Brownian motion,Lp properties of Schrödinger operators and the localization of binding

We use Brownian motion ideas to study Schrödinger operators $\text{H =}\text{built?1}\text{2}\text{Δ + V}\text{on}\text{L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{v}}\text{)}$. In particular: (a) We prove that lim_t→∞t^?1In ∥ e^?tH ∥_p,p is p-independent for a very large class of V's where ∥ A ∥_p,p = norm of A as an operator from L^pto L^p. (b) For v ? 3 and $\text{V ? L}{}^{\mathrm{<mtext>v</mtext><mtext>2\; ?\; ?</mtext>}}\text{∩ L}{}^{\mathrm{<mtext>v</mtext><mtext>2\; +\; ?</mtext>}}$, we show that sup ∥ e^?tH ∥_∞,∞ < ∞ if and only if H has no negative eigenvalues or zero energy resonances. (c) We relate the “localization of binding” recently noted by Sigal to Brownian hitting probabilities.     

A family of difference sets in non-cyclic groups

A construction is given for difference sets in certain non-cyclic groups with the parameters $\text{v = q}{}^{\mathrm{s+1}}\text{{[}\text{(q}{}^{\mathrm{s+1}}\text{? 1)}\text{(q ? 1)}\text{] + 1}}$, $\text{k = q}{}^{\mathrm{s}}\text{(q}{}^{\mathrm{s+1}}\text{? 1)}\text{(q ? 1)}$, $\text{λ = q}{}^{\mathrm{s}}\text{(q}{}^{\mathrm{s}}\text{? 1)}\text{(q ? 1)}$, n = q^2s for every prime power q and every positive integer s. If q^s is odd, the construction yields at least $\text{1}\text{2}\text{(q}{}^{\mathrm{s}}\text{+ 1)}$ inequivalent difference sets in the same group. For q = 5, s = 2 a difference set is obtained with the parameters (v, k, λ, n) = (4000, 775, 150, 625), which has minus one as a multiplier.     

An interpolation principle for martingale inequalities

A general principle for martingale inequalities is illustrated by deriving $\text{∥M∥}{}_{\mathrm{p}}\text{? [p + (p}{}^2\text{? 2p)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{] ∥S∥}{}_{\mathrm{p}}$ for all p ? 2 from the case p = 2 where M is the maximal function of a martingale and S its square function.     

Sharp nonlogarithmic Kneser theorems for fourth-order elliptic equations

Let D(?) be the Doob's class containing all functions f(z) analytic in the unit disk Δ such that f(0) = 0 and lim $\text{inf}\text{¦f(z) ¦ ? 1}$ on an arc A of ?Δ with length $\text{¦A ¦? ?}$. It is first proved that if f?D(?) then the spherical norm ∥ f ∥ = sup_z?Δ$\text{(}\text{1 ? ¦z¦}{}^2\text{)¦f′(z)¦}\text{(1 + ¦f(z)¦}{}^2\text{) ? C}{}_1\text{sin}\text{(π ? (}\text{?}\text{2}\text{))/ (π ? (}\text{g9}\text{2}\text{))}$, where C₁ = lim_n→∞$\text{∥ z}{}^{\mathrm{n}}\text{∥}\text{and}\text{1}\text{2}\text{< C}{}_1\text{<}\text{2}\text{e}$. Next, U represents the Seidel's class containing all non-constant functions f(z) bounded analytic in Δ such that $\text{¦tf(e}{}^{\mathrm{i0}}\text{)¦ = 1}$ almost everywhere. It is proved that inf_f?U∥f∥ = 0, and if f has either no singularities or only isolated singularities on ?Δ, then ∥f∥ ? C₁. Finally, it is proved that if f is a function normal in Δ, namely, the norm ∥f∥< ∞, then we have the sharp estimate ∥f^p∥ ? p∥f∥, for any positive integer p.     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

Estimates on the first eigenvalue of an integral equation

We show how inequalities of the type $\text{∥F∥}{}_{\mathrm{p}}\text{? C(p, q) a}{}^{\mathrm{<mtext>1\; +\; (</mtext><mtext>1</mtext><mtext>p</mtext><mtext>)?\; (</mtext><mtext>1</mtext><mtext>q</mtext><mtext>)</mtext>}}\text{∥ F ′ ∥}{}_{\mathrm{q\prime }}$ when F(0) = 0 can be used to find lower bounds of the first eigenvalue of the integral equation F(z) = λ ∝₀^ak(s, z)F(s) ds.     

Uncertainty principle and Lp–Lq-sufficient pairs on noncompact real symmetric spaces

We consider a real semi-simple Lie group G with finite center and a maximal compact sub-group K of G. Let $\text{G=K}\text{exp}\text{(}\text{a}{}_{\mathrm{+}}\text{)K}$ be a Cartan decomposition of G. For x∈G denote ∥x∥ the norm of the $\text{a}{}_{\mathrm{+}}$-component of x in the Cartan decomposition of G. Let $\text{a>0,}\text{b>0}$ and 1?p,q?∞. In this Note we give necessary and sufficient conditions on $\text{a,}\text{b}$ such that for all K-bi-invariant measurable function f on G, if e^a∥x∥2f∈L^p(G) and then f=0 almost everywhere. To cite this article: S. Ben Farah, K. Mokni, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

The quartic characters of certain quadratic units

Criteria are obtained for the quartic residue character of the fundamental unit of the real quadratic field $\text{Q((2q)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where q is prime and either q ≡ 7(mod 8), or q ≡ 1(mod 8) and X² ? 2qY² = ?2 is solvable in integers X and Y.     

Inclusion mappings between lp spaces

It is shown that the inclusion mapping from $\text{l}{}^{\mathrm{p}}\text{into}\text{l}{}^{\mathrm{q}}\text{is}\text{(}\text{2pq}\text{(pq ? 2p + 2q)}\text{, 1)}$-absolutely summing if 1 ? p ? q ? 2 and (p, 1)-absolutely summing if 1 ? p ? 2 ? q ? ∞ or 2 ? p ? q ? ∞. As a corollary, it follows that a (p, 1)-absolutely summing operator, p > 1, need not have the Dunford-Pettis property; this observation answering a question raised by Pelczynski.     

New bounds on exponential sums related to the Diffie–Hellman distributions

Given $\text{θ∈}\text{F}{}_{\mathrm{p}}{}^1$ (p prime) of multiplicative order t>p^δ, we obtain nontrivial bounds on exponential sums

$\text{∑}\text{s′=1}\text{t}\text{∑}\text{s=1}\text{t}\text{e}{}_{\mathrm{p}}\text{aθ}{}^{\mathrm{s}}\text{+cθ}{}^{\mathrm{ss\prime }}$

as well as the corresponding incomplete sums. These estimates are of relevance to several issues, such as the Diffie–Hellman distributions in cryptography, prime divisors of ‘sparse integers’, the distribution $\text{mod}\text{p}$ of Mersenne numbers M_q=2^q?1 (q prime). The method is closely related to that of Bourgain and Konyagin (C. R. Acad. Sci. Paris, Ser. I 337 (2) (2003) 75–80). To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

Certain conjugacy classes of units in integral group rings of metacyclic groups

Let G be the metacyclic group of order pq given by

$\text{G = 〈σ, τ: σ}{}^{\mathrm{p}}\text{= 1 = τ}{}^{\mathrm{q}}\text{, τστ}{}^{\mathrm{?}}\text{= σ}{}^{\mathrm{j}}\text{〉}$

where p is an odd prime, q ≥ 2 a divisor of p ? 1, and where j belongs to the exponent q mod p. Let V denote the group of units of augmentation 1 in the integral group ring $\text{Z}$G of G. In this paper it is proved that the number of conjugacy classes of elements of order p in V is

$\text{(p ? 1)}{}^{\mathrm{q?1}}\text{μ}{}_0\text{H}\text{vq}$

where ν, μ₀, and H are suitably defined numbers.     

Operator properties of Sobolev imbeddings over unbounded domains

For an open set Ω ? $\text{R}$^N, 1 ? p ? ∞ and λ ∈ $\text{R}$⁺, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators $\text{U}$, 1 ? p, q ? ∞ and a quasibounded domain Ω ? $\text{R}$^N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the given Banach ideal $\text{U}$: Assume the quasibounded domain fulfills condition C_k^l for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any $\text{x ? Ω}$ to the boundary ?Ω tends to zero as $\text{O(¦ x ¦}{}^{\mathrm{?l}}\text{)}$ for $\text{¦ x ¦ → ∞}$, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ $\text{N}$, μ > λ _S($\text{U}$; p,q:N) and v > N/l · λ_D($\text{U}$;p,q), one has that $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ belongs to the Banach ideal $\text{U}$. Here λ_D($\text{U}$;p,q;N)∈$\text{R}$⁺ and λ_S($\text{U}$;p,q;N)∈$\text{R}$⁺ are the D-limit order and S-limit order of the ideal $\text{U}$, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings l_pⁿ → l_qⁿ for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω fulfills condition C₁^l.For an open set Ω in $\text{R}$^N, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in $\text{R}$^N and give sufficient conditions on λ such that the Sobolev imbedding operator $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded open set in $\text{R}$^N.     

Hilbertian and complemented finite-dimensional subspaces of Banach lattices and unitary ideals

Let 1 < p ? 2 ? q < ∞ and X be either a Banach lattice which is p-convex and q-concave or a unitary ideal of operators on l₂ which is modeled on a symmetric space which is p-convex and q-concave. If E ?X is any n-dimensional subspace, then both the distance from E to $\text{l}{}_2{}^{\mathrm{n}}$ and the relative projection constant of E in X are dominated by $\text{cn}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext><mtext>\; ?\; </mtext><mtext>1</mtext><mtext>q</mtext>}}$.     

Weak-1 generators for a class of subalgebras of l1

Let α ? 0 and let $\text{D(}\text{α}\text{) = {f(z) = ∑}{}_0{}^{\mathrm{\infty }}\text{α}{}_{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{¦ ∑}{}_0{}^{\mathrm{\infty }}\text{(n + 1)}{}^{\mathrm{<mtext>\alpha </mtext>}}\text{¦ a}{}_{\mathrm{n}}\text{¦ < ∞}}$. Then D(α) is a subalgebra of l¹. We discuss the weak-1 generators of D(α). We use some of our techniques to prove that if ? is a weak-1 generator of H^∞ and ∥ ? ∥_∞ ? 1, then the composition operator C_? on the Dirichlet space has dense range.     

Almost commuting matrices

It is shown that if A and B are n × n complex matrices with $\text{A = A}{}^1\text{and}\text{∥AB ? BA∥</}\text{2?}{}^2\text{(n ? 1)}$, then there exist n × n matrices A′ and B′ with $\text{A′ = A′}{}^1\text{such that}\text{A′B′ = B′A′}\text{and}\text{∥A ? A′∥? ?, ∥B ? B′∥? ?}$.     

Computing the number of totally positive circular units which are squares

An efficient method for computing the number of invariants of the quotient group $\text{C}\text{C}\text{? T}$, where C is the group of circular units in the maximal real subfield of the qth cyclotomic field, q is a prime, and T is the group of totally positive units, is developed by establishing an isomorphism between $\text{C}\text{C}\text{? T}$ and a specific ideal in the algebra $\text{F[x]}\text{〈x}{}^{\mathrm{p}}\text{+ 1〉}$, where F is the Galois field of two elements and $\text{p =}\text{(q ? 1)}\text{2}$. Based on computations using this method, it is conjectured that every totally positive circular unit is a square if p is a prime.     

On some t−(2k, k, λ) designs

t?(2k, k, λ) designs having a property similar to that of Hadamard 3-designs are studied. We consider conditions (i), (ii), or (iii) for t?(2k, k, λ) designs: (i) The complement of each block is a block. (ii) If A and B are a complementary pair of blocks, then ∥ A ∩ C ∥ = ∥ B ∩ C ∥ ± u holds for any block C distinct from A and B, where u is a positive integer. (iii) if A and B are a complementary pair of blocks, then ∥ A ∩ C ∥ = ∥ B ∩ C ∥ or ∥ A ∩ C ∥ = ∥ B ∩ C ∥ ± u holds for any block C distinct from A and B, where u is a positive integer. We show that a t?(2k, k, λ) design with t ? 2 and with properties (i) and (ii) is a 3?(2u(2u + 1), u(2u + 1), u(2u² + u ? 2)) design, and that a t?(2k, k, λ) design with t ? 4 and with properties (i) and (iii) is the 5-(12, 6, 1) design, the 4-(8, 4, 1) design, a $\text{5?(2u}{}^2\text{, u}{}^2\text{,}\text{1}\text{4}\text{(u}{}^2\text{? 3) (u}{}^2\text{? 4))}$ design, or a $\text{5?(}\text{2}\text{3}\text{u(2u + 1),}\text{1}\text{3}\text{u(2u = 1),}\text{1}\text{5 4}\text{u(2u}{}^2\text{+ u ? 9) (2u}{}^2\text{+ u ? 12))}$ design.     

An intersection problem whose extremum is the finite projective space

Suppose that $\text{A}$ is a finite set-system of N elements with the property |A ∩ A′| = 0, 1 or k for any two different A, A′ ?A. We show that for N > k¹⁴

$\text{|a|=?}\text{N(N?1)(N?k)}\text{(k}{}^2\text{?k+1)(k}{}^2\text{?2k+1)}\text{+}\text{N(N?1)}\text{k(k?1)}\text{+N+1}$

where equality holds if and only if k = q + 1 (q is a prime power) $\text{N =}\text{(q}{}^{\mathrm{t+1}}\text{? 1)}\text{(q ? 1)}$ and $\text{A}$ is the set of subspaces of dimension at most two of the t-dimensional finite projective space of order q.