Generalized Hardy–Morrey Spaces (original) (raw)

Zeitschrift für Analysis und ihre Anwendungen, 2017

The generalized Morrey space M p,φ (R n) was defined by Mizuhara 1991 and Nakai in 1994. It is equipped with a parameter 0 < p < ∞ and a function φ : R n × (0, ∞) → (0, ∞). Our experience shows that M p,φ (R n) is easy to handle when 1 < p < ∞. However, when 0 < p ≤ 1, the function space M p,φ (R n) is difficult to handle as many examples show. We propose a way to deal with M p,φ (R n) for 0 < p ≤ 1, in particular, to obtain some estimates of the Hardy-Littlewood maximal operator on these spaces. Especially, the vector-valued estimates obtained in the earlier papers are refined. The key tool is the weighted dual Hardy operator. Much is known on the weighted dual Hardy operator.

A note on vector-valued Hardy and Paley inequalities

Proceedings of the American Mathematical Society, 1992

The values of p and q for Lp(Lq) that satisfy the extension of Paley and Hardy inequalities for vector-valued //' functions are characterized. In particular, it is shown that Li(L\) is a Paley space that fails Hardy inequality. A complex Banach space X is said to verify vector-valued Hardy inequality (for short X is a (Hl)-space) if (H) gJ!^M<c||/||, for all fe Hx (X), ton + l where HX(X) = {f e LX(T, X):/(«) = 0 for n < 0}. Both inequalities can be regarded in the framework of vector-valued extensions of multipliers from 771 to /'. Recall that a sequence (m") isa (771-/1)multiplier, to be denoted by mn e (Hx-/'), if Tm"(f) = (f(n)m") defines a bounded operator from 77 ' into /'. The (771-/^-multipliers were characterized by C. Fefferman in the following way (see [SW] for a proof): (*)

Hardy-Littlewood-Stein-Weiss inequality in the variable exponent Morrey spaces

We prove the boundedness of the weighted Hardy-Littlewood maximal operator and the singular integral operator on variable Morrey spaces L p(•),λ(•),|•| γ (Ω) over a bounded open set Ω ⊂ R n and a Hardy-Littlewood-Stein-Weiss type L p(•),λ(•),|x−x 0 | γ (Ω) to L q(•),λ(•),|x−x 0 | µ (Ω)-theorem for the potential operators I α(•) , x 0 ∈ Ω , also of variable order. In the case of constant α , the limiting case is also studied when the potential operator I α acts into BM O |•| γ .

Norm Inequalities in Generalized Morrey Spaces

Journal of Fourier Analysis and Applications, 2014

We prove that Calderón-Zygmund operators, Marcinkiewicz operators, maximal operators associated to Bochner-Riesz operators, operators with rough kernel as well as commutators associated to these operators which are known to be bounded on weighted Morrey spaces under appropriate conditions, are bounded on a wide family of function spaces.

2 Norm Inequalities in a Class of Function Spaces Including Weighted Morrey Spaces

2016

Boundedness of the Maximal, Potential and Singular Operators in the Generalized Morrey Spaces

Journal of Inequalities and Applications, 2009

We consider generalized Morrey spaces M p,ω R n with a general function ω x, r defining the Morrey-type norm. We find the conditions on the pair ω 1 , ω 2 which ensures the boundedness of the maximal operator and Calderón-Zygmund singular integral operators from one generalized Morrey space M p,ω1 R n to another M p,ω2 R n , 1 < p < ∞, and from the space M 1,ω1 R n to the weak space WM 1,ω2 R n. We also prove a Sobolev-Adams type M p,ω1 R n → M q,ω2 R n-theorem for the potential operators I α. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω 1 , ω 2 , which do not assume any assumption on monotonicity of ω 1 , ω 2 in r. As applications, we establish the boundedness of some Schrödinger type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class. As an another application, we prove the boundedness of various operators on generalized Morrey spaces which are estimated by Riesz potentials.

Weighted Hardy and singular operators in Morrey spaces

Journal of Mathematical Analysis and Applications, 2009

We study the weighted boundedness of the Cauchy singular integral operator S Γ in Morrey spaces L p,λ (Γ) on curves satisfying the arc-chord condition, for a class of "radial type" almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces L p,λ (0, ℓ), ℓ > 0. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves.

The Hardy and Heisenberg Inequalities in Morrey Spaces

Bulletin of the Australian Mathematical Society, 2018

We use the Morrey norm estimate for the imaginary power of the Laplacian to prove an interpolation inequality for the fractional power of the Laplacian on Morrey spaces. We then prove a Hardy-type inequality and use it together with the interpolation inequality to obtain a Heisenberg-type inequality in Morrey spaces.

Generalized Hardy–Morrey Spaces (original) (raw)

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