Real Hardy Space Research Articles

Hardy-Goldberg operator and its conjugate one in hardy spaces and BMO $$(\mathbb{T})$$

The Hardy operator is well-known in harmonic analysis. It transforms the sequence of Fourier coefficients of a function to a the sequence of its arithmetic means. In the paper we consider the Hardy-Goldberg operator generalizing the Hardy operator and its conjugate one. We prove the boundedness of the Hardy-Goldberg operator in the real Hardy space and of its analog in the Hardy space on the disc. We establish the boundedness of the conjugate Hardy-Goldberg operator in the periodic BMO andVMO spaces.

Factorization of sequences in discrete Hardy spaces

The purpose of this paper is to obtain a discrete version for the Hardy spaces H p (Z) of the weak factorization results obtained for real Hardy spaces H p (R n ) by Coifman, Rochberg and Weiss, in R n in the range p > n=(n + 1), and by Miyachi, for p n=(n + 1). It represents an extension, in the one dimensional case, of the corresponding result by A. Uchiyama who obtained a factorization theorem in the general context of spaces X of homogeneous type with some restrictions on the measure that exclude the case of points of positive measure on X and, hence, Z. In order to obtain the factorization theorem, we previously study the boundedness of some bilinear maps dened on discrete Hardy spaces.

A characterization of Triebel–Lizorkin spaces and Besov spaces by BMO families expansion

P. Tchamitchian has constructed an atomic decomposition of the real Hardy space with Lp-atoms () by rearrangements of wavelet expansion. In this paper, we give a characterization of Triebel–Lizorkin spaces and Besov spaces by a BMO family expansion by applying the arguments above.

Hardy's inequalities for Hermite and Laguerre expansions revisited

We show that Hardy's inequalities for Laguerre expansions hold on the space $L^1(0,\infty)$ when the Laguerre parameters $\alpha$ are positive, and we prove that although the inequality holds on the real Hardy space $H^1(0,\infty)$ if $\alpha= 0$, it does not hold on $L^1(0,\infty)$. Further, Hardy's inequality for Hermite expansion is established on $L^1(0,\infty)$.

Convergence and Stability of the Calderón Reproducing Formula in H 1 and BMO

We prove that a general form of the Calderon reproducing formula converges in H1(Rd) (the real Hardy space of Fefferman and Stein) as a natural limit of approximating integrals. We show that this convergence is H1-stable with respect to small errors in dilation and translation. Using duality, we show that the Calderon reproducing formula converges, in a stable fashion, weak-∗ in BMO. We give quantitative estimates of the formula’s stability and rate of convergence. These theorems generalize results of the author on the convergence and stability of the Calderon reproducing formula in Lp(w), where 1<p<∞ and w is a Muckenhoupt Ap weight.

Atomic decomposition of a real Hardy space for Jacobi analysis

Let (ℝ+, Δ(x)dx) be a Jacobi hypergroup with weight function Δ(x) = c(sinh x)2α+1(cosh x)2β+1. As in the Euclidean case, the real Hardy space H1(Δ) for (ℝ+, Δ(x)dx) is defined as the set of all locally integrable functions on ℝ+ whose radial maximal functions belong to L1(Δ). In this paper we give a characterization of H1(Δ) in terms of weighted Triebel–Lizorkin spaces on ℝ via the Abel transform. As an application, we introduce three types of atoms for (ℝ+, Δ), one of them is smooth, and give an atomic decomposition of H1(Δ).

H 1-estimates of Littlewood-Paley and Lusin functions for Jacobi analysis

For α ≥ β ≥ −1/2 let $$ \Delta (x) = (2shx)^{2\alpha + 1} (2chx)^{2\beta + 1} $$ denote the weight function on R+ and L1(Δ) the space of integrable functions on R+ with respect to Δ(x)dx, equipped with a convolution structure. For a suitable ϕ ∈ L1(Δ), we put $$ \varphi _t (x) = t^{ - 1} \Delta (x)^{ - 1} \Delta (x/t)\varphi (x/t) $$ for t > 0 and define the radial maximal operator Mϕ as usual manner. We introduce a real Hardy space H1(Δ) as the set of all locally integrable functions f on R+ whose radial maximal function Mϕ(f) belongs to L1(Δ). In this paper we obtain a relation between H1(Δ) and H1(R). Indeed, we characterize H1(Δ) in terms of weighted H1 Hardy spaces on R via the Abel transform of f. As applications of H1(Δ) and its characterization, we shall consider (H1(Δ),L1(Δ))-boundedness of some operators associated to the Poisson kernel for Jacobi analysis: the Poisson maximal operator MP, the Littlewood-Paley g-function and the Lusin area function S. They are bounded on Lp(Δ) for p > 1, but not true for p = 1. Instead, MP, g and a modified Sa, γ are bounded from H1(Δ) to L1(Δ).

Multidimensional Hausdorff operators on the real Hardy space

AbstractFor a wide family of multivariate Hausdorff operators, the boundedness of an operator from this family i s proved on the real Hardy space. By this we extend and strengthen previous results due to Andersen and Móricz.

Tolokonnikov’s Lemma for Real H∞ and the Real Disc Algebra

We prove Tolokonnikov’s Lemma and the inner-outer factorization for the real Hardy space \({{H^{\infty}_{\mathbb{R}}}}\) , the space of bounded holomorphic (possibly operator-valued) functions on the unit disc all of whose matrix-entries (with respect to fixed orthonormal bases) are functions having real Fourier coefficients, or equivalently, each matrix entry f satisfies \({\overline{f(\overline{z})}} = f(z)\) for all z ∈ \({\mathbb{D}}\).

The maximal conjugate and Hilbert operators on real Hardy spaces

We prove that the maximal conjugate and Hilbert operators are not bounded from the real Hardy space H1 to L1, where the underlying spaces may be over T or R. We also draw corollaries for the corresponding spaces over T2 and R2.

Trigonometric Multipliers on H2π

AbstractIn this paper we consider multipliers on the real Hardy space H2π. It is known that the Marcinkiewicz and the Hörmander–Mihlin conditions are sufficient for the corresponding trigonometric multiplier to be bounded on , 1 &lt; p &lt; ∞. We show among others that the Hörmander– Mihlin condition extends to H2π but the Marcinkiewicz condition does not.

Hardy spaces H1 for Schrödinger operators with compactly supported potentials

Let L=-Δ+V be a Schrodinger operator on ℝd, d≥3, where V is a non-negative compactly supported potential that belongs to Lp for some p>d/2. Let {Kt}t>0 denote the semigroup of linear operators generated by -L. For a function f we define its H1L-norm by \(\| f\|_{H^1_L}=\| \sup_{t>0} |K_t f(x)|\|_{L^1(dx)}\). It is proved that for a properly defined weight w a function f belongs to H1L if and only if wf∈H1(ℝd), where H1(ℝd) is the classical real Hardy space.

A transplantation theorem for the Hankel transform on the Hardy space

The transplantation operators for the Hankel transform are considered and their boundedness on the real Hardy space is established. As its application, we obtain the Hormander-Mihlin type multiplier theorem for the Hankel transform on the real Hardy space.

Ltivariate Hausdorff operators on the spaces H1(Rn) and BMO(Rn)

A multivariate Hausdorff operator H = H(μ, c, A) is defined in terms of a σ-finite Borel measure μ on R n , a Borel measurable function c on R n , and an n × n matrix A whose entries are Borel measurable functions on r n and such that A is nonsingular μ-a.e. The operator H*:= H (μ, c | det A-1|, A-1) is the adjoint to H in a well-defined sense. Our goal is to prove sufficient conditions for the boundedness of these operators on the real Hardy space H1(R n ) and BMO (R n ). Our main tool is proving commuting relations among H, H*, and the Riesz transforms R j . We also prove commuting relations among H, H*, and the Fourier transform.

Strong Approximation by Cesàro Means with Critical Index in the Hardy Spaces H p $$ (\mathbb{S}^{{d - 1}} ) $$ (0 &lt; p ≤ 1)

Let $$ (\mathbb{S}^{{d - 1}} ) = {\left\{ {x:{\left| x \right|} = 1} \right\}} $$ be a unit sphere of the d–dimensional Euclidean space ℝ d and let $$ H^{p} \equiv H^{p} (\mathbb{S}^{{d - 1}} ) $$ (0 < p ≤ 1) denote the real Hardy space on $$ \mathbb{S}^{{d - 1}} .$$ For 0 < p ≤ 1 and $$ f \in H^{p} (\mathbb{S}^{{d - 1}} ), $$ let E j (f,H p ) (j = 0, 1, ...) be the best approximation of f by spherical polynomials of degree less than or equal to j, in the space $$ H^{p} (\mathbb{S}^{{d - 1}} ). $$ Given a distribution f on $$ \mathbb{S}^{{d - 1}} , $$ its Cesaro mean of order δ > –1 is denoted by $$ \sigma ^{\delta }_{k} (f). $$ For 0 < p ≤ 1, it is known that $$ \delta (p): = \frac{{d - 1}} {p} - \frac{d} {2} $$ is the critical index for the uniform summability of $$ \sigma ^{\delta }_{k} $$ in the metric H p . In this paper, the following result is proved: Theorem Let 0<p<1 and $$ \delta = \delta (p): = \frac{{d - 1}} {p} - \frac{d} {2}. $$ Then for $$ f \in H^{p} (\mathbb{S}^{{d - 1}} ), $$ $$ {\sum\limits_{j = 1}^N {\frac{1} {j}{\left\| {\sigma ^{\delta }_{j} (f) - f} \right\|}^{p}_{{H^{p} }} \approx {\sum\limits_{j = 1}^N {\frac{1} {j}E^{p}_{j} {\left( {f,H^{p} } \right)}} }} }, $$ where A N (f)≈B N (f) means that there’s a positive constant C, independent of N and f, such that $$ C^{{ - 1}} A_{N} (f) \leqslant B_{N} (f) \leqslant C\;A_{N} (f). $$ In the case d = 2, this result was proved by Belinskii in 1996.

The Maximal Riesz, Fej�r, and Ces�ro Operators on Real Hardy Spaces

We prove that the maximal Riesz operator $\sigma^{\alpha,\gamma}_*$ is of strong type from $L^1(\R) \cap H^p$ $ (\R)$ to $L^p (\R)$ for $\alpha, \gamma>0$ and $1/(1+\alpha) 0$ and $1/(1+\alpha) = p$, and these results are best possible. The proofs are based on sharp estimates of the derivatives of the Riesz kernel. We characterize the real Hardy space $H^p(\R)$ in terms of $\sigma^{\alpha,1}_*$ for $1/(1+ \alpha) < p \le 1$, and draw consequences for real Hardy spaces on $\R^2$, as well. For example, an integrable function $f$ belongs to $H^1(\R)$ if and only if the maximal Fej\’er operator $\sigma^{1,1}_*$ applied to $f$ belongs to $L^1(\R)$. We also establish analogous results for real Hardy spaces on $\T$ and $\T^2$.

The maximal Fejér operator on real Hardy spaces

We prove that the maximal Fej'er operator is not bounded on the real Hardy spaces H1, which may be considered over \(\mathbb{T}\) and \(\mathbb{R}\). We also draw corollaries for the corresponding Hardy spaces over \(\mathbb{T}\)2 and \(\mathbb{R}\)2.

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R2, and acting on the real Hardy space H1(R), or the product Hardy space H11(R×R), or one of the hybrid Hardy spaces H10(R2) and H01(R2), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform.

Continuity and differentiability of Nemytskii operators on the Hardy space $\mathcal{H}^{1,1} (T^1 )$

Let\(\mathcal{H}^{1,1} (T^1 )\) denote the Hardy space of real-valued functions on the unit circle with weak derivatives in the usual real Hardy space\(\mathcal{H}^1 (T^1 )\). It is shown that when the weak derivative of a locally Lipschitz continuous functionf has bounded variation on compact sets the Nemytskii operatorF, defined byF(u)=f·u, maps\(\mathcal{H}^{1,1} (T^1 )\) continuously into itself. A further condition sufficient for the continuous Frechet differentiability ofF is then added.

A transplantation theorem for ultraspherical polynomials at critical index

We investigate the behaviour of Fourier coefficients with respect to the system of ultraspherical polynomials. This leads us to the study of the boundary Lorentz space ℒλ corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients {c(λ)n(f)} of ℒλ-functions turn out to behave like the Fourier coefficients of functions in the real Hardy space Re H1. Namely, we prove that for any f ∈ ℒλ the series ∑∞n=1 c(λ)n(f) cos nθ is the Fourier series of some function φ ∈ Re H1 with ∥φ∥Re H1 ≤ c∥f∥ℒλ.