Explore the vast range of titles available.

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let In another article to appear, we will prove that when

Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an embedding between the two? A decomposition space We establish readily verifiable criteria which ensure the existence of a continuous inclusion (“an embedding”) In a nutshell, in order to apply the embedding results presented in this article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings. These sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of We also prove a The resulting embedding theory is illustrated by applications to

The spectral theory on the S -spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional heat diffusion and to the spectral theory for the Dirac operator on manifolds. In this seminal paper we introduce the harmonic functional calculus based on the S -spectrum and on an integral representation of axially harmonic functions. This calculus can be seen as a bridge between harmonic analysis and the spectral theory. The resolvent operator of the harmonic functional calculus is the commutative version of the pseudo S -resolvent operator. This new calculus also appears, in a natural way, in the product rule for the F -functional calculus.

The goal of this article is to define, explore and analyze two new families of meromorphically harmonic functions by applying the concept of a certain generalized convolution q -operator along with the idea of convolution. We investigate convolution properties and sufficiency criteria for these families of meromorphically harmonic functions. Some of the interesting consequences of our investigation are also included.

Suppose α , β ∈ R ∖ Z − such that α + β > − 1 and 1 ≤ p ≤ ∞ . Let u = P α , β [ f ] be an ( α , β ) -harmonic function on D , the unit disc of C , with the boundary f being absolutely continuous and f ˙ ∈ L p ( 0 , 2 π ) , where f ˙ ( e i θ ) : = d d θ f ( e i θ ) . In this paper, we investigate the membership of the partial derivatives ∂ z u and ∂ z ‾ u in the space H G p ( D ) , the generalized Hardy space. We prove, if α + β > 0 , then both ∂ z u and ∂ z ‾ u are in H G p ( D ) . For α + β < 0 , we show if ∂ z u or ∂ z ‾ u ∈ H G 1 ( D ) then u = 0 or u is a polyharmonic function.

In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of Brunn-Minkowski type for a nonlinear capacity, In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: In the second part of this article we study a Minkowski problem for a certain measure associated with a compact convex set

Recently, it is proven that positive harmonic functions defined in the unit disc or the upper half-plane in $\\mathbb{C}$ are contractions in hyperbolic metrics [14]. Furthermore, the same result does not hold in higher dimensions as shown by given counterexamples [16]. In this paper, we shall show that positive (or bounded) harmonic functions defined in the unit ball in $\\mathbb{R}^{n}$ are Lipschitz in hyperbolic metrics. The involved method in main results allows to establish essential improvements of Schwarz type inequalities for monogenic functions in Clifford analysis [24, 25] and octonionic analysis [21] in a unified approach.

In this paper, we propose composition products in the class of complex harmonic functions so that the composition of two such functions is again a complex harmonic function. From here, we begin the study of the iterations of the functions of this class showing briefly its potential to be a topic of future research. In parallel, we define and study composition operators on a Hardy type space denoted by H H 2 ( D ) of complex harmonic functions also introduced for us in the present work. The symbols of these composition operators have of form χ + π ¯ where χ , π are analytic functions from D into D . We also analyze the space of bounded linear operators on H H 2 ( D ) .

Wegive a complete characterization of functions which are at the same time harmonic and M-harmonic in the unit polydisc. These are precisely functions which are harmonic in each of the variables, or equivalently those that can be written as a linear combination of functions which are in each of the variables either holomorphic or conjugate holomorphic. As a consequence we obtain characterization of functions u s uch that both u and us (an integer s ? 2) have this property. If we additionally assume that u is real valued, then u is constant. Our results stand in contrast to results known for such functions in the unit ball in Cn .

We verify a long-standing conjecture on the membership of univalent harmonic mappings in the Hardy space, whenever the functions have a “nice” analytic part. We also produce a coefficient estimate for these functions, which is in a sense best possible. The problem is then explored in a new direction, without the additional hypothesis. Interestingly, our ideas extend to certain classes of locally univalent harmonic mappings. Finally, we prove a Baernstein-type extremal result for the function $\\log (h'+cg')$ , when $f=h+\\overline {g}$ is a close-to-convex harmonic function, and c is a constant. This leads to a sharp coefficient inequality for these functions.