Sharp weighted norm inequalities for Littlewood–Paley operators and singular integrals

We prove sharp Lp(w) norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the Ap characteristic of w for all 1<p<∞. This implies the same sharp inequalities for the classical Lusin area integral S(f), the Littlewood–Paley g-function, and their continuous analogs Sψ and gψ. Also, as a corollary, we obtain sharp weighted inequalities for any convolution Calderón–Zygmund operator for all 1<p?3/2 and 3?p<∞, and for its maximal truncations for 3?p<∞.     

Weighted Hardy and potential operators in the generalized Morrey spaces

We study the weighted p→q-boundedness of the multi-dimensional Hardy type operators in the generalized Morrey spaces Lp,φ(Rn,w) defined by an almost increasing function φ(r) and radial type weight w(|x|). We obtain sufficient conditions, in terms of some integral inequalities imposed on φ and w, for such a p→q-boundedness. In some cases the obtained conditions are also necessary. These results are applied to derive a similar weighted p→q-boundedness of the Riesz potential operator.     

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators

This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. For L in some class of elliptic operators, we study weighted norm Lp inequalities for singular “non-integral” operators arising from L; those are the operators φ(L) for bounded holomorphic functions φ, the Riesz transforms ∇L−1/2 (or (−Δ)^1/2L−1/2) and its inverse L1/2(−Δ)^−1/2, some quadratic functionals gL and GL of Littlewood-Paley-Stein type and also some vector-valued inequalities such as the ones involved for maximal Lp-regularity. For each, we obtain sharp or nearly sharp ranges of p using the general theory for boundedness of Part I and the off-diagonal estimates of Part II. We also obtain commutator results with BMO functions.     

Stopped Doob inequality for p-th moment, 0 <p<∞, stochastic convolution integrals

We give stopped Doob inequalities for p-th moment, 0<p<∞, of stochastic convolution integrals ∫_(0,t] U(t,s)φ _{s ^?} dM _s in a Hilbert space, where M is a Hilbert space-valued cadlag square integrable martingale, φ is an operator-valued predictable process and U(t,s) is a contraction-type evolution operator. We also generalize the previous results and try to get smaller constants.

     

Keller-Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group

We study the qualitative behavior of non-negative entire solutions of differential inequalities with gradient terms on the Heisenberg group. We focus on two classes of inequalities: Δφu?f(u)l(|∇u|) and Δφu?f(u)−h(u)g(|∇u|), where f, l, h, g are non-negative continuous functions satisfying certain monotonicity properties. The operator Δφ, called the φ-Laplacian, generalizes the p-Laplace operator considered by various authors in this setting. We prove some Liouville theorems introducing two new Keller-Osserman type conditions, both extending the classical one which appeared long ago in the study of the prototype differential inequality Δu?f(u) in Rm. We show sharpness of our conditions when we specialize to the p-Laplacian. While proving these results we obtain a strong maximum principle for Δφ which, to the best of our knowledge, seems to be new. Our results continue to hold, with the obvious minor modifications, also for Euclidean space.     

Functions of operators under perturbations of class Sp

This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα(R) and 1<p<∞, the operator f(A)−f(B) belongs to Sp/α, whenever A and B are self-adjoint operators with A−B∈Sp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α_‖ in terms of ‖A−BSp_‖ and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences . We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f(A)−f(B) to belong to Sq under the assumption that A−B∈Sp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)R−Rf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K).     

Some Estimates for the Norm of the Bergman Projection on Besov Spaces

In this paper we obtain an estimate of the norm of the Bergman projection from L ^p (D, dλ) onto the Besov space B _p , 1 < p < + ∞. The result is asymptotically sharp when p → + ∞. Further for the case P : L ¹(D, dλ) → B ₁, we consider some weak type inequalities with the corresponding spaces.     

Selection over classes of ordinals expanded by monadic predicates

A monadic formula ψ(Y) is a selector for a monadic formula φ(Y) in a structure M if ψ defines in M a unique subset P of the domain and this P also satisfies φ in M. If C is a class of structures and φ is a selector for ψ in every M∈C, we say that φ is a selector for φ over C.For a monadic formula φ(X,Y) and ordinals α≤ω₁ and δ<ω^ω, we decide whether there exists a monadic formula ψ(X,Y) such that for every P⊆αof order-type smaller thanδ, ψ(P,Y) selects φ(P,Y) in (α,<). If so, we construct such a ψ.We introduce a criterion for a class C of ordinals to have the property that every monadic formula φ has a selector over it. We deduce the existence of S⊆ω^ω such that in the structure (ω^ω,<,S) every formula has a selector.Given a monadic sentence π and a monadic formula φ(Y), we decide whether φ has a selector over the class of countable ordinals satisfying π, and if so, construct one for it.     

A note on Keller-Osserman conditions on Carnot groups

This paper deals with the study of differential inequalities with gradient terms on Carnot groups. We are mainly focused on inequalities of the form Δ_φu≥f(u)l(|∇₀u|), where f, l and φ are continuous functions satisfying suitable monotonicity assumptions and Δ_φ is the φ-Laplace operator, a natural generalization of the p-Laplace operator which has recently been studied in the context of Carnot groups. We extend to general Carnot groups the results proved in Magliaro et al. (2011) [7] for the Heisenberg group, showing the validity of Liouville-type theorems under a suitable Keller-Osserman condition. In doing so, we also prove a maximum principle for inequality Δ_φu≥f(u)l(|∇₀u|). Finally, we show sharpness of our results for a general φ-Laplacian.     

Clarkson–McCarthy Inequalities for l p -Spaces of Operators in Schatten Ideals

In this paper we obtain generalized Clarkson–McCarthy inequalities for spaces l _q (S ^p ) of operators from Schatten ideals S ^p . We show that all Clarkson–McCarthy type inequalities are, in fact, some estimates on the norms of operators acting on the spaces l _q (S ^p ) or from one such space into another. We also extend some inequalities for partitioned operators and for Cartesian decomposition of operators.     

On functions of bounded p-variation

We obtain estimates of the total p-variation (1<p<∞) and other related functionals for a periodic function f∈Lp[0,1] in terms of its Lp-modulus of continuity ωp(f;δ). These estimates are sharp for any rate of the decay of ωp(f;δ). Moreover, the constant coefficients in them depend on parameters in an optimal way.     

On weighted inequalities for martingale transform operators

For 1 < p < ∞, the almost surely finiteness of is a necessary and sufficient condition in order to have almost surely convergence of the sequences {E(f|?_n)} with f ∈ L^p(v dP). This condition is also equivalent to have weighted inequalities from L^p(v dP) into L^p(u dP) for some weight u for Doob's maximal function, square function and generalized Burkholder martingale transforms. Similarly, E(u|?₁) < ∞ turns out to be necessary and sufficient for the above weighted inequalities to hold for some v.     

The resolvent of a dilation-analytic three-particle system

If the potential in a three-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has a dilation-analytic continuation H(φ). The continuous spectrum of H(φ) consists of half-lines Y(λ_p, φ) starting at the thresholds λ_p of scattering channels and making angles 2φ with the positive real axis. If the interaction is the sum of local two-body potentials in suitable $\text{L}$^p-spaces, each half-line Y(λ_p, φ) is associated with an operator P(λ_p, φ) that projects onto an invariant subspace of H(φ). Suppose Y(λ_p, φ) does not pass through any two- or three-particle eigenvalues λ ≠ λ_p when φ runs through some interval $\text{0 < α ? φ ? β <}\text{π}\text{2}$. For φ in [α, β], this paper shows that the resolvent R(λ, φ) has smoothness properties near Y(λ_p, φ) that are sufficient for P(λ_p, φ)[H(φ) ? λ_p] e^?2iφ to be spectral and to generate a strongly differentiable group. The projection, the group, and the spectral resolution operators are norm continuous in φ. These results are not affected by any spurious poles of the resolvent equation. At a spurious pole λ = λ_p + ze^2iφ, the resolvent R(λ_p + ze^2iφ,φ) is examined by a method that uses two resolvent equations in succession and shows that there is norm continuity in z, φ. The case of spurious poles on Y(λ_p, φ) is included.     

Sharp weighted bounds without testing or extrapolation

We give a short proof of the sharp weighted bound for sparse operators that holds for all p,1?<?p?< ??. By recent developments this implies the bounds hold for any Calderón?CZygmund operator. The novelty of our approach is that we avoid two techniques that are present in other proofs: two weight inequalities and extrapolation. Our techniques are applicable to fractional integral operators as well.     

An elementary proof of the blow-up for semilinear wave equation in high space dimensions

This paper concerns the blow-up of solutions to u_tt−Δu=|u|^p in high dimensions for n?4 and 1<p<p₀(n), where p₀(n) is a critical exponent. We proved that the solutions blow up in finite time by estimating the solutions near the wave front using elementary inequalities.     

Classification of some metric automorphisms defined by Standish

Let F be a finite set with a probability distribution {P_i: i?F} and (Ω $\text{F}$, P) denote the product space of countably many copies of (F, P). A permutation (bijection) φ of the integers induces an invertible measure preserving transformation T_φ on (Ω $\text{F}$, P) given by the equation (T_φw)_i = w_φ(j). Such metric automorphisms we call S-automorphisms.We show in this paper that S-automorphisms are ergodic if and only they are Bernoulli shifts and two ergodic S-automorphisms are isomorphic if and only if their associated permutations are conjugate.We also show that S-automorphisms have discrete spectrum if and only if they have zero entropy and every S-automorphism is either a Bernoulli shift, has discrete spectrum, or is a product of a Bernoulli shift and an automorphism with discrete spectrum.S-automorphism with discrete spectrum are those whose associated permutations contain only cycles of finite length. These automorphisms are studied according to the number of such finite cycles. Those whose permutations have infinitely many finite cycles with unbounded lengths are shown to be antiperiodic and those whose permutations have infinitely many finite cycles of bounded length are periodic with almost no fixed points. An example is given of two automorphisms of this latter type which are isomorphic but whose permutations are not conjugate.A complete isomorphism invariant is given for S-automorphisms whose associated permutations consist of finitely many finite cycles. Using this invariant we show that if φ is either a product of k disjoint cycles of prime power p^α, or a single cycle of length pq where p and q are primes, or a product of k disjoint cycles of prime lengths p₁ < p₂ < ··· < p_kand if ψ is a permutation such that T_ψand T_φ are isomorphic then ψ is conjugate to φ.     

A global Lipschitz continuity result for a domain-dependent Neumann eigenvalue problem for the Laplace operator

Let Ω be an open connected subset of Rⁿ of finite measure for which the Poincaré-Wirtinger inequality holds. We consider the Neumann eigenvalue problem for the Laplace operator in the open subset φ(Ω) of Rⁿ, where φ is a locally Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then, we show Lipschitz-type inequalities for the reciprocals of the eigenvalues delivered by the Rayleigh quotient. Then, we further assume that the imbedding of the Sobolev space W^1,2(Ω) into the space L²(Ω) is compact, and we prove the same type of inequalities for the projections onto the eigenspaces upon variation of φ.     

Toeplitz operators in several complex variables

Let S be the unit sphere in Cⁿ. We investigate the properties of Toeplitz operators on S, i.e., operators of the form Tφf = P(φf) where φ?L^∞(S) and P denotes the projection of L²(S) onto H²(S). The aim of this paper is to determine how far the extensive one-variable theory remains valid in higher dimensions. We establish the spectral inclusion theorem, that the spectrum of T_φ contains the essential range of φ, and obtain a characterization of the Toeplitz operators among operators on H²(S) by an operator equation. Particular attention is paid to the case where φ ? H^∞(S) + C(S) where C(S) denotes the algebra of continuous functions on S. Finally we describe a class of Toeplitz operators useful for providing counterexamples—in particular, Widom's theorem on the connectedness of the spectrum fails when n > 1.     

On a generalization of the Euler totient function

For a general polynomial Euler product F(s) we define the associated Euler totient function φ(n, F) and study its asymptotic properties. We prove that for F(s) belonging to certain subclass of the Selberg class of L-functions, the error term in the asymptotic formula for the sum of φ(n, F) over positive integers n ≤ x behaves typically as a linear function of x. We show also that for the Riemann zeta function the square mean value of the error term in question is minimal among all polynomial Euler products from the Selberg class, and that this property uniquely characterizes ζ(s).     

Integer points in backward orbits

A theorem of J. Silverman states that a forward orbit of a rational map φ(z) on P¹(K) contains finitely many S-integers in the number field K when (φ°φ)(z) is not a polynomial. We state an analogous conjecture for the backward orbits using a general S-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map φ(z)=zd, and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for zn−β when β≠0 is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for φn(z)−β is bounded independently of n.