On a variant of the Hardy inequality between weighted Orlicz spaces

Agnieszka Kałamajska; Katarzyna Pietruska-Pałuba

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Displaying similar documents to “On a variant of the Hardy inequality between weighted Orlicz spaces”

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Let $B_{p}$ be the Ariõ-Muckenhoupt weight class which controls the weighted $L^{p}$ -norm inequalities for the Hardy operator on non-increasing functions. We replace the constant p by a function p(x) and examine the associated $L^{p (x)}$ -norm inequalities of the Hardy operator.

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In this paper we completely characterize those weighted Hardy spaces that are Poletsky-Stessin Hardy spaces $H_{u}^{p}$ . We also provide a reduction of $H^{\infty}$ problems to $H_{u}^{p}$ problems and demonstrate how such a reduction can be used to make shortcuts in the proofs of the interpolation theorem and corona problem.

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The canonical injection of the Hardy-Orlicz space $H^{Ψ}$ into the Bergman-Orlicz space $^{Ψ}$

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We study the canonical injection from the Hardy-Orlicz space $H^{Ψ}$ into the Bergman-Orlicz space $^{Ψ}$ .

The maximal theorem for weighted grand Lebesgue spaces

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We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality ${| | M f | |}_{p), w} {\leq c | | f | |}_{p), w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_{p}$ , and therefore iff ${| | M f | |}_{p, w} {\leq c | | f | |}_{p, w}$ for some c independent of f. Some results of similar type are discussed for the case of small...

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Monge-Ampère measures and Poletsky-Stessin Hardy spaces on bounded hyperconvex domains

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Poletsky-Stessin Hardy (PS-Hardy) spaces are the natural generalizations of classical Hardy spaces of the unit disc to general bounded, hyperconvex domains. On a bounded hyperconvex domain Ω, the PS-Hardy space $H_{u}^{p} (Ω)$ is generated by a continuous, negative, plurisubharmonic exhaustion function u of the domain. Poletsky and Stessin considered the general properties of these spaces and mainly concentrated on the spaces $H_{u}^{p} (Ω)$ where the Monge-Ampère measure $(d d^{c} u) ⁿ$ has compact support for the associated...

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We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spaces $H^{p}$ and $H^{q}$ for 1 ≤ p,q ≤ ∞. In particular we give some estimates for the cases 1 = p ≤ q ≤ ∞ and 1 ≤ q < p ≤ ∞.

Gagliardo-Nirenberg inequalities in logarithmic spaces

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba (2006)

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We obtain interpolation inequalities for derivatives: $\int M_{q, α} (| \nabla f (x) |) d x \leq C [\int M_{p, β} (Φ ₁ (x, | f |, | \nabla^{(2)} f |)) d x + \int M_{r, γ} (Φ ₂ (x, | f |, | \nabla^{(2)} f |)) d x]$ , and their counterparts expressed in Orlicz norms: ||∇f||²(q,α) ≤ C||Φ₁(x,|f|,|∇(2)f|)||(p,β) ||Φ₂(x,|f|,|∇(2)f|)||(r,γ) $,$ where ${| | \cdot | |}_{(s, κ)}$ is the Orlicz norm relative to the function $M_{s, κ} (t) = t^{s} {(l n (2 + t))}^{κ}$ . The parameters p,q,r,α,β,γ and the Carathéodory functions Φ₁,Φ₂ are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces...

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We study continuity envelopes of function spaces $B_{p, q}^{s} (ℝ ⁿ, w)$ and $F_{p, q}^{s} (ℝ ⁿ, w)$ where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

Factorization of sequences in discrete Hardy spaces

Santiago Boza (2012)

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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 the real Hardy spaces $H^{p} (ℝ ⁿ)$ by Coifman, Rochberg and Weiss for 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, but with some restrictions on the measure that exclude the case...

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Necessary and sufficient conditions governing two-weight $L^{p}$ norm estimates for multiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given.

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A real-valued Hardy space $H ¹_{\sqrt} () \subseteq L ¹ ()$ related to the square root of the Poisson kernel in the unit disc is defined. The space is shown to be strictly larger than its classical counterpart H¹(). A decreasing function is in $H ¹_{\sqrt} ()$ if and only if the function is in the Orlicz space LloglogL(). In contrast to the case of H¹(), there is no such characterization for general positive functions: every Orlicz space strictly larger than L log L() contains positive functions which do not belong to $H ¹_{\sqrt} ()$ , and no Orlicz...

On the spectrum of the p-biharmonic operator involving p-Hardy's inequality

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In this paper, we study the spectrum for the following eigenvalue problem with the p-biharmonic operator involving the Hardy term: ${Δ (| Δ u |}^{p - 2} {Δ u) = λ (| u |}^{p - 2} u) / (δ {(x)}^{2 p})$ in Ω, $u \in W ₀^{2, p} (Ω)$ . By using the variational technique and the Hardy-Rellich inequality, we prove that the above problem has at least one increasing sequence of positive eigenvalues.

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We prove a conjecture of Wojtaszczyk that for 1 ≤ p < ∞, p ≠ 2, $H_{p} ()$ does not admit any norm one projections with dimension of the range finite and greater than 1. This implies in particular that for 1 ≤ p < ∞, p ≠ 2, $H_{p}$ does not admit a Schauder basis with constant one.

The metric space $H^{p} (A)$ , 0

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The Hardy-Littlewood maximal function ℳ and the trigonometric function sin x are two central objects in harmonic analysis. We prove that ℳ characterizes sin x in the following way: Let $f \in C^{α} (ℝ, ℝ)$ be a periodic function and α > 1/2. If there exists a real number 0 < γ < ∞ such that the averaging operator $(A_{x} f) (r) = 1 / 2 r \int_{x - r}^{x + r} f (z) d z$ has a critical point at r = γ for every x ∈ ℝ, then f(x) = a + bsin(cx+d) for some a,b,c,d ∈ ℝ. This statement can be used to derive a characterization of trigonometric functions as...

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The continuity of pseudo-differential operators on weighted local Hardy spaces

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We first show that a linear operator which is bounded on $L ²_{w}$ with w ∈ A₁ can be extended to a bounded operator on the weighted local Hardy space $h ¹_{w}$ if and only if this operator is uniformly bounded on all $h ¹_{w}$ -atoms. As an application, we show that every pseudo-differential operator of order zero has a bounded extension to $h ¹_{w}$ .

A new characterization of the Sobolev space

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The purpose of this paper is to provide a new characterization of the Sobolev space $W^{1, 1} (ℝ ⁿ)$ . We also show a new proof of the characterization of the Sobolev space $W^{1, p} (ℝ ⁿ)$ , 1 ≤ p < ∞, in terms of Poincaré inequalities.

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The Hardy spaces of Dirichlet series, denoted by $^{p}$ (p ≥ 1), have been studied by Hedenmalm et al. (1997) when p = 2 and by Bayart (2002) in the general case. In this paper we study some $L^{p}$ -generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces, denoted $^{p}$ and $ℬ^{p}$ . Each could appear as a “natural” way to generalize the classical case of the unit disk. We recover classical properties of spaces of analytic functions: boundedness of point evaluation, embeddings...

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Many problems in analysis are described as weighted norm inequalities that have given rise to different classes of weights, such as $A_{p}$ -weights of Muckenhoupt and $B_{p}$ -weights of Ariño and Muckenhoupt. Our purpose is to show that different classes of weights are related by means of composition with classical transforms. A typical example is the family $M_{p}$ of weights w for which the Hardy transform is $L_{p} (w)$ -bounded. A $B_{p}$ -weight is precisely one for which its Hardy transform is in $M_{p}$ , and also a weight...

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We study the holomorphic Hardy-Orlicz spaces $^{Φ} (Ω)$ , where Ω is the unit ball or, more generally, a convex domain of finite type or a strictly pseudoconvex domain in ℂⁿ. The function Φ is in particular such that $¹ (Ω) \subset^{Φ} (Ω) \subset^{p} (Ω)$ for some p > 0. We develop maximal characterizations, atomic and molecular decompositions. We then prove weak factorization theorems involving the space BMOA(Ω). As a consequence, we characterize those Hankel operators which are bounded from $^{Φ} (Ω)$ into ¹(Ω).

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For any locally integrable f on ℝⁿ, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: $S f (x) = 1 / | B (0, | x |) | \int_{B (0, | x |)} f (t) d t$ , $T f (x) = 1 / | B (x, | x |) | \int_{B (x, | x |)} f (t) d t$ for x ∈ ℝⁿ. The purpose of the paper is to establish sharp localized LlogL estimates for S and T. The proof rests on a corresponding one-weight estimate for a martingale maximal function, a result which is of independent interest.

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