Morrey Spaces are Embedded Between Weak Morrey Spaces and Stummel Classes

Abstract

In this paper, we show that the Morrey spaces $ L^{1,\left( \frac{\lambda}{p} -\frac{n}{p} + n \right) } \left( \mathbb{R}^{n} \right) $ are embedded betweenweak Morrey spaces $ wL^{p,\lambda}\left( \mathbb{R}^{n} \right) $ and Stummelclasses $ S_{\alpha}\left( \mathbb{R}^{n} \right) $ under some conditions on$ p, \lambda $ and $ \alpha $. More precisely, we prove that $ wL^{p,\lambda}\left(\mathbb{R}^{n} \right) \subseteq L^{1,\left( \frac{\lambda}{p} - \frac{n}{p} + n\right) } \left( \mathbb{R}^{n} \right) \subseteq S_{\alpha}\left( \mathbb{R}^{n}\right) $ where $ 1p\infty, 0\lambdan $ and $ \frac{n-\lambda}{p}\alphan $.We also show that these inclusion relations under the above conditions are proper.Lastly, we present an inequality of Adams' type \cite{A}

An Existence and Uniqueness of the Weak Solution of the Dirichlet Problem With the Data in Morrey Spaces

“…By the linearity of the weak derivative and integration, it is easy to show that 𝑩, defined by ( 6), is a bilinear mapping. Notice that, according to (5), (6), and (7), 𝑢 ∈ 𝑊 0 1,2 (Ω) is a weak solution of (4) if 𝑩(𝑢, 𝜙) = 𝐹 𝑓 (𝜙), (8) for every 𝜙 ∈ 𝑊 0 1,2 (Ω).…”

“…for 1 ≤ 𝑝 < ∞ and 0 ≤ 𝜆 ≤ 𝑛. This Morrey spaces were introduced by C. B. Morrey [1] and still attracted the attention of many researcher to investigate its inclusion properties or application in partial differential equation [2,3,4,5,6,7,8].…”

Characterization of the boundedness of generalized fractional integral and maximal operators on Orlicz–Morrey and weak Orlicz–Morrey spaces

“…For their generalization, see, for example, [33, 36, 48, 49]. Weak Morrey spaces were studied in , for example,[13, 14, 53, 58, 63]. The Orlicz–Morrey space $$L^{(\Phi ,\varphi )}(\mathbb {R}^n)$$ was first studied in [41].…”

Weak‐type Fefferman–Stein inequality and commutators on weak Orlicz–Morrey spaces

“…For their generalization, see [38,42,57,58], etc. Weak Morrey spaces were studied in [16,17,62,70,76], etc. The Orlicz-Morrey space 𝐿 (Φ,𝜑) (ℝ 𝑛 ) was first studied in [47].…”