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Addressing the intricate challenges in plane elasticity, especially with non-vanishing traction and complex geometries, requires innovative methods. This paper offers a novel approach, drawing inspiration from the Neumann problem for the inhomogeneous Cauchy-Riemann equations. Our method applies to domains conformally equivalent to a unit disk or an annulus, focusing on deriving explicit solutions for the displacement field rather than the stress tensor, which distinguishes it from most traditional approaches. We explore solutions for specific classical cases to demonstrate its efficacy, such as a cardioid domain, a ring domain with a shifted hole, and a gear-like structure. This work enhances the toolkit for researchers and practitioners tackling isotropic planar elastostatic challenges with a unified and flexible approach.

It is shown that the classical Bloch space and the Bergman space Aηp, induced by a radial doubling weight η, can be characterized by using the fractional derivative (Formula presented.) induced by two radial weights admitting certain doubling conditions. Here (Formula presented.) are the odd moments of ω, and f^(k) stands for the Maclaurin coefficient of the analytic function f in the unit disc D. The main findings of this paper generalize and complete in part recent results by Moreno, Peláez and de la Rosa [Fractional derivative description of the Bloch space, Potential Anal. 61 (2024), no. 3, 555–571], and Peláez and de la Rosa [Littlewood-Paley inequalities for fractional derivative on Bergman spaces, Ann. Fenn. Math. 47 (2022), no. 2, 1109–1130]. The arguments employed here rely on integral representations via Bergman reproducing kernels of the operator Rν,ω and hence differ from those used in the said papers.

We establish that the Volterra-type integral operator J_b on the Hardy spaces H^p of the unit ball Bn exhibits a rather strong rigid behavior. More precisely, we show that the compactness, strict singularity and ℓ^p-singularity of J_b are equivalent on H^p for any 1≤p<∞. Moreover, we show that the operator J_b acting on H^p cannot fix an isomorphic copy of ℓ² when p≠2.

We define essentially positive operators on Hilbert space as a class of self-adjoint operators whose essential spectra is contained in the non-negative real numbers and describe their basic properties. Using Toeplitz operators and the Berezin transform, we further illustrate the notion of essential positivity in the Hardy space and the Bergman space.

The boundedness of the maximal operator on the upper half-plane pi+ is established. Here pi+ is equipped with a positive Borel measure d omega(y)dx satisfying the doubling property omega ((0, 2t)) <= C omega ((0, t)). This result is connected to the Carleson embedding theorem, which we use to characterize the boundedness and compactness of the Volterra type integral operators on the Bergman spaces Ap omega(pi+).

In this paper, two specific sub-Hilbert spaces are studied. They arise from the action of a Toeplitz operator on Fock–Sobolev type spaces, in-duced by a general Gaussian type weight. The argument is based on analysing the reproducing kernel of the corresponding sub-Hilbert space.

We establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure dω(y) dx with the doubling property ω(0 , 2 t) ≤ Cω(0 , t). The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

This paper is concerned with the L-p integrability of N-harmonic functions with respect to the standard weights (1 - vertical bar x vertical bar(2))(alpha) on the unit ball B of R-n, n >= 2. More precisely, our goal is to determine the real (negative) parameters alpha for which (1 - vertical bar x vertical bar(2))(alpha/p) u(x) is an element of L-p(B) implies that u equivalent to 0 whenever u is a solution of the N-Laplace equation on B. This question is motivated by the uniqueness considerations of the Dirichlet problem for the N-Laplacian Delta(N).

Our study is inspired by a recent work of Borichev and Hedenmalm (Adv. Math. 264 (2014), 464-505), where a complete answer to the above question in the case n D 2 is given for the full scale 0 < p < infinity. When n >= 3, we obtain an analogous characterization for n-2/n-1 <= p < infinity and remark that the remaining case can be genuinely more difficult. Also, we extend the remarkable cellular decomposition theorem of Borichev and Hedenmalm to all dimensions.