A bridge connecting Lebesgue and Morrey spaces via Riesz norms
- PDF / 2,158,802 Bytes
- 29 Pages / 439.37 x 666.142 pts Page_size
- 50 Downloads / 184 Views
Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00106-6 ORIGINAL PAPER
A bridge connecting Lebesgue and Morrey spaces via Riesz norms Jin Tao1 · Dachun Yang1 · Wen Yuan1 Received: 21 April 2020 / Accepted: 21 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract In this article, via combining Riesz norms with Morrey norms, the authors introduce and study the so-called Riesz–Morrey space, which differs from the John–Nirenberg–Campanato space in subtracting integral means. These spaces provide a bridge connecting both Lebesgue spaces and Morrey spaces which prove to be the endpoint spaces of Riesz–Morrey spaces. Moreover, the authors introduce a block-type space which proves to be the predual space of the Riesz–Morrey space. Keywords Euclidean space · Cube · Lebesgue space · Morrey space · John– Nirenberg–Campanato space · Duality · Riesz–Morrey space Mathematics subject classification 42B35 · 42B30 · 46E30 · 46E35
1 Introduction In this article, a cube always means the cube with its edges parallel to the coordinate axes, which is not necessary to be closed or open. Let X be either ℝn or a cube Q0 in ℝn with finite side length. It is well known that the Lebesgue space Lq (X) with q ∈ [1, ∞] plays a vital role in modern mathematics. It is defined to be the set of all measurable functions f such that, when q ∈ [1, ∞), Communicated by Mieczyslaw Mastylo. * Dachun Yang [email protected] Jin Tao [email protected] Wen Yuan [email protected] 1
Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China Vol.:(0123456789)
J. Tao et al.
� ‖f ‖Lq (X) ∶=
∫X
�1 �f (x)� dx q
q
(4n −2n )k𝜖 }
which implies that, for any integer k > (4n − 2n )k𝜖, p
‖f ‖L∞ (Q ) < 𝜖. k
⋃∞ Letting U(𝟎n , 𝜖) ∶= ( k=(4n −2n )k +1 Qk ) ∪ {𝟎n } = [−2−k𝜖 , 2−k𝜖 ]n , then we have 𝜖
p
‖f ‖L∞ (U(𝟎 ,𝜖)) < 𝜖. n
Similarly, for any x ∈ U(x, 𝜖) such that
Q◦0
(the set of all interior points of Q0 ), there exists a cube p
‖f ‖L∞ (U(x,𝜖)) < 𝜖. Observe that
Q◦0 ⊂
⋃{
} U(x, 𝜖) ∶ x ∈ Q◦0 .
Then p
p
‖f ‖L∞ (Q ) = ‖f ‖L∞ (Q◦ ) < 𝜖. 0
0
By this and the arbitrariness of both Q0 and 𝜖 , we conclude that ‖f ‖L∞ (X) = 0 and hence f = 0 almost everywhere in X .
J. Tao et al.
Now, we consider the case q ∈ (p, ∞) . Let f ∈ RMp,q, 1 − 1 (X) with p ∈ [1, ∞) and p
q
q ∈ (p, ∞) . We claim that ‖f ‖Lq (Q) = 0 for any cube Q ⊂ X . Indeed, if not, then there exists some cube Q0 ⊂ X such that ∫Q |f (x)|q dx =∶ C0 ∈ (0, ∞) . Without loss of 0 ∑∞ generality, we may assume that Q0 ∶= [−1, 1]n and C0 ∶= k=1 k−q∕p . For any t ∈ (0, 1) , let tQ0 ∶= [−t, t]n . Then, from the absolute continuity of the Lebesgue integral, we deduce that there exist {tk }∞ ⊂ (0, 1) such that 1 =∶ t0 > t1 > t2 > … k=1 and, for any k ∈ ℕ, ∫tk−1 Q0 ⧵tk Q0
|f (x)|q dx = k−q∕p .
(13)
We now label the cubes with pairwise disjoint interiors in each ring-like domain tk−1 Q0 ⧵tk Q0 as follows. By a geometrical observation, for
Data Loading...
