Growth rate of the functions in Bergman type spaces

For 0<p,α<∞, let ‖f‖_p,α be the L^p-norm with respect the weighted measure . We define the weighted Bergman space A_α^p(D) consisting of holomorphic functions f with ‖f‖_p,α<∞. For any σ>0, let A^−σ(D) be the space consisting of holomorphic functions f in D with . If D has C² boundary, then we have the embedding A_α^p(D)⊂A^−(n+α)/p(D). We show that the condition of C²-smoothness of the boundary of D is necessary by giving a counter-example of a convex domain with C^1,λ-smooth boundary for 0<λ<1 which does not satisfy the embedding.     

On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line

In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the Dunkl-type fractional maximal operator Mβ, and the Dunkl-type fractional integral operator Iβ from the spaces Lp,α(R) to the spaces Lq,α(R), 1<p<q<∞, and from the spaces L1,α(R) to the weak spaces WLq,α(R), 1<q<∞. In the case , we prove that the operator Mβ is bounded from the space Lp,α(R) to the space L∞,α(R), and the Dunkl-type modified fractional integral operator is bounded from the space Lp,α(R) to the Dunkl-type BMO space BMOα(R). By this results we get boundedness of the operators Mβ and Iβ from the Dunkl-type Besov spaces to the spaces , 1<p<q<∞, 1/p−1/q=β/(2α+2), 1?θ?∞ and 0<s<1.     

Hadamard products with power functions and multipliers of Hardy spaces

We consider Hadamard products of power functions P(z)=(1−z)^−b with functions analytic in the open unit disk in the complex plane, and an integral representation is obtained when 0<Reb<2. Let where μ is a complex-valued measure on the closed unit disk Such sequences are shown to be multipliers of H^p for 1?p?∞. Moreover, if the support of μ is contained in a finite set of Stolz angles with vertices on the unit circle, we prove that {μ_n} is a multiplier of H^p for every p>0. When the support of μ is [0,1] we get the multiplier sequence which provides more concrete applications. We show that if the sequences {μ_n} and {ν_n} are related by an asymptotic expansion

     

Connection coefficients for Laguerre-Sobolev orthogonal polynomials

Laguerre-Sobolev polynomials are orthogonal with respect to an inner product of the form , where α>−1, λ?0, and , the linear space of polynomials with real coefficients. If dμ(x)=xαe−xdx, the corresponding sequence of monic orthogonal polynomials {Q_n^(α,λ)(x)} has been studied by Marcellán et al. (J. Comput. Appl. Math. 71 (1996) 245-265), while if dμ(x)=δ(x)dx the sequence of monic orthogonal polynomials {L_n^(α)(x;λ)} was introduced by Koekoek and Meijer (SIAM J. Math. Anal. 24 (1993) 768-782). For each of these two families of Laguerre-Sobolev polynomials, here we give the explicit expression of the connection coefficients in their expansion as a series of standard Laguerre polynomials. The inverse connection problem of expanding Laguerre polynomials in series of Laguerre-Sobolev polynomials, and the connection problem relating two families of Laguerre-Sobolev polynomials with different parameters, are also considered.     

Cores for Feller semigroups with an invariant measure

Let be an elliptic differential operator with unbounded coefficients on RN and assume that the associated Feller semigroup (T(t))_t?0 has an invariant measure μ. Then (T(t))_t?0 extends to a strongly continuous semigroup (Tp(t))_t?0 on Lp(μ)=Lp(RN,μ) for every 1?p<∞. We prove that, under mild conditions on the coefficients of A, the space of test functions is a core for the generator (Ap,Dp) of (Tp(t))_t?0 in Lp(μ) for 1?p<∞.     

Duality in Segal-Bargmann spaces

For α>0, the Bargmann projectionPα is the orthogonal projection from L²(γα) onto the holomorphic subspace , where γα is the standard Gaussian probability measure on Cn with variance (2α)⁻ⁿ. The space is classically known as the Segal-Bargmann space. We show that Pα extends to a bounded operator on Lp(γαp/2), and calculate the exact norm of this scaled Lp Bargmann projection. We use this to show that the dual space of the Lp-Segal-Bargmann space is an Lp^′ Segal-Bargmann space, but with the Gaussian measure scaled differently: (this was shown originally by Janson, Peetre, and Rochberg). We show that the Bargmann projection controls this dual isomorphism, and gives a dimension-independent estimate on one of the two constants of equivalence of the norms.     

Estimations anisotropes dans les convexes de type fini

Let Ω be a smoothly bounded convex domain of finite type m and f be a (0,1)-form -closed in Ω. It is proved that the equation admits a solution u belonging to the space Λ1(Ω) (respectively to the anisotropic space Γ^α_(ρ) of McNeal-Stein, for all α,0<α<1/m) if the anisotropic norm - introduced by Bruna-Charpentier-Dupain - is finite (respectively if the Euclidian norm ‖f‖_∞ of the form f is finite).     

Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights

We completely describe those positive Borel measures μ in the unit disc D such that the Bergman space Ap(w)⊂Lq(μ), 0<p,q<∞, where w belongs to a large class W of rapidly decreasing weights which includes the exponential weights , α>0, and some double exponential type weights.As an application of that result, we characterize the boundedness and the compactness of Tg:Ap(w)→Aq(w), 0<p,q<∞, w∈W, where Tg is the integration operator

     

The Riemannian product structures of spacelike hypersurfaces with constant k-th mean curvature in the de Sitter spaces

In this paper, we investigate complete spacelike hypersurfaces in the de Sitter space with constant k-th mean curvature and two distinct principal curvatures one of which is simple. We obtain some characterizations of the Riemannian product H¹(c₁)×Sn−1(c₂) or Hn−1(c₁)×S¹(c₂) in the de Sitter space .     

A new proof of Poltoratskii's theorem

We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure μ on and any a.e. w.r.t. μ_sing, where μ_sing is the part of μ which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, and F_μ(z)=∫(x−z)⁻¹dμ(x).     

Weight characterization of an averaging operator

Let 0<α<1 and , x?0. A factorization theorem is given, which provides a weight characterization of the space of all positive functions f such that T_αf belongs to L^p_w, 1<p<∞, w a weight function. This theorem yields a two-sided estimate for the norm of T_αf. An analogous result holds for α=0. In the latter case, it is also shown that the averaging Hardy operator T₀ and its dual  are comparable in L^p_w, 1<p<∞, if w belongs to the Muckenhoupt weight class A_p.     

Heat-diffusion maximal operators for Laguerre semigroups with negative parameters

We prove L^p boundedness for the maximal operator of the heat semigroup associated to the Laguerre functions, , when the parameter α is greater than -1. Namely, the maximal operator is of strong type (p,p) if p>1 and , when -1<α<0. If α?0 there is strong type for 1<p?∞. The behavior at the end points is studied in detail.     

Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).