Introduction to Potential Theory. By L. L. Helms. Pp. ix, 282. £7. 1970. (Wiley.)

Let $\varphi : \mathbb{R}^n\times [0,\,\infty)\to[0,\infty)$ be a Musielak-Orlicz function and $A$ an expansive dilation. Let $H^\varphi_A({\mathbb {R}}^n)$ be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. Its atomic characterization and some other maximal function characterizations of $H^\varphi_A({\mathbb {R}^n})$, in terms of the radial, the non-tangential and the tangential maximal functions, are known. In this article, the authors further obtain their characterizations in terms of the Lusin-area function, the $g$-function or the $g^\ast_\lambda$-function via first establishing an anisotropic Peetre's inequality of Musielak-Orlicz type. Moreover, the range of $\lambda$ in the $g^\ast_\lambda$-function characterization of $H^\varphi_A({\mathbb {R}}^n)$ coincides with the known best conclusions in the case when $H^\varphi_A({\mathbb {R}}^n)$ is the classical Hardy space $H^p({\mathbb{R}}^n)$ or the anisotropic Hardy space $H^p_A({\mathbb {R}}^n)$ or their weighted variants, where $p\in(0,\,1]$.

/pdf/littlewood-paley-characterizations-of-anisotropic-hardy-2ni70c72kz.pdf

Littlewood-paley characterizations of anisotropic hardy spaces of musielak-orlicz type

Let $$\vec A: = \left( {{A_1},{A_2}} \right)$$
 be a pair of expansive dilations and φ: ℝ
 n
 ×ℝ
 m
 ×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$
 via the anisotropic Lusin-area function and establish its atomic characterization, the $$\vec g$$
 
-function characterization, the $$\vec g_\lambda ^*$$
 -function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type Moreover, we prove that finite atomic decomposition norm on a dense subspace of $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$
 is equivalent to the standard infinite atomic decomposition norm As an application, we show that, for a given admissible triplet (
 $$\left( {\varphi ,q,\vec s} \right)$$
 ), if T is a sublinear operator and maps all (
 $$\left( {\varphi ,q,\vec s} \right)$$
 )-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$
 to B Another application is that we obtain the boundedness of anisotropic product singular integral operators from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$
 to L
 φ
 (R
 n
 × R
 m
 ) and from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$
 to itself, whose kernels are adapted to the action of $$\vec A$$
  The results of this article essentially extend the existing results for weighted product Hardy spaces on ℝ
 n
 × ℝ
 m
 and are new even for classical product Orlicz-Hardy spaces

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Given a>0, we construct a weighted Lebesgue measure on R^n for which the family of non constant curves has p-modulus zero for p\leq 1+a but the weight is a Muckenhoupt A_p weight for p>1+a. In particular, the p-weak gradient is trivial for small p but non trivial for large p. This answers an open question posed by several authors. We also give a full description of the p-weak gradient for any locally finite Borel measure on the real line.

The p-Weak Gradient Depends on p

Maximal Function Methods for Sobolev Spaces

https://www.ams.org/proc/2015-143-06/S0002-9939-2015-12416-1/S0002-9939-2015-12416-1.pdf

Lattice property of p -admissible weights

Given an open hyperbolic Riemannian manifold, we show that various vector spaces of harmonic functions coincide if and only if they are finite dimensional.

/pdf/harmonic-hardy-orlicz-spaces-4joimhq638.pdf

Harmonic hardy-orlicz spaces

Let W be an open Riemann surface and W a p-sheeted (1 < ρ < oo) unlimited covering surface of W. Denote by Δ, (resp. Δ/) the set of all minimal Kuramochi boundary points of W (resp. W). For ζ e Δ, , let!%(() be the (cardinal) number of the set of points ζ Ε Δι which lie over ζ and Mc the class of subregions Μ of W such that W \\ Μ is thin at ζ. Our main result is the following: ^ ( 0 = maxM£M('nw(M), where η^,(Μ) is the number of components π~'(Μ) and π is the projection of W onto W. Moreover, some applications of the above results are discussed when W is the unit disc. AMS-Classification: Primary 30F25, secondary 30F15

Kuramochi boundary of unlimited covering surfaces

/pdf/hyperbolic-riemann-surfaces-without-unbounded-positive-2u3ya5ue1w.pdf

Hyperbolic Riemann surfaces without unbounded positive harmonic functions

Consider an end Ω in the sense of Heins [7]: Ω is a relatively noncompact subregion of an open Riemann surface R, which is of null boundary and has a single ideal boundary component, and the relative boundary dΩ(^ 0) of Ω consists of finitely many analytic closed Jordan curves. We denote by P(Ω) the class of nonnegative harmonic functions on Ω with vanishing boundary values on dΩ. The minimum number of elements o f P ( Ω ) generating P(Ω) provided that such a finite set exists, otherwise oo, is referred to as the harmonic dimension of β, dimP(Ω) in notation. In terms of Martin compactification, it is known that dίmP(Ω) coincides with the number of minimal boundary points in the Martin compactification of R (cf. e.g. [4]), and hence dimP(Ω) = dimP(Ω') for any pair (β, Ω') of ends of R. Denote by D the punctured unit disc {0 < \z\ < 1} and let W be a p-sheeted (1 < p < oo) unlimited covering surface of D such that the projection of branch points of W accumulates only at z = 0. Then W is naturally considered as a subregion of an open Riemann surface R which is a p-sheeted unlimited covering surface of {0 < \z\ < oo}. I f R has a single ideal boundary component, it is seen that W is an end. We denote by ep the class of ends W of this kind. In this paper we are especially concerned with ends belonging to £p. First of all, it is noted that 1 < dimP(W) < p for each W € £p (cf. Heins [7]). Roughly speaking, if each sheet of W is closely connected with any of the other sheets, then dίmP(W) = I and if each sheet of W is faintly connected with the other sheets, then dimP(W) — p. This intuition is realized as follows. Consider two positive decreasing sequences {αn} and {bn} satisfying 6n+ι < an < bn < I and lim^^ an = 0. Set G = {0 < \z\ < 1} /, where / = U^=lln and In = [αn, bn]. We take p copies G\, , Gp of G and join the upper edge of In on Gj with the lower edge of In on Gj+i (j mod p) for every n. Then we obtain a p-sheeted covering surface W\ of D which belongs to 8P. For this end W\ we have showed the following (cf. [9], [11] and [14]).

/pdf/harmonic-dimension-of-covering-surfaces-and-minimal-fine-4ciilpbiao.pdf

Hiroaki Masaoka

Papers

Lattice property of p -admissible weights

Harmonic hardy-orlicz spaces

Kuramochi boundary of unlimited covering surfaces

Hyperbolic Riemann surfaces without unbounded positive harmonic functions

Harmonic dimension of covering surfaces and minimal fine neighborhood