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1.
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
2.
In this paper, we prove the commutator T
b
generated by the strongly singular integral operator T and the function b is bounded from L
p
(w) to L
q
(w
1−q
) if and only if b ∈ Lip
β
(w), where w ∈ A
1, 0 < β < 1, 1 < p < n/β and 1/q = 1/p − β/n. To do this, we first show a maximal function estimate for the commutator. 相似文献
3.
For p ∈ R, the generalized logarithmic mean Lp(a,b) and Seiffert's mean T(a,b) of two positive real numbers a and b are defined in (1.1) and (1.2) below respectively. In this paper, we find the greatest p and least q such that the double-inequality Lp(a,b) < T(a,b) < Lq(a,b) holds for all a,b > 0 and a ≠ b. 相似文献
4.
We find the greatest value p and least value q such that the double inequality L p (a, b)?<?T(a, b)?<?L q (a, b) holds for all a, b?>?0 with a?≠ b, and give a new upper bound for the complete elliptic integral of the second kind. Here ${T(a,b)=\frac{2}{\pi}\int\nolimits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}d\theta}$ and L p (a, b)?=?(a p+1?+?b p+1)/(a p ?+?b p ) denote the Toader and p-th Lehmer means of two positive numbers a and b, respectively. 相似文献
5.
K. Kazarian V. N. Temlyakov 《Proceedings of the Steklov Institute of Mathematics》2013,280(1):181-190
We consider a weighted L p space L p (w) with a weight function w. It is known that the Haar system H p normalized in L p is a greedy basis of L p , 1 < p < ∞. We study a question of when the Haar system H p w normalized in L p (w) is a greedy basis of L p (w), 1 < p < ∞. We prove that if w is such that H p w is a Schauder basis of L p (w), then H p w is also a greedy basis of L p (w), 1 < p < ∞. Moreover, we prove that a subsystem of the Haar system obtained by discarding finitely many elements from it is a Schauder basis in a weighted norm space L p (w); then it is a greedy basis. 相似文献
6.
Jacob Burbea 《Rendiconti del Circolo Matematico di Palermo》1978,27(2):259-269
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. 相似文献
7.
We determine the smallest Schatten class containing all integral operators with kernels inL p(Lp', q)symm, where 2 <p∞ and 1≦q≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1]. 相似文献
8.
The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained. 相似文献
9.
Fang Gensun 《中国科学A辑(英文版)》2001,44(9):1126-1131
The main result of this paper asserts that if a function f is in the class Bπ,p, 1 <p < ∞; that is, those p-integrable functions whose Fourier transforms are supported in the interval [ - π, π], then f and its
derivatives f(j) j = 1, 2, …, can be recovered from its sampling sequence{f(k)} via the cardinal interpolating spline of degree m in the metric
ofL
q(ℝ)), 1 <p=q < ∞, or 11 <p=q < ⩽ ∞. 相似文献
10.
Let 0<p≤1<q<0, andw
1
,w
2
∈ A
1
(Muckenhoupt-class). In this paper the authors prove that the strongly singular convolution operators are bounded from the
homogeneous weighted Herz-type Hardy spacesH Kα, p
q(w1; w2) to the homogeneous weighted Herz spacesK
α, p
q
(w1; w2), provided α=n(1−1/q). Moreover, the boundedness of these operators on the non-homogeneous weighted Herz-type Hardy spacesH K
α, p
q
(w
1;w
2) is also investigated.
Supported by the National Natural Science Foundation of China 相似文献
11.
In an exterior domain Ω??n, n ? 2, we consider the generalized Stokes resolvent problem in Lq-space where the divergence g = div u and inhomogeneous boundary values u = ψ with zero flux ∫?Ωψ·N do = 0 may be prescribed. A crucial step in our approach is to find and to analyse the right space for the divergence g. We prove existence, uniqueness and a priori estimates of the solution and get new results for the divergence problem. Further, we consider the non-stationary Stokes system with non-homogeneous divergence and boundary values and prove estimates of the solution in L5(0, T;Lq(Ω)) for 1 < s, q < ∞. 相似文献
12.
Djordjije Vujadinović 《Integral Equations and Operator Theory》2013,76(2):213-224
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. 相似文献
13.
In this paper, we study integral operators of the form Tαf(x)=∫Rn|x-A1y|-α1 ··· |x-Amy|-αmf(y)dy,where Ai are certain invertible matrices, αi 0, 1 ≤ i ≤ m, α1 + ··· + αm = n-α, 0 ≤α n. For 1/q = 1/p-α/n , we obtain the Lp (Rn, wp)-Lq(Rn, wq) boundedness for weights w in A(p, q) satisfying that there exists c 0 such that w(Aix) ≤ cw(x), a.e. x ∈ Rn , 1 ≤ i ≤ m.Moreover, we obtain theappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators. 相似文献
14.
Bei Hu 《Journal of Mathematical Analysis and Applications》2008,340(1):598-605
In this paper we show that b∈Lipβ,μ if and only if the commutator [b,T] of the multiplication operator by b and the singular integral operator T is bounded from Lp(μ) to Lq(μ1−q), where 1<p<q<∞, 0<β<1 and 1/q=1/p−β/n. Also we will obtain that b∈Lipβ,μ if and only if the commutator [b,Iα] of the multiplication operator by b and the fractional integral operator Iα is bounded from Lp(μ) to Lr(μ1−(1−α/n)r), where 1<p<∞, 0<β<1 and 1/r=1/p−(β+α)/n with 1/p>(β+α)/n. 相似文献
15.
《Journal of Complexity》2001,17(2):467-492
We investigate optimal non-linear approximations of multivariate periodic functions with mixed smoothness. In particular, we study optimal approximation using sets of finite cardinality (as measured by the classical entropy number), as well as sets of finite pseudo-dimension (as measured by the non-linear widths introduced by Ratsaby and Maiorov). Approximation error is measured in the Lq(Td)-sense, where Td is the d-dimensional torus. The functions to be approximated are in the unit ball SBrp, θ of the mixed smoothness Besov space or in the unit ball SWrp of the mixed smoothness Sobolev space. For 1<p, q<∞, 0<θ⩽∞ and r>0 satisfying some restrictions, we establish asymptotic orders of these quantities, as well as construct asymptotically optimal approximation algorithms. We particularly prove that for either r>1/p and θ⩾p or r>(1/p−1/q)+ and θ⩾min{q, 2}, the asymptotic orders of these quantities for the Besov class SBrp, θ are both n−r(log n)(d−1)(r+1/2−1/θ). 相似文献
16.
Maria De Natividade 《Monatshefte für Mathematik》2011,164(1):87-114
Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces L ??(w) in terms of the fundamental function of L ??(w). In particular, we prove that these bases are greedy in L ??(w) if and only if L ??(w) =?L p (w), 1?<?p?<???. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For L p (w) the approximation spaces are identified with weighted Besov spaces. 相似文献
17.
A. M. Sedletskii 《Mathematical Notes》2006,79(5-6):697-706
Let g be a given function in L 1 = L 1(0, 1), and let B be one of the spaces L p (0, 1), 1 ≤ p < ∞, or C 0[0, 1]. We prove that the set of all convolutions f * g, f ∈ B, is dense in B if and only if g is nontrivial in an arbitrary right neighborhood of zero. Under an additional restriction on g, we prove the equivalence in B of the systems f n * g and I f n , where f n ∈ L 1, n ∈ ?, and I f = f * 1 is the antiderivative of f. As a consequence, we obtain criteria for the completeness and basis property in B of subsystems of antiderivatives of g. 相似文献
18.
《Journal of Complexity》1998,14(4):448-453
LetP⊂[0, 1]dbe ann-point set and letw: P→[0, ∞) be a weight function withw(P)=∑z∈P w(z)=1. TheL2-discrepancy of the weighted set (P, w) is defined as theL2-average ofD(x)=vol(Bx)−w(P∩Bx) overx∈[0, 1]d, where vol(Bx) is the volume of thed-dimensional intervalBx=∏dk=1 [0, xk). The exponent of discrepancyp* is defined as the infimum of numberspsuch that for all dimensionsd⩾1 and allε>0 there exists a weighted set of at mostKε−ppoints in [0, 1]dwithL2-discrepancy at mostε, whereK=K(p) is a suitable number independent ofεandd. Wasilkowski and Woźniakowski proved thatp*⩽1.4779, by combining known bounds for the error of numerical integration and using their relation toL2-discrepancy. In this note we observe that a careful treatment of a classical lower- bound proof of Roth yieldsp*⩾1.04882, and by a slight modification of the proof we getp*⩾1.0669. Determiningp* exactly seems to be quite a difficult problem. 相似文献
19.
We investigate two problems concerning uniform approximation by weighted rationals {w nrn~ ∞ n=1 }, wherer n=pn Namely, forw(x):=e x we prove that uniform convergence to 1 ofw nrn is not possible on any interval [0,a] witha>2π. Forw(x):=x ?, ?>1, we show that uniform convergence to 1 ofw nrn is not possible on any interval [b, 1] withb<tan 4(π(??1)/4?). (The latter result can be interpreted as a rational analogue of results concerning “incomplete polynomials.”) More generally, for α≥0, β≥0, α+β>0, we investigate forw(x)=e x andw(x)=x ?, the possibility of approximation byw n pn/qn~ ∞ n=1 , where depp n≤αn, degq n≤βn. The analysis utilizes potential theoretic methods. These are essentially sharp results though this will not be established in this paper. 相似文献
20.
Richard A Alò André de Korvin Vo Van Tho 《Journal of Mathematical Analysis and Applications》1978,63(3):563-590
For a Banach space E and for 1 ? p < ∞ let ?p<∞ let LEp(μ) = LEp(S,B,μ) denote all Bochner p-integrable E-valued functions on a measure space (S,B,μ). Under study are convergence theorems for integrals of functions in LEp(μ) with respect to Nemytskii measures. Weak integrals are then denoted to Hammerstein operators, and a study of topologies generated by vector measures leads to a characterization of compact Hammerstein operators. 相似文献