The wavelet characterization of Herz-type Hardy spaces with variable exponent

In this paper, using the atomic decomposition of the Herz-Morrey-Hardy spaces with variable exponent, the wavelet characterization by means of a local version of the discrete tent spaces with variable exponent is established. As an application, the boundedness of the fractional integral operators from variable exponent Herz-Morrey-Hardy spaces into variable exponent Herz-Morrey spaces is obtained.

In this paper, we introduce Hardy spaces with variable exponents defined on a probability space and develop the martingale theory of variable Hardy spaces. We prove the weak-type and strong-type inequalities on Doob’s maximal operator, and we get a (1,p(⋅),∞)-atomic decomposition for Hardy martingale spaces associated with conditional square functions. As applications, we obtain a dual theorem and the John–Nirenberg inequalities in the frame of variable exponents. The key ingredient is that we find a condition with a probabilistic characterization of p(⋅) to replace the so-called log-Hölder continuity condition in Rn.

This paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

Hardy spaces with variable exponents and generalized Campanato spaces

Via the boundedness of intrinsic g-functions from the Hardy spaces with variable exponent, ℋp(·)(ℝn), into Lebesgue spaces with variable exponent, Lp(·)(ℝn), and establishing some estimates on a discrete Littlewood-Paley g-function and a Peetre-type maximal function, we obtain several equivalent characterizations of ℋp(·)(ℝn) in terms of wavelets, which extend the wavelet characterizations of the classical Hardy spaces. The main ingredients are that, we overcome the difficulties of the quasi-norms of ℋp(·)(ℝn) by elaborately using an observation and the Fefferman-Stein vector-valued maximal inequality on Lp(·)(ℝn), and also overcome the difficulty of the failure of q = 2 in the atomic decomposition of ℋp(·)(ℝn) by a known idea.

The theory of Hardy spaces has close connections to many branches of mathematics including Fourier analysis, harmonic analysis, singular integrals, potential theory and operator theory, and has found essential applications in robust control engineering. For each application, the ability to represent elements of these classes by series or integral formulas is of utmost importance. This self-contained text provides an introduction to a wide range of representation theorems and provides a complete description of the representation theorems with direct proofs for both classes of Hardy spaces: Hardy spaces of the open unit disc and Hardy spaces of the upper half plane. With over 300 exercises, many with accompanying hints, this book is ideal for those studying Advanced Complex Analysis, Function Theory or Theory of Hardy Spaces. Advanced undergraduate and graduate students will find the book easy to follow, with a logical progression from basic theory to advanced research.

Some new function spaces and their applications to Harmonic analysis

In this paper, a certain Herz-type Hardy spaces with variable exponent are introduced, and characterizations of these spaces are established in terms of atomic and molecular decompositions. Using these decompositions, the authors obtain the boundedness of some operators on the Herz-type Hardy spaces with variable exponent.

Traces of functions in Hardy and Besov spaces on Lipschitz domains with applications to compensated compactness and the theory of Hardy and Bergman type spaces

In their 1977 Bulletin of AMS paper [CoWe77], Ronald Coifman and Guido Weiss managed to develop a theory of Hardy spaces on spaces of homogeneous type by taking the atomic characterization of H p(X) as a definition. This was the starting point in generalizing the theory of Hardy spaces in abstract settings. The main goal of this chapter is to explore the relationship between the Hardy spaces developed in this monograph and those in [CoWe77]. Understanding this connection is an important step towards unifying the theory of Hardy spaces.KeywordsHardy SpaceMaximal FunctionDensity ResultNotational ConventionHomogeneous TypeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The aim of this paper is to establish the boundedness of the commutator $$[b_{1}, b_{2}, T_{\sigma }]$$ generated by the bilinear pseudo-differential operator $$T_{\sigma }$$ with smooth symbols and $$b_{1},\ b_{2}\in \mathrm {BMO}({\mathbb {R}}^{n})$$ on product of local Hardy spaces with variable exponents. By applying the refined atomic decomposition result, the authors prove that the bilinear pseudo-differential operator $$T_{\sigma }$$ is bounded from the Lebesgue space $$L^{p}({\mathbb {R}}^{n})$$ into $$h^{p_{1}(\cdot )}({\mathbb {R}}^{n})\times h^{p_{2}(\cdot )}({\mathbb {R}}^{n})$$ . Moreover, the boundedness of the commutator $$[b_{1}, b_{2}, T_{\sigma }]$$ on product of local Hardy spaces with variable exponents is also obtained.

Let T be the singular integral operator with variable kernel defined by T f ( x ) = p . v . ∫ R n Ω ( x , x − y ) | x − y | n f ( y ) d y $$\begin{array}{} \displaystyle Tf(x)= p.v. \int\limits_{\mathbb{R}^{n}}\frac{{\it\Omega}(x,x-y)}{|x-y|^{n}}f(y)\text{d}y \end{array} $$ and Dγ (0 ≤ γ ≤ 1) be the fractional differentiation operator. Let T ∗ and T ♯ be the adjoint of T and the pseudo-adjoint of T, respectively. The aim of this paper is to establish some boundedness for TDγ − DγT and (T ∗ − T ♯)Dγ on the homogeneous Morrey-Herz-type Hardy spaces with variable exponents H M K ˙ p ( ⋅ ) , λ α ( ⋅ ) , q $\begin{array}{} HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda} \end{array} $ via the convolution operator T m, j and Calderón-Zygmund operator, and then establish their boundedness on these spaces. The boundedness on H M K ˙ p ( ⋅ ) , λ α ( ⋅ ) , q $\begin{array}{} HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda} \end{array} $ (ℝ n ) is shown to hold for TDγ − DγT and (T ∗ − T ♯)Dγ . Moreover, the authors also establish various norm characterizations for the product T 1 T 2 and the pseudo-product T 1 ∘ T 2.

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a measurable function satisfying some decay condition and some locally log-H\"older continuity. In this article, via first establishing characterizations of the variable exponent Hardy space $H^{p(\cdot)}(\mathbb R^n)$ in terms of the Littlewood-Paley $g$-function, the Lusin area function and the $g_\lambda^\ast$-function, the authors then obtain its intrinsic square function characterizations including the intrinsic Littlewood-Paley $g$-function, the intrinsic Lusin area function and the intrinsic $g_\lambda^\ast$-function. The $p(\cdot)$-Carleson measure characterization for the dual space of $H^{p(\cdot)}(\mathbb R^n)$, the variable exponent Campanato space $\mathcal{L}_{1,p(\cdot),s}(\mathbb R^n)$, in terms of the intrinsic function is also presented.

We show the atomic decomposition of molecule on discrete Hardy spaces without using the theory of continuous Hardy spaces

In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.