On a New Characterization of Besov Spaces with Negative Exponents

2009 ◽  
pp. 273-284 ◽  
Author(s):  
Moshe Marcus ◽  
Laurent Véron
Keyword(s):  
Besov Spaces

Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems

1981 ◽  
Vol 90 (1-2) ◽  
pp. 63-70 ◽  
Author(s):  
Bernd Carl
Keyword(s):  
Banach Space ◽  
Asymptotic Behaviour ◽  
Besov Spaces ◽  
Eigenvalue Problems ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Entropy Numbers ◽  
Diagonal Operators

SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.

scholarly journals Characterization of nonlinear Besov spaces

2019 ◽  
Vol 373 (1) ◽  
pp. 529-550 ◽  
Author(s):  
Chong Liu ◽  
David J. Prömel ◽  
Josef Teichmann
Keyword(s):  
Besov Spaces

scholarly journals Harmonic Besov spaces on the ball

2016 ◽  
Vol 27 (09) ◽  
pp. 1650070 ◽  
Author(s):  
Seçil Gergün ◽  
H. Turgay Kaptanoğlu ◽  
A. Ersin Üreyen
Keyword(s):  
Besov Spaces ◽  
Bergman Spaces ◽  
Reproducing Kernels ◽  
Integral Representations ◽  
Kernel Estimates ◽  
Harmonic Bergman Spaces ◽  
Infinite Sums ◽  
Bergman Projections ◽  
Derivatives Of

We initiate a detailed study of two-parameter Besov spaces on the unit ball of [Formula: see text] consisting of harmonic functions whose sufficiently high-order radial derivatives lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels are weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel. Estimates of the growth of kernels lead to characterization of integral transformations on Lebesgue classes. The transformations allow us to conclude that the order of the radial derivative is not a characteristic of a Besov space as long as it is above a certain threshold. Using kernels, we define generalized Bergman projections and characterize those that are bounded from Lebesgue classes onto Besov spaces. The projections provide integral representations for the functions in these spaces and also lead to characterizations of the functions in the spaces using partial derivatives. Several other applications follow from the integral representations such as atomic decomposition, growth at the boundary and of Fourier coefficients, inclusions among them, duality and interpolation relations, and a solution to the Gleason problem.

Characterization of weighted analytic Besov spaces in terms of operators of fractional differentiation

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Alexey Karapetyants ◽  
Ferdos Kodzoeva
Keyword(s):  
Kernel Function ◽  
Besov Spaces ◽  
Hadamard Product ◽  
Fractional Differentiation ◽  
Bergman Kernel Function ◽  
Similar Weight ◽  
Weighted Besov Space ◽  
Weighted Bergman Projection ◽  
Weighted Bergman Kernel

AbstractLet $$\mathbb{D}$$ stand for the unit disc in the complex plane ℂ. Given 0 < p < ∞, −1 < λ < ∞, the analytic weighted Besov space $$B_p^\lambda \left( \mathbb{D} \right)$$ is defined to consist of analytic in $$\mathbb{D}$$ functions such that $$\int\limits_\mathbb{D} {\left( {1 - \left| z \right|^2 } \right)^{Np - 2} \left| {f^{\left( N \right)} \left( z \right)} \right|^p d\mu _\lambda \left( z \right) < \infty ,}$$ where dμ λ(z) = (λ + 1)(1 − |z|2)λ dμ(z), $$d\mu (z) = \tfrac{1} {\pi }dxdy$$, and N is an arbitrary fixed natural number, satisfying N p > 1 − λ.We provide a characterization of weighted analytic Besov spaces $$B_p^\lambda \left( \mathbb{D} \right)$$, 0 < p < ∞, in terms of certain operators of fractional differentiation R zα,t of order t. These operators are defined in terms of construction known as Hadamard product composition with the function b. The function b is calculated from the condition that R zα,t (uniquely) maps the weighted Bergman kernel function $$\left( {1 - z\bar w} \right)^{ - 2 - \alpha }$$ to the similar (weight parameter shifted) kernel function $$\left( {1 - z\bar w} \right)^{ - 2 - \alpha - t}$$, t > 0. We also show that $$B_p^\lambda \left( \mathbb{D} \right)$$ can be thought as the image of certain weighted Lebesgue space $$L^p \left( {\mathbb{D},d\nu _\lambda } \right)$$ under the action of the weighted Bergman projection $$P_\mathbb{D}^\alpha$$.

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