Wavelet characterization of weighted spaces

We give a characterization of weighted Hardy spaces H ^p (w), valid for a rather large collection of wavelets, 0 <p ≤ 1,and weights w in the Muckenhoupt class A _∞ We improve the previously known results and adopt a systematic point of view based upon the theory of vector-valued Calderón-Zygmund operators. Some consequences of this characterization are also given, like the criterion for a wavelet to give an unconditional basis and a criterion for membership into the space from the size of the wavelet coefficients.     

On some Hilbert spaces of harmonic functions

Two kinds of spaces of harmonic functions defined on m-dimensional domains are considered. In the first case, the spherical domain is a m-dimensional ball B_r in the second case B_r is replaced by B_r1,r2:=B_r2B_r1. In both cases, several inner products are considered and Hilbert space properties are proved. The reproducing kernel functions of these spaces and hence some new representations for harmonic functions are also obtained     

A characterization of harmonic functions

Gauss' mean value characterization of harmonic functions involves circles (or spheres) centered at every point of the domain of the function. The present paper gives a criterion of this type which involves only one point of the domain; to make up for this, circles or spheres are replaced by a larger family of convex curves or surfaces which surround this point.     

Wavelet characterization of weighted Triebel-Lizorkin spaces

In this paper we use wavelets to characterize weighted Triebel-Lizorkin spaces. Our weights belong to the Muckenhoupt class Aq and our weighted Triebel-Lizorkin spaces are weighted atomic Triebel-Lizorkin spaces.     

Equivalent norms onL p spaces of harmonic functions

This note gives necessary and sufficient conditions for a measurable setG in the unit ballB in ? ⁿ to satisfy the following property: there exists a constantC>0 such that $$\int\limits_B {|f|^2 } dm \leqslant C\int\limits_G {|f|^2 } dm$$ for everyf∈L ² (B, dm) which is harmonic inB. Herem is the Lebesgue measure of dimensionn. The same condition is sufficient if any exponentp>0 replaces 2 in (^*), and if certain weighted measures replacem. Applications to the problem of representing harmonic functions inL ² (B) as a sum of kernel functions are indicated.     

Some embedding theorems for spaces of harmonic functions

For the domains of the space R ⁿ ,n2, with a finite number of conical points, one proves embedding theorems for the spaces of harmonic functions which generalize the Littlewood-Paley and Carleson theorems. Let ·_p, be a norm which is transferred in some natural manner to the space of harmonic functions in the domain and which in the unit circle of the space ² turns into the norm of the Hardy space H^p and let ^p() be the space of harmonic functions in with this norm. One establishes, in particular, sufficient conditions on the measureV, for which one has the inequality.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 191–194, 1976.     

A note onL p spaces of harmonic functions

The following result is proved: Letp>0,a>?1. Suppose thatG is a measurable subset ofB, the unit ball in ? ^N , for which there exists a positive constantA ₁, so that $$\int\limits_B {\left( {1 - \left| x \right|} \right)^a \left| {f(x)} \right|^p dm \leqslant A_1 } \int\limits_G {\left( {1 - \left| x \right|} \right)^a \left| {f(x)} \right|^p dm}$$ for each function that is harmonic inB and for which the left-hand side of the above inequality is finite. Then there is a positive constantA ₂ so that for each ballK with center on ?B, $$m\left( {K \cap B} \right) \leqslant A_2 m\left( {K \cap G} \right).$$ Herem denotes Lebesgue measure in ? ^N . This result answers a question left open byDan Luecking [2].     

WAVELET CHARACTERIZATION OF WEIGHTED TRIEBEL－LIZORKIN SPACES

In this paper we use wavelets to characterize weighted Triebel-Lizorkin spaces,Our weights belong to the Muckenhoupt class Aq and our weighted Triebel-Lizorkin spaces are weighted atomic Triebel-Lizorkin spaces.     

Hardy spaces and partial derivatives of conjugate harmonic functions

In this paper we give necessary and sufficient conditions for a harmonic vector and all its partial derivatives to belong to for all .

     

Spectrum,harmonic functions,and hyperbolic metric spaces

The main result of the paper says, in particular, that ifM is a complete simply connected Riemannian manifold with Ricci curvature bounded from below and without focal points, which is also a hyperbolic metric space in the sense of Gromov, then the top λ of theL ²-spectrum of the Laplace-Beltrami operator Δ is negative, the Martin boundary ofM corresponding to Δ is homeomorphic to the sphere at infinityS(∞), and the harmonic measures onS(∞) have positive Hausdorff dimensions. These generalize the results of [AS], [An1], [Ki], [KL] and [BK]. Moreover, if dimM=2, then in the presence of the other conditions the hyperbolicity is also necessary for λ<0. The machinery consists of a combination of geometrical and probabilistic means. Partially supported by U.S.-Israel BSF. Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

EXISTENCE OF RADIAL LIMITS OF HARMONIC FUNCTIONS IN BANACH SPACES

The following result is established:let X be a Banach space without the Radon-Nikodym property,there exists a uniformly bounded harmonic function f defined onthe open unit disk of C with values in X,such that for almost allθ∈[0,2π],(?)f(re~(iθ))does not exist.     

Compactness of the Hardy-Littlewood operator on some spaces of harmonic functions

We study the compactness of the Hardy-Littlewood operator on several spaces of harmonic functions on the unit ball in ? ⁿ such as: a-Bloch, weighted Hardy, weighted Bergman, Besov, BMO _p , and Dirichlet spaces.     

Operators of harmonic analysis in weighted spaces with non-standard growth

Last years there was increasing an interest to the so-called function spaces with non-standard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia's extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. There are also given their vector-valued analogues.