x w|^{p-2}\nabla_x w \right)}$ , 1 < p ≤ 2. In our study we impose neither restrictions on the behaviour of entire weak sub- and super-solutions of (*) on the hyper-plane t = 0, nor any growth conditions on their behaviour and on that of any of their partial derivatives at infinity. We prove that if ${1< q \leq p-1+\frac {p}{n}}$ and u and v are, respectively, an entire weak super-solution and an entire weak sub-solution of (*) in ${\mathbb {S}}$ which belong, only locally in ${\mathbb {S}}$ , to the corresponding Sobolev space and are such that u ≥  v, then u ≡ v. The result is sharp. As direct corollaries we obtain known Fujita-type and Liouville-type theorems.     

Total curvature and L 2 harmonic 1-forms on complete submanifolds in space forms

Let M ⁿ be an n-dimensional complete noncompact oriented submanifold with finite total curvature, i.e., ${\int_M(|A|^2-n|H|^2)^{\frac n2} < \infty}$ , in an (n + p)-dimensional simply connected space form N ^n+p (c) of constant curvature c, where |H| and |A|² are the mean curvature and the squared length of the second fundamental form of M, respectively. We prove that if M satisfies one of the following: (i) n ≥ 3, c = 0 and ${\int_M|H|^n < \infty}$ ; (ii) n ≥ 5, c = ?1 and ${|H| < 1-\frac{2}{\sqrt n}}$ ; (iii) n ≥ 3, c = 1 and |H| is bounded, then the dimension of the space of L ² harmonic 1-forms on M is finite. Moreover, in the case of (i) or (ii), M must have finitely many ends.     

Weighted Hardy Spaces Associated to Discrete Laplacians on Graphs and Applications

Let Γ be a infinite graph with a weight μ and let d and m be the distance and the measure associated with μ such that (Γ,d,m) is a space of homogeneous type. Let p(·,·) be the natural reversible Markov kernel on (Γ,d,m) and its associated operator be defined by \(Pf(x) = \sum _{y} p(x, y)f(y)\) . Then the discrete Laplacian on L ²(Γ) is defined by L=I?P. In this paper we investigate the theory of weighted Hardy spaces \({H^{p}_{L}}(\Gamma , w)\) associated to the discrete Laplacian L for 0<p≤1 and \(w\in A_{\infty }\) . Like the classical results, we prove that the weighted Hardy spaces \({H^{p}_{L}}(\Gamma , w)\) can be characterized in terms of discrete area operators and atomic decompositions as well. As applications, we study the boundedness of singular integrals on (Γ,d,m) such as square functions, spectral multipliers and Riesz transforms on these weighted Hardy spaces \({H^{p}_{L}}(\Gamma ,w)\) .     

Interpolation of Besov B pτ σq and Lizorkin-Triebel F pτ σq spaces

An interpolation method is introduced for anisotropic spaces which generalizes the method by D. L. Fernandez [4]. By means of this method, interpolation properties of Besov B _pτ ^σq and Lizorkin-Triebel F _pτ ^σq spaces are investigated. Among others, the completeness of the scale of these spaces is proved with respect to the considered interpolation method.     

Finite Index Operators on Surfaces

We consider differential operators L acting on functions on a Riemannian surface, Σ, of the form $$L = \Delta+ V -a K,$$ where Δ is the Laplacian of Σ, K is the Gaussian curvature, a is a positive constant, and V∈C ^∞(Σ). Such operators L arise as the stability operator of Σ immersed in a Riemannian three-manifold with constant mean curvature (for particular choices of V and a). We assume L is nonpositive acting on functions compactly supported on Σ. If the potential, V:=c+P with c a nonnegative constant, verifies either an integrability condition, i.e., P∈L ¹(Σ) and P is nonpositive, or a decay condition with respect to a point p ₀∈Σ, i.e., |P(q)|≤M/d(p ₀,q) (where d is the distance function in Σ), we control the topology and conformal type of Σ. Moreover, we establish a Distance Lemma. We apply such results to complete oriented stable H-surfaces immersed in a Killing submersion. In particular, for stable H-surfaces in a simply-connected homogeneous space with 4-dimensional isometry group, we obtain:

• There are no complete stable H-surfaces Σ??²×?, H>1/2, so that either $K_{e}^{+}:=\max \left \{0,K_{e}\right \} \in L^{1} (\Sigma)$ or there exist a point p ₀∈Σ and a constant M so that |K _e (q)|≤M/d(p ₀,q); here K _e denotes the extrinsic curvature of Σ.
• Let $\Sigma\subset \mathbb{E}(\kappa, \tau)$ , τ≠0, be an oriented complete stable H-surface so that either ν ²∈L ¹(Σ) and 4H ²+κ≥0, or there exist a point p ₀∈Σ and a constant M so that |ν(q)|²≤M/d(p ₀,q) and 4H ²+κ>0. Then:
• In $\mathbb{S}^{3}_{\text{Berger}}$ , there are no such a stable H-surfaces.
• In Nil₃, H=0 and Σ is either a vertical plane (i.e., a vertical cylinder over a straight line in ?²) or an entire vertical graph.
• In $\widetilde{\mathrm{PSL}(2,\mathbb{R})}$ , $H=\sqrt{-\kappa }/2$ and Σ is either a vertical horocylinder (i.e., a vertical cylinder over a horocycle in ?²(κ)) or an entire graph.