Higher order Riesz transforms associated with Bessel operators

In this paper we investigate Riesz transforms R _μ ^(k) of order k≥1 related to the Bessel operator Δ_μ f(x)=-f”(x)-((2μ+1)/x)f’(x) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We obtain that for every k≥1, R _μ ^(k) is a principal value operator of strong type (p,p), p∈(1,∞), and weak type (1,1) with respect to the measure dλ(x)=x ^2μ+1  dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which R _μ ^(k) maps L ^p (ω) into itself and L ¹(ω) into L ^1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman.     

The Bellman function,the two-weight Hilbert transform,and embeddings of the model spaces<Emphasis Type="Italic">K</Emphasis><Subscript>θ</Subscript>

This paper is devoted to embedding theorems for the spaceK _θ , where θ is an inner function in the unit disc D. It turns out that the question of embedding ofK _θ into L²(Μ) is virtually equivalent to the boundedness of the two-weight Hilbert transform. This makes the embedding question quite difficult (general boundedness criteria of Hunt-Muckenhoupt-Wheeden type for the twoweight Hilbert transform have yet to be found). Here we are not interested in sufficient conditions for the embedding ofKg into L²(Μ) (equivalent to a certain two-weight problem for the Hilbert transform). Rather, we are interested in the fact that a certain natural set of conditions is not sufficient for the embedding ofK _θ intoL ² (Μ) (equivalently, a certain set of conditions is not sufficient for the boundedness in a two-weight problem for the Hilbert transform). In particular, we answer (negatively) certain questions of W. Cohn about the embedding ofK _θ into L²(Μ). Our technique leads naturally to the conclusion that there can be a uniform embedding of all the reproducing kernels ofK _θ but the embedding of the wholeK _θ intoL ²(Μ) may fail. Moreover, it may happen that the embedding into a potentially larger spaceL ²(μ) fails too. Both authors are supported by the NSF grant DMS 9970395 and joint American-Israeli grant BSF 00030.     

Riesz transforms for a non-symmetric Ornstein-Uhlenbeck semigroup

Let (ℋ _t ) _t≥0 be the Ornstein-Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix −λ(I+R), where λ>0 and R is a skew-adjoint matrix and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L ²(γ _∞). We investigate the weak type 1 estimate of the Riesz transforms for (ℋ _t ) _t≥0. We prove that if the matrix R generates a one-parameter group of periodic rotations then the first order Riesz transforms are of weak type 1 with respect to the invariant measure γ _∞. We also prove that the Riesz transforms of any order associated to a general Ornstein-Uhlenbeck semigroup are bounded on L ^p (γ _∞) if 1<p<∞. The authors have received support by the Italian MIUR-PRIN 2005 project “Harmonic Analysis” and by the EU IHP 2002-2006 project “HARP”.     

Singular Integrals with Bilinear Phases

We prove the boundedness from Lp（T2） to itself, 1 〈 p 〈∞, of highly oscillatory singular integrals Sf（x, y） presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of integration, roughly speaking, defined by ｜y′｜ 〉 ｜x′｜, and presenting phases λ（Ax ＋ By） with 0≤ A, B ≤ 1 and λ≥ 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A1 B and A involved. Our method extends to a more general family of phases. These results are relevant to problems of almost everywhere convergence of double Fourier and Walsh series.     

On semigroups generated by restrictions of elliptic operators to invariant subspaces

Let A be the closed unbounded operator inL _p(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ ₁≦argλ≦θ ₂} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1<p<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2≦θ<θ ₂≦3π/2 ; and there exists a constantc such that (I)││(λ-A)^-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE ⁻ andE ⁺ such thatA is completely reduced by the direct sum decompositionL _p(G)=E⁻L_p (G) ⊕E⁺L_p (G) and each of the operatorsAE ⁻ and—AE ⁺ is the infinitestimal generator of an analytic semigroup.     

Boundedness criteria for singular integrals in weighted grand lebesgue spaces

Boundedness criteria for the Calderón singular integral, Riesz transform and Cauchy singular integral in generalized weighted grand Lebesgue spaces L ^p),θ _w , 1 < p < ∞, are studied. It is shown that an operator K of this type is bounded in L ^p),θ _w if and only if the weight w satisfies the Muckenhoupt A _p condition. Bibliography: 15 titles.     

On an oppenheim-type conjecture for systems of quadratic forms

In this paper we establish dimension freeL ^p (ℝ ⁿ ,|x|^α) norm inequalities (1<p<∞) for the oscillation and variation of the Riesz transforms in ℝ ⁿ . In doing so we findA _p -weighted norm inequalities for the oscillation and the variation of the Hilbert transform in ℝ. Some weighted transference results are also proved. Partially supported by European Commission via the TMR network “Harmonic Analysis”.     

Hölder continuity for spatial and path processes via spectral analysis

For ν(dθ), a σ-finite Borel measure on R ^d , we consider L ²(ν(dθ))-valued stochastic processes Y(t) with te property that Y(t)=y(t,·) where y(t,θ)=∫ ^t ₀ e ^{−λ(θ)( t − s )} dm(s,θ) and m(t,θ) is a continuous martingale with quadratic variation [m](t)=∫ ^t ₀ g(s,θ)ds. We prove timewise H?lder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions of the form dX=AXdt+GdW with W a cylindrical Brownian motion. Maximal inequalities and H?lder continuity results are also provenfor the path process _t (τ)≗Y(τt∧t). Received: 25 June 1999 / Revised version: 28 August 2000 /?Published online: 9 March 2001     

The ρ-variation of the Hermitian Riesz transform

In this paper we prove the behaviour in weighted Lp spaces of the oscillation and variation of the Hilbert transform and the Riesz transform associated with the Hermite operator of dimension 1. We prove that this operator maps LP（R, w（x）dx） into itself when w is a weight in the Ap class for 1 〈 p 〈 ∞. For p = 1 we get weak type for the A1 class. Weighted estimated are also obtained in the extreme case p = ∞.     

Dimension free estimates for Riesz transforms of some Schr?dinger operators

We prove dimension free L ^∞ → L ^∞-estimates for the Riesz transform T = V L ⁻¹, L = −Δ + V, where Δ is the Laplacian in ℝ ^d , and the polynomial V ≥ 0 satisfies C. L. Fefferman conditions; see [7]. As a corollary we get dimension free L ^p → L p(ℓ ²)-estimates, 1 < p < ∞, for the vector of Riesz transforms.